Parametrized Homotopy Theory
J. P. May
J. Sigurdsson
Department of Mathematics, The University of Chicago, Chicago,
IL 60637
Department of Mathematics, The University of Notre Dame, Notre
Dame, IN, 465564618
Email address: may@math.uchicago.edu
Email address: jsigurds@nd.edu
1991 Mathematics Subject Classification. Primary 19D99, 55N20, 55P42;
Secondary 19L99, 55N22, 55T25
Key words and phrases. exspace, exspectrum, model category, parametrized
spectrum, parametrized homotopy theory, equivariant homotopy theory,
parametrized stable homotopy theory, equivariant stable homotopy theory
May was partially supported by the NSF.
Abstract.Part I: We set the stage for our homotopical work with preliminary
chapters on the pointset topology necessary to parametrized homotopy theo*
*ry,
the base change and other functors that appear in over and under categorie*
*s,
and generalizations of several classical results about equivariant bundles*
* and
fibrations to the context of proper actions of noncompact Lie groups.
Part II: Despite its long history, the homotopy theory of exspaces re
quires further development before it can serve as the starting point for a*
* rig
orous modern treatment of parametrized stable homotopy theory. We give a
leisurely account that emphasizes several issues that are of independent i*
*nterest
in the theory and applications of topological model categories. The essent*
*ial
point is to resolve problems about the homotopy theory of exspaces that a*
*re
absent from the homotopy theory of spaces. In contrast to previously encou*
*n
tered situations, model theoretic techniques are intrinsically insufficien*
*t to a
full development of the basic foundational properties of the homotopy cate*
*gory
of exspaces. Instead, a rather intricate blend of model theory and classi*
*cal ho
motopy theory is required. However, considerable new material on the gener*
*al
theory of topologically enriched model categories is also required.
Part III: We give a systematic treatment of the foundations of parametr*
*ized
stable homotopy theory, working equivariantly and with highly structured
smash products and function spectra. The treatment is based on equivari
ant orthogonal spectra, which are simpler for the purpose than alternative
kinds of spectra. Again, the parametrized context introduces many difficul*
*ties
that have no nonparametrized counterparts and cannot be dealt with using
standard model theoretic techniques. The space level techniques of Part II
only partially extend to the spectrum level, and many new twists are encou*
*n
tered. Most of the difficulties are already present in the nonequivariant *
*special
case. Equivariantly, we show how change of universe, passage to fixed poin*
*ts,
and passage to orbits behave in the parametrized setting.
Part IV: We give a fiberwise duality theorem that allows fiberwise reco*
*g
nition of dualizable and invertible parametrized spectra. This allows dire*
*ct
application of the formal theory of duality in symmetric monoidal categori*
*es
to the construction and analysis of transfer maps. The relationship between
transfer for general Hurewicz fibrations and for fiber bundles is illumina*
*ted
by the construction of fiberwise bundles of spectra, which are like bundle*
*s of
tangents along fibers, but with spectra replacing spaces as fibers. Using *
*this
construction, we obtain a simple conceptual proof of a generalized Wirthm"*
*uller
isomorphism theorem that calculates the right adjoint to base change along*
* an
equivariant bundle with manifold fibers in terms of a shift of the left ad*
*joint.
Due to the generality of our bundle theoretic context, the Adams isomorphi*
*sm
theorem relating orbit and fixed point spectra is a direct consequence.
Contents
Prologue 1
Part I. Pointset topology, change functors, and proper actions 7
Chapter 1. The pointset topology of parametrized spaces 8
Introduction 8
1.1. Convenient categories of topological spaces 8
1.2. Topologically bicomplete categories and exobjects 11
1.3. Convenient categories of exspaces 14
1.4. Convenient categories of exGspaces 17
1.5. Appendix: nonassociativity of smash products in T op* 19
Chapter 2. Change functors and compatibility relations 21
Introduction 21
2.1. The base change functors f!, f*, and f* 22
2.2. Compatibility relations 23
2.3. Change of group and restriction to fibers 25
2.4. Normal subgroups and quotient groups 28
2.5. The closed symmetric monoidal category of retracts 30
Chapter 3. Proper actions, equivariant bundles, and fibrations 33
Introduction 33
3.1. Proper actions of locally compact groups 33
3.2. Proper actions and equivariant bundles 37
3.3. Spaces of the homotopy types of GCW complexes 37
3.4. Some classical theorems about fibrations 39
3.5. Quasifibrations 41
Part II. Model categories and parametrized spaces 43
Introduction 44
Chapter 4. Topologically bicomplete model categories 45
Introduction 45
4.1. Model theoretic philosophy: h, q, and mmodel structures 46
4.2. Strong Hurewicz cofibrations and fibrations 47
4.3. Towards classical model structures in topological categories 50
4.4. Classical model structures in general and in K and U 53
4.5. Compactly generated qtype model structures 55
Chapter 5. Wellgrounded topological model categories 59
iii
iv CONTENTS
Introduction 59
5.1. Over and under model structures 60
5.2. The specialization to over and under categories of spaces 63
5.3. Wellgrounded topologically bicomplete categories 66
5.4. Wellgrounded categories of weak equivalences 68
5.5. Wellgrounded compactly generated model structures 72
5.6. Properties of wellgrounded model categories 73
Chapter 6. The qfmodel structure on KB 77
Introduction 77
6.1. Some of the dangers in the parametrized world 78
6.2. The qf model structure on the category K =B 79
6.3. Statements and proofs of the thickening lemmas 82
6.4. The compatibility condition for the qfmodel structure 84
6.5. The quasifibration and right properness properties 86
Chapter 7. Equivariant qftype model structures 88
Introduction 88
7.1. Families and noncompact Lie groups 89
7.2. The equivariant q and qfmodel structures 90
7.3. External smash product and base change adjunctions 94
7.4. Change of group adjunctions 97
7.5. Fiber adjunctions and Brown representability 100
Chapter 8. Exfibrations and exquasifibrations 102
8.1. Exfibrations 102
8.2. Preservation properties of exfibrations 104
8.3. The exfibrant approximation functor 106
8.4. Preservation properties of exfibrant approximation 108
8.5. Quasifibrant exspaces and exquasifibrations 110
Chapter 9. The equivalence between Ho GKB and hGWB 112
Introduction 112
9.1. The equivalence of Ho GKB and hGWB 112
9.2. Derived functors on homotopy categories 114
9.3. The functors f* and FB on homotopy categories 115
9.4. Compatibility relations for smash products and base change 117
Part III. Parametrized equivariant stable homotopy theory 121
Introduction 122
Chapter 10. Enriched categories and Gcategories 125
Introduction 125
10.1. Parametrized enriched categories 125
10.2. Equivariant parametrized enriched categories 127
10.3. Gtopological model Gcategories 129
Chapter 11. The category of orthogonal Gspectra over B 132
Introduction 132
11.1. The category of IG spaces over B 132
CONTENTS v
11.2. The category of orthogonal Gspectra over B 136
11.3. Orthogonal Gspectra as diagram exGspaces 139
11.4. The base change functors f*, f!, and f* 140
11.5. Change of groups and restriction to fibers 143
11.6. Some problems concerning noncompact Lie groups 145
Chapter 12. Model structures for parametrized Gspectra 148
Introduction 148
12.1. The level model structure on GSB 149
12.2. Some Quillen adjoint pairs relating level model structures 152
12.3. The stable model structure on GSB 153
12.4. The ss*isomorphisms 156
12.5. Proofs of the model axioms 159
12.6. Some Quillen adjoint pairs relating stable model structures 163
Chapter 13. Adjunctions and compatibility relations 167
Introduction 167
13.1. Brown representability and the functors f* and FB 168
13.2. The category GEB of excellent prespectra over B 170
13.3. The level exfibrant approximation functor P on prespectra 172
13.4. The auxiliary approximation functors K and E 175
13.5. The equivalence between Ho GPB and hGEB 177
13.6. Derived functors on homotopy categories 178
13.7. Compatibility relations for smash products and base change 180
Chapter 14. Module categories, change of universe, and change of groups 184
Introduction 184
14.1. Parametrized module Gspectra 184
14.2. Change of universe 188
14.3. Restriction to subgroups 192
14.4. Normal subgroups and quotient groups 195
Part IV. Duality, transfer, and base change isomorphisms 197
Chapter 15. Fiberwise duality and transfer maps 198
Introduction 198
15.1. The fiberwise duality theorem 199
15.2. Duality and transfer maps 201
15.3. The bundle construction on parametrized spectra 205
15.4. free parametrized spectra 208
15.5. The fiberwise transfer for ( ; )bundles 209
15.6. Sketch proofs of the compatible triangulation axioms 212
Chapter 16. The Wirthm"uller and Adams isomorphisms 214
Introduction 214
16.1. A natural comparison map f!! f* 215
16.2. The Wirthm"uller isomorphism for manifolds 216
16.3. The fiberwise Wirthm"uller isomorphism 219
16.4. The Adams isomorphism 221
16.5. Proof of the Wirthm"uller isomorphism for manifolds 222
vi CONTENTS
Bibliography 231
Prologue
What is this book about and why is it so long? Parametrized homotopy theory
concerns systems of spaces and spectra that are parametrized as fibers over poi*
*nts of
a given base space B. Parametrized spaces, or "exspaces", are just spaces over*
* and
under B, with a projection, often a fibration, and a section. Parametrized spec*
*tra
are analogous but considerably more sophisticated objects. They provide a world*
* in
which one can apply the methods of stable homotopy theory without losing track
of fundamental groups and other unstable information. Parametrized homotopy
theory is a natural and important part of homotopy theory that is implicitly ce*
*ntral
to all of bundle and fibration theory. Results that make essential use of it ar*
*e widely
scattered throughout the literature. For classical examples, the theory of tran*
*sfer
maps is intrinsically about parametrized homotopy theory, and EilenbergMoore
type spectral sequences are parametrized K"unneth theorems. Several new and
current directions, such as "twisted" cohomology theories and parametrized fixed
point theory cry out for the rigorous foundations that we shall develop.
On the foundational level, homotopy theory, and especially stable homotopy
theory, has undergone a thorough reanalysis in recent years. Systematic use of
Quillen's theory of model categories has illuminated the structure of the subje*
*ct
and has done so in a way that makes the general methodology widely applicable
to other branches of mathematics. The discovery of categories of spectra with
associative and commutative smash products has revolutionized stable homotopy
theory. The systematic study and application of equivariant algebraic topology *
*has
greatly enriched the subject.
There has not been a thorough and rigorous study of parametrized homotopy
theory that takes these developments into account. It is the purpose of this mo*
*no
graph to provide such a study, although we shall leave many interesting loose e*
*nds.
We shall also give some direct applications, especially to equivariant stable h*
*omo
topy theory where the new theory is particularly essential. The reason this stu*
*dy
is so lengthy is that, rather unexpectedly, many seemingly trivial nonparametri*
*zed
results fail to generalize, and many of the conceptual and technical obstacles *
*to
a rigorous treatment have no nonparametrized counterparts. For this reason, the
resulting theory is considerably more subtle than its nonparametrized precursor*
*s.
We indicate some of these problems here.
The central conceptual subtlety in the theory enters when we try to prove th*
*at
structure enjoyed by the pointset level categories of parametrized spaces desc*
*ends
to their homotopy categories. Many of our basic functors occur in Quillen adjoi*
*nt
pairs, and such structure descends directly to homotopy categories. Recall that
an adjoint pair of functors (T, U) between model categories is a Quillen adjoint
pair, or a Quillen adjunction, if the left adjoint T preserves cofibrations and*
* acyclic
cofibrations or, equivalently, the right adjoint U preserves fibrations and acy*
*clic
1
2 PROLOGUE
fibrations. It is a Quillen equivalence if, further, the induced adjunction on*
* ho
motopy categories is an adjoint equivalence. We originally hoped to find a model
structure on parametrized spaces in which all of the relevant adjunctions are Q*
*uillen
adjunctions. It eventually became clear that there can be no such model structu*
*re,
for altogether trivial reasons. Therefore, it is intrinsically impossible to la*
*y down
the basic foundations of parametrized homotopy theory using only the standard
methodology of model category theory.
The force of parametrized theory largely comes from base change functors ass*
*o
ciated to maps f :A ! B. The existing literature on fiberwise homotopy theory
says surprisingly little about such functors. This is especially strange since*
* they
are the most important feature that makes parametrized homotopy theory useful
for the study of ordinary homotopy theory: such functors are used to transport
information from the parametrized context to the nonparametrized context. One
of the goals of our work is to fill this gap.
On the pointset level, there is a pullback functor f* from exspaces (or sp*
*ectra)
over B to exspaces (or spectra) over A. That functor has a left adjoint f! and*
* a
right adjoint f*. We would like both of these to be Quillen adjunctions, but th*
*at
is not possible unless the model structures lead to trivial homotopy categories*
*. We
mean literally trivial: one object and one morphism. We explain why. It will be
clear that the explanation is generic and applies equally well to a variety of *
*sheaf
theoretic situations where one encounters analogous base change functors.
Counterexample 0.0.1. Consider the following diagram.
OE
; ______//_B
OE i0
fflffl fflffl
B __i1_//B x I
Here ; is the empty set and OE is the initial (empty) map into B. This diagram *
*is a
pullback since B x{0}\B x{1} = ;. The category of exspaces over ; is the trivi*
*al
category with one object, and it admits a unique model structure. Let *B denote
the exspace B over B, with section and projection the identity map. Both (OE!,*
* OE*)
and (OE*, OE*) are Quillen adjoint pairs for any model structure on the categor*
*y of
exspaces over B. Indeed, OE! and OE* preserve weak equivalences, fibrations, *
*and
cofibrations since both take *; to *B . We have (i0)* O (i1)! ~=OE!O OE* since*
* both
composites take any exspace over B to *B . If (i1)!and (i0)* were both Quillen*
* left
adjoints, it would follow that this isomorphism descends to homotopy categories.
If, further, the functors (i1)!and (i0)* on homotopy categories were equivalenc*
*es of
categories, this would imply that the homotopy category of exspaces over B with
respect to the given model structure is equivalent to the trivial category.
Information in ordinary homotopy theory is derived from results in parametri*
*zed
homotopy theory by use of the base change functor r!associated to the trivial m*
*ap
r :B ! *. For this and other reasons, we choose our basic model structure to
be one such that (f!, f*) is a Quillen adjoint pair for every map f :A ! B and
is a Quillen equivalence when f is a homotopy equivalence. Then (f*, f*) cannot
be a Quillen adjoint pair in general. However, it is essential that we still h*
*ave
the adjunction (f*, f*) after passage to homotopy categories. For example, taki*
*ng
f to be the diagonal map on B, this adjunction is used to obtain the adjunction
PROLOGUE 3
on homotopy categories that relates the fiberwise smash product functor ^B on
exspaces over B to the function exspace functor FB . To construct the homotopy
category level right adjoints f*, we shall have to revert to more classical met*
*hods,
using Brown's representability theorem. However, it is not clear how to verify *
*the
hypotheses of Brown's theorem in the model theoretic framework.
Counterexample 0.0.1 also illustrates the familiar fact that a commutative d*
*ia
gram of functors on the pointset level need not induce a commutative diagram of
functors on homotopy categories. When commuting left and right adjoints, this i*
*s a
problem even when all functors in sight are parts of Quillen adjunctions. There*
*fore,
proving that compatibility relations that hold on the pointset level descend t*
*o the
homotopy category level is far from automatic. In fact, proving such "compatibi*
*l
ity relations" is often a highly nontrivial problem, but one which is essentia*
*l to
the applications. We do not know how to prove the most interesting compatibility
relations working only model theoretically.
Even in the part of the theory in which model theory works, it does not work*
* as
expected. There is an obvious naive model structure on exspaces over B in which
the weak equivalences, fibrations, and cofibrations are the exmaps whose maps *
*of
total spaces are weak equivalences, fibrations, and cofibrations of spaces in t*
*he usual
Quillen model structure. This "qmodel structure" is the natural starting point*
* for
the theory, but it turns out to have severe drawbacks that limit its space level
utility and bar it from serving as the starting point for the development of a *
*useful
spectrum level stable model structure. In fact, it has two opposite drawbacks. *
*First,
it has too many cofibrations. In particular, the model theoretic cofibrations n*
*eed
not be cofibrations in the intrinsic homotopical sense. That is, they fail to s*
*atisfy
the fiberwise homotopy extension property (HEP) defined in terms of parametrized
mapping cylinders. This already fails for the sections of cofibrant objects and*
* for
the inclusions of cofibrant objects in their cones. Therefore the classical the*
*ory of
cofiber sequences fails to mesh with the model category structure.
Second, it also has too many fibrations. The fibrant exspaces are Serre fib*
*ra
tions, and Serre fibrations are not preserved by fiberwise colimits. Such coli*
*mits
are preserved by a more restrictive class of fibrations, namely the wellsectio*
*ned
Hurewicz fibrations, which we call exfibrations. Such preservation properties *
*are
crucial to resolving the problems with base change functors that we have indica*
*ted.
In model category theory, decreasing the number of cofibrations increases the
number of fibrations, so that these two problems cannot admit a solution in com
mon. Rather, we require two different equivalent descriptions of our homotopy
categories of exspaces. First, we have another model structure, the "qfmodel
structure", which has the same weak equivalences as the qmodel structure but h*
*as
fewer cofibrations, all of which satisfy the fiberwise HEP. Second, we have a d*
*escrip
tion in terms of the classical theory of exfibrations, which does not fit natu*
*rally
into a model theoretic framework. The former is vital to the development of the
stable model structure on parametrized spectra. The latter is vital to the solu*
*tion
of the intrinsic problems with base change functors.
Before getting to the issues just discussed, we shall have to resolve various
others that also have no nonparametrized analogues. Even the point set topology
requires care since function exspaces take us out of the category of compactly
generated spaces. Equivariance raises further problems, although most of our new
foundational work is already necessary nonequivariantly. Passage to the spectrum
4 PROLOGUE
level raises more serious problems. The main source of difficulty is that the u*
*nder
lying total space functor is too poorly behaved, especially with respect to sma*
*sh
products and fibrations, to give good control of homotopy groups as one passes
from parametrized spaces to parametrized spectra. Moreover, the resolution of
base change problems requires a different set of details on the spectrum level *
*than
on the space level.
The end result may seem intricate, but it gives a very powerful framework in
which to study homotopy theory. We illustrate by showing how fiberwise duality
and transfer maps work out and by showing that the basic change of groups isomo*
*r
phisms of equivariant stable homotopy theory, namely the generalized Wirthm"ull*
*er
and Adams isomorphisms, drop out directly from the foundations. Costenoble and
Waner [28] have already given other applications in equivariant stable homotopy
theory, using our foundations to study Poincar'e duality in ordinary RO(G)grad*
*ed
cohomology. Further applications are work in progress.
The theory here gives perhaps the first worked example in which a model theo
retic approach to derived homotopy categories is intrinsically insufficient and*
* must
be blended with a quite different approach even to establish the essential stru*
*ctural
features of the derived category. Such a blending of techniques seems essential*
* in
analogous sheaf theoretic contexts that have not yet received a modern model th*
*e
oretic treatment. Even nonequivariantly, the basic results on base change, smash
products, and function exspaces that we obtain do not appear in the literature.
Such results are essential to serious work in parametrized homotopy theory.
Much of our work should have applications beyond the new parametrized the
ory. The model theory of topological enriched categories has received much less
attention in the literature than the model theory of simplicially enriched cate
gories. Despite the seemingly equivalent nature of these variants, the topologi*
*cal
situation is actually quite different from the simplicial one, as our applicati*
*ons make
clear. In particular, the interweaving of htype and qtype model structures th*
*at
pervades our work seems to have no simplicial counterpart. This interweaving do*
*es
also appear in algebraic contexts of model categories enriched over chain compl*
*exes,
where foundations analogous to ours can be developed. One of our goals is to gi*
*ve
a thorough analysis and axiomatization of how this interweaving works in general
in topologically enriched model categories.
History. This project began with unpublished notes, dating from the summer
of 2000, of the first author [64]. He put the project aside and returned to it*
* in
the fall of 2002, when he was joined by the second author. Some of Parts I and *
*II
was originally in a draft of the first author that was submitted and accepted f*
*or
publication, but was later withdrawn. That draft was correct, but it did not in*
*clude
the "qfmodel structure", which comes from the second author's 2004 PhD thesis
[88]. The first author's notes [64] claimed to construct the stable model struc*
*ture on
parametrized spectra starting from the qmodel structure on exspaces. Following
[64], the monograph [47] of Po Hu also takes that starting point and makes that
claim. The second author realized that, with the obvious definitions, the axiom*
*s for
the stable model structure cannot be proven from that starting point and that a*
*ny
naive variant would be disconnected with cofiber sequences and other essential *
*needs
of a fully worked out theory. His qfmodel structure is the crucial new ingredi*
*ent
that is used to solve this problem. Although implemented quite differently, the
applications of Chapter 16 were inspired by Hu's work.
PROLOGUE 5
Thanks. We thank the referee of the partial first version for several helpf*
*ul
suggestions, Gaunce Lewis and Peter Booth for help with the point set topology,
Mike Cole for sharing his remarkable insights about model categories, and Mike
Mandell for much technical help. We thank Kathleen Lewis for working out the
counterexample in Theorem 1.1.1 and Victor Ginzburg for giving us the striking
Counterexample 11.6.2. We are especially grateful to Kate Ponto for a meticulou*
*sly
careful reading that uncovered many obscurities and infelicities. Needless to s*
*ay,
she is not to blame for those that remain.
Part I
Pointset topology, change
functors, and proper actions
CHAPTER 1
The pointset topology of parametrized spaces
Introduction
We develop the basic pointset level properties of the category of exspaces
over a fixed base space B in this chapter. In x1.1, we discuss convenient categ*
*ories
of topological spaces. The usual category of compactly generated spaces is not
adequate for our study of exspaces, and we shall see later that the interplay *
*between
model structures and the relevant convenient categories is quite subtle. In x1.*
*2, we
give basic facts about based and unbased topologically bicomplete categories. T*
*his
gives the language that is needed to describe the good formal properties of the
various categories in which we shall work. We discuss convenient categories of
exspaces in x1.3, and we discuss convenient categories of exGspaces in x1.4.
As a matter of recovery of lost folklore, x1.5 is an appendix, the substance*
* of
which is due to Kathleen Lewis. It is only at her insistence that she is not na*
*med as
its author. It documents the nonassociativity of the smash product in the ordin*
*ary
category of based spaces, as opposed to the category of based kspaces. When wr*
*it
ing the historical paper [70], the first author came across several 1950's refe*
*rences
to this nonassociativity, including an explicit, but unproven, counterexample i*
*n a
1958 paper of Puppe [82]. However, we know of no reference that gives details, *
*and
we feel that this nonassociativity should be documented in the modern literatur*
*e.
We are very grateful to Gaunce Lewis for an extended correspondence and
many details about the material of this chapter, but he is not to be blamed for
the point of view that we have taken. We are also much indebted to Peter Booth.
He is the main pioneer of the theory of fibered mapping spaces (see [5, 6, 7]) *
*and
function exspaces, and he sent us several detailed proofs about them.
1.1. Convenient categories of topological spaces
We recall the following by now standard definitions.
Definition 1.1.1. Let B be a space and A a subset. Let f :K ! B run over
all continuous maps from compact Hausdorff spaces K into B.
(i)A is compactly closed if each f1 (A) is closed.
(ii)B is weak Hausdorff if each f(K) is closed.
(iii)B is a kspace if each compactly closed subset is closed.
(iv)B is compactly generated if it is a weak Hausdorff kspace.
Let Top be the category of all topological spaces and let K , wH , and U =
K \wH be its full subcategories of kspaces, weak Hausdorff spaces, and compact*
*ly
generated spaces. The kification functor k :Top ! K assigns to a space X the
same set with the finer topology that is obtained by requiring all compactly cl*
*osed
subsets to be closed. It is right adjoint to the inclusion K ! T op. The weak
8
1.1. CONVENIENT CATEGORIES OF TOPOLOGICAL SPACES 9
Hausdorffication functor w :T op ! wH assigns to a space X its maximal weak
Hausdorff quotient. It is left adjoint to the inclusion wH  ! Top.
From now on, we work in K , implicitly kifying any space that is not a ksp*
*ace
to begin with. In particular, products and function spaces are understood to be
kified. With this convention, B is weak Hausdorff if and only if the diagonal *
*map
embeds it as a closed subspace of B x B. Let A xcB denote the classical cartesi*
*an
product in T op and recall that B is Hausdorff if and only if the diagonal embe*
*ds
it as a closed subspace of B xcB. The following result is proven in [56, App.x2*
*].
Proposition 1.1.2. Let A and B be kspaces. If one of them is locally compact
or if both of them are first countable, then
A x B = A xcB.
Therefore, if B is either locally compact or first countable, then B is Hausdor*
*ff if
and only if it is weak Hausdorff.
We would have preferred to work in U rather than K since there are many
counterexamples which reveal the pitfalls of working without a separation prope*
*rty.
However, as we will explain in x1.3, several inescapable facts about exspaces *
*force
us out of that convenient category. Like U , the category K is closed cartesi*
*an
monoidal. This means that it has function spaces Map(X, Y ) with homeomorphisms
Map (X x Y, Z) ~=Map (X, Map(Y, Z)).
This was proven by Vogt [94], who uses the term compactly generated for our k
spaces. See also [99]. An early unpublished preprint by Clark [20] also showed
this, and an exposition of exspaces based on [20] was given by Booth [6].
Philosophically, we can justify a preference for K over U by remarking that
the functor w is so poorly behaved that we prefer to minimize its use. In U , c*
*olimits
must be constructed by first constructing them in K and then applying the funct*
*or
w, which changes the underlying point set and loses homotopical control. Howeve*
*r,
this justification would be more persuasive were it not that colimits in K that*
* are
not colimits in U can already be quite badly behaved topologically. For example,
w itself is a colimit construction in K . We describe a relevant situation in w*
*hich
colimits behave better in U than in K in Remark 1.1.4 below.
More persuasively, w is a formal construction that only retains formal contr*
*ol
because both colimits and the functor w are left adjoints. We will encounter ri*
*ght
adjoints constructed in K that do not preserve the weak Hausdorff property when
restricted to U , and in such situations we cannot apply w without losing the
adjunction. In fact, when restricted to U , the relevant left adjoints do not c*
*ommute
with colimits and so cannot be left adjoints there. We shall encounter other re*
*asons
for working in K later. An obvious advantage of K is that U sits inside it, so *
*that
we can use K when it is needed, but can restrict to the better behaved category*
* U
whenever possible. Actually, the situation is more subtle than a simple dichoto*
*my.
In our study of exspaces, it is essential to combine use of the two categories,
requiring base spaces to be in U but allowing total spaces to be in K .
We have concomitant categories K* and U* of based spaces in K and in U .
We generally write T for U* to mesh with a number of relevant earlier papers.
Using duplicative notations, we write Map (X, Y ) for the space K (X, Y ) of ma*
*ps
X  ! Y and F (X, Y ) for the based space K*(X, Y ) of based maps X  ! Y
between based spaces. Both K* and T are closed symmetric monoidal categories
10 1. THE POINTSET TOPOLOGY OF PARAMETRIZED SPACES
under ^ and F [56, 94, 99]. This means that the smash product is associative,
commutative, and unital up to coherent natural isomorphism and that ^ and F are
related by the usual adjunction homeomorphism
F (X ^ Y, Z) ~=F (X, F (Y, Z)).
The need for kification is illustrated by the nonassociativity of the smash pr*
*oduct
in Top*; see x1.5.
We need a few observations about inclusions and colimits. Recall that a map
is an inclusion if it is a homeomorphism onto its image. Of course, inclusions
need not have closed image. As noted by Strom [91], the simplest example of
a nonclosed inclusion in K is the inclusion i: {a} {a, b}, where {a, b} has
the indiscrete topology. Here i is both the inclusion of a retract and a Hurewi*
*cz
cofibration (satisfies the homotopy extension property, or HEP). As is wellkno*
*wn,
such pathology cannot occur in U .
Lemma 1.1.3. Let i: A ! X be a map in K .
(i)If there is a map r :X ! A such that r O i = id, then i is an inclusion. I*
*f,
further, X is in U , then i is a closed inclusion.
(ii)If i is a Hurewicz cofibration, then i is an inclusion. If, further, X is *
*in U ,
then i is a closed inclusion.
Proof. Inclusions i: A ! X are characterized by the property that a func
tion j :Y ! A is continuous if and only if iOj is continuous. This implies the*
* first
statement in (i). Alternatively, one can note that a map in K is an inclusion i*
*f and
only if it is an equalizer in K , and a map in U is a closed inclusion if and o*
*nly if it
is an equalizer in U [56, 7.6]. Since i is the equalizer of iOr and the identit*
*y map of
X, this implies both statements in (i). For (ii), let Mi be the mapping cylinde*
*r of i.
The canonical map j :Mi ! X x I has a left inverse r and is thus an inclusion *
*or
closed inclusion in the respective cases. The evident closed inclusions i1: A *
*! Mi
and i1: X ! X x I satisfy j O i1 = i1 O i, and the conclusions of (ii) follow.
The following remark, which we learned from Mike Cole [22] and Gaunce Lewis,
compares certain colimits in K and U . It illuminates the difference between th*
*ese
categories and will be needed in our later discussion of htype model structure*
*s.
Remark 1.1.4. Suppose given a sequence of inclusions gn :Xn ! Xn+1 and
maps fn :Xn ! Y in K such that fn+1gn = fn. Let X = colimXn and let
f :X ! Y be obtained by passage to colimits. Fix a map p: Z ! Y . The maps
Z xY Xn ! Z xY X induce a map
ff: colim(Z xY Xn) ! Z xY X.
Lewis has provided counterexamples showing that ff need not be a homeomorphism
in general. However, if Y 2 U , then a result of his [56, App. 10.3] shows that*
* ff
is a homeomorphism for any p and any maps gn. In fact, as in Proposition 2.1.3
below, if Y 2 U , then the pullback functor p*: K =Y  ! K =Z is a left adjoint
and therefore commutes with all colimits. To see what goes wrong when Y is not
1.2. TOPOLOGICALLY BICOMPLETE CATEGORIES AND EXOBJECTS 11
in U , consider the diagram
colim(Z xY Xn) __ff//_Z xY X
' 
fflffl fflffl
colim(Z x Xn)______//Z x X.
Products commute with colimits, so the bottom arrow is a homeomorphism, and
the top arrow ff is a continuous bijection. The right vertical arrow is an incl*
*usion
by the construction of pullbacks. If the left vertical arrow ' is an inclusion,*
* then the
diagram implies that ff is a homeomorphism. The problem is that ' need not be an
inclusion. One point is that the maps Z xY Xn ! Z x Xn are closed inclusions if
Y is weak Hausdorff, but not in general otherwise. Now assume that all spaces in
sight are in U . Since the gn are inclusions, the relevant colimits, when compu*
*ted in
K , are weak Hausdorff and thus give colimits in U . Therefore the commutation *
*of
p* with colimits (which is a result about colimits in K ) applies to these part*
*icular
colimits in U to show that ff is a homeomorphism.
The following related observation will be needed for applications of Quillen*
*'s
small object argument to qtype model structures in x4.5 and elsewhere.
Lemma 1.1.5. Let Xn ! Xn+1, n 0, be a sequence of inclusions in K with
colimit X. Suppose that X=X0 is in U . Then, for a compact Hausdorff space C,
the natural map
colimK (C, Xn) ! K (C, X)
is a bijection.
Proof. The point is that X0 need not be in U . Let f :C ! X be a map.
Then the composite of f with the quotient map X ! X=X0 takes image in some
Xn=X0, hence f takes image in Xn. The conclusion follows.
Scholium 1.1.6. One might expect the conclusion to hold for colimits of se
quences of closed inclusions Xn1 ! Xn such that Xn  Xn1 is a T1 space. This
is stated as [49, 4.2], whose authors got the statement from May. However, Lewis
has shown us a counterexample.
1.2.Topologically bicomplete categories and exobjects
We need some standard and some not quite so standard categorical language.
All of our categories C will be topologically enriched, with the enrichment giv*
*en
by a topology on the underlying set of morphisms. We therefore agree to write
C (X, Y ) for the space of morphisms X ! Y in C . Enriched category theory
would have us distinguish notationally between morphism spaces and morphism
sets, but we shall not do that. A topological category C is said to be topologi*
*cally
bicomplete if, in addition to being bicomplete in the usual sense of having all*
* limits
and colimits, it is bitensored in the sense that it is tensored and cotensored *
*over
K . We shall denote the tensors and cotensors by X x K and Map (K, X) for a
space K and an object X of C . The defining adjunction homeomorphisms are
(1.2.1) C (X x K, Y ) ~=K (K, C (X, Y )) ~=C (X, Map(K, Y )).
By the Yoneda lemma, these have many standard implications. For example,
(1.2.2) X x * ~=X and Map (*, Y ) ~=Y,
12 1. THE POINTSET TOPOLOGY OF PARAMETRIZED SPACES
(1.2.3)X x (K x L) ~=(X x K) x L and Map(K, Map(L, X)) ~=Map (K x L, X).
We say that a bicomplete topological category C is based if the unique map
from the initial object ; to the terminal object * is an isomorphism. In that c*
*ase,
C is enriched in the category K* of based kspaces, the basepoint of C (X, Y ) *
*being
the unique map that factors through *. We then say that C is based topologically
bicomplete if it is tensored and cotensored over K*. We denote the tensors and
cotensors by X ^ K and F (K, X) for a based space K and an object X of C . The
defining adjunction homeomorphisms are
(1.2.4) C (X ^ K, Y ) ~=K*(K, C (X, Y )) ~=C (X, F (K, Y )).
The based versions of (1.2.2) and (1.2.3) are
(1.2.5) X ^ S0 ~=X and F (S0, Y ) ~=Y,
(1.2.6) X ^ (K ^ L) ~=(X ^ K) ^ L and F (K, F (L, X)) ~=F (K ^ L, X).
Although not essential to our work, a formal comparison between the based
and unbased notions of bicompleteness is illuminating. The following result all*
*ows
us to interpret topologically bicomplete to mean based topologically bicomplete
whenever C is based, a convention that we will follow throughout.
Proposition 1.2.7. Let C be a based and bicomplete topological category. Then
C is topologically bicomplete if and only if it is based topologically bicomple*
*te.
Proof. Suppose given tensors and cotensors for unbased spaces K and write
them as X n K and Map (K, X)* as a reminder that they take values in a based
category. We obtain tensors and cotensors X ^ K and F (K, X) for based spaces K
as the pushouts and pullbacks displayed in the respective diagrams
X n * _____//X n K and F (K, X)_____//Map(K, X)*
   
   
fflffl fflffl fflffl fflffl
* _______//X ^ K * ________//Map(*, X)*.
Conversely, given tensors and cotensors X ^ K and F (K, X) for based spaces K,
we obtain tensors and cotensors X n K and Map (K, X)* for unbased spaces K by
setting
X n K = X ^ K+ and Map (K, X)* = F (K+ , X),
where K+ is the union of K and a disjoint basepoint.
As usual, for any category C and object B in C , we let C =B denote the
category of objects over B. An object X = (X, p) of C =B consists of a total ob*
*ject
X together with a projection map p: X ! B to the base object B. The morphisms
of C =B are the maps of total objects that commute with the projections.
Proposition 1.2.8. If C is a topologically bicomplete category, then so is C*
* =B.
Proof. The product of objects Yi over B, denoted xB Yi, is constructed by
taking the pullback of the product of the projections Yi ! B along the diagonal
B ! xiB. Pullbacks and arbitrary colimits of objects over B are constructed by
taking pullbacks and colimits on total objects and giving them the induced proj*
*ec
tions. General limits are constructed as usual from products and pullbacks. If *
*X is
an object over B and K is a space, then the tensor X xB K is just X x K together
1.2. TOPOLOGICALLY BICOMPLETE CATEGORIES AND EXOBJECTS 13
with the projection X x K ! B x * ~=B induced by the projection of X and
the projection of K to a point. Note that this makes sense even though the tens*
*or
x need have nothing to do with cartesian products in general; see Remark 1.2.10
below. The cotensor Map B(K, X) is the pullback of the diagram
B __'_//_Map(K, B)oo__Map (K, X)
where ' is the adjoint of B x K ! B x * ~=B.
The terminal object in C =B is (B, id). Let CB denote the category of based
objects in C =B, that is, the category of objects under (B, id) in C =B. An obj*
*ect
X = (X, p, s) in CB , which we call an exobject over B, consists of on object *
*(X, p)
over B together with a section s: B ! X. We can therefore think of the exobje*
*cts
as retract diagrams
p
B __s__//X____//_B.
The terminal object in CB is (B, id, id), which we denote by *B ; it is also an*
* initial
object. The morphisms in CB are the maps of total objects X that commute with
the projections and sections.
Proposition 1.2.9. If C is a topologically bicomplete category, then the cat*
*e
gory CB is based topologically bicomplete.
Proof. The coproduct of objects Yi 2 CB , which we shall refer to as the
"wedge over B" of the Yiand denote by _B Yi, is constructed by taking the pusho*
*ut
of the coproduct qB ! qYi of the sections along the codiagonal qiB ! B.
Pushouts and arbitrary limits of objects in CB are constructed by taking pushou*
*ts
and limits on total objects and giving them the evident induced sections and pr*
*o
jections. The tensor X ^B K of X = (X, p, s) and a based space K is the pushout
of the diagram
B oo___ (X x *) [B (B x K)____//X x K,
where the right map is induced by the basepoint of K and the section of X. The
cotensor FB (K, X) is the pullback of the diagram
B _s__//_Xo"o_Map B(K, X),
where " is evaluation at the basepoint of K, that is, the adjoint of the eviden*
*t map
X x K ! X over B.
Remark 1.2.10. Notationally, it may be misleading to write X xK and X ^K
for unbased and based tensors. It conjures up associations that are appropriate
for the examples on hand but that are inappropriate in general. The tensors in a
topologically bicomplete category C may bear very little relationship to cartes*
*ian
products or smash products. The standard uniform notation would be X K.
However, we have too many relevant examples to want a uniform notation. In
particular, we later use the notations X xB K and X ^B K in the parametrized
context, where a notation such as X B K would conjure up its own misleading
associations.
14 1. THE POINTSET TOPOLOGY OF PARAMETRIZED SPACES
1.3.Convenient categories of exspaces
We need a convenient topologically bicomplete category of exspaces1 over a
space B, where "convenient" requires that we have smash product and function ex
space functors ^B and FB under which our category is closed symmetric monoidal.
Denoting the unit B x S0 of ^B by S0B, a formal argument shows that we will then
have isomorphisms
(1.3.1) X ^B K ~=X ^B (S0B^B K) and FB (K, Y ) ~=FB (S0B^B K, Y )
relating tensors and cotensors to the smash product and function exspace funct*
*ors.
In particular, S0B^B K is just the product exspace B x K with section determin*
*ed
by the basepoint of K.
The pointset topology leading to such a convenient category is delicate, and
there are quite a few papers devoted to this subject. They do not give exactly
what we need, but they come close enough that we shall content ourselves with a
summary. It is based on the papers [5, 6, 7, 10, 11, 57] of Booth, Booth and
Brown, and Lewis; see also James [51, 52].
We assume once and for all that our base spaces B are in U . We allow the
total spaces X of spaces over B to be in K . We let K =B and U =B denote the
categories of spaces over B with total spaces in K or U . Similarly, we let KB *
*and
UB denote the respective categories of exspaces over B.
Remark 1.3.2. The section of an exspace in UB is closed, by Lemma 1.1.3.
Quite reasonably, references such as [29, 51] make the blanket assumption that
sections of exspaces must be closed. We have not done so since we have not
checked that all constructions in sight preserve this property.
Both the separation property on B and the lack of a separation property on
X are dictated by consideration of the function spaces Map B(X, Y ) over B that
we shall define shortly. These are only known to exist when B is weak Hausdorff.
However, even when B, X and Y are weak Hausdorff, Map B(X, Y ) is generally
not weak Hausdorff unless the projection p: X ! B is an open map. Categori
cally, this means that the cartesian monoidal category U =B is not closed carte*
*sian
monoidal. Wishing to retain the separation property, Lewis [57] proposed the fo*
*l
lowing as convenient categories of spaces and exspaces over a compactly genera*
*ted
space B.
Definition 1.3.3. Let O(B) and O*(B) be the categories of those compactly
generated spaces and exspaces over B whose projection maps are open.
Remark 1.3.4. Bundle projections over B are open maps. Hurewicz fibrations
over B are open maps if the diagonal B ! B x B is a Hurewicz cofibration [57,
2.3]; this holds, for example, if B is a CW complex.
However, the categories O(B) and O*(B) are insufficient for our purposes.
Working in these categories, we only have the base change adjunction (f*, f*) o*
*f x2.1
below for open maps f :A ! B, which is unduly restrictive. For example, we need
the adjunction ( *, *), where : B ! BxB is the diagonal map. Moreover, the
generating cofibrations of our qtype model structures do not have open project*
*ion
____________
1Presumably the prefix "ex" stands for "cross", as in "cross section". The *
*unlovely term "ex
space" has been replaced in some recent literature by "fiberwise pointed space"*
*. Used repetitively,
that is not much of an improvement. The term "retractive space" has also been u*
*sed.
1.3. CONVENIENT CATEGORIES OF EXSPACES 15
maps. This motivates us to drop the weak Hausdorff condition on total spaces and
to focus on KB as our preferred convenient category of exspaces over B. The
cofibrant exspaces in our qtype model structures are weak Hausdorff, hence th*
*is
separation property is recovered upon cofibrant approximation. Therefore, use of
K can be viewed as scaffolding in the foundations that can be removed when doing
homotopical work.
We topologize the set of exmaps X ! Y as a subspace of the space K (X, Y )
of maps of total spaces. It is based, with basepoint the unique map that factors
through *B . Therefore the category KB is enriched over K*. It is based topo
logically bicomplete by Proposition 1.2.8. Recall that we write xB Yi and _B Yi
for products and wedges over B. We also write Y=BX for quotients, which are
understood to be pushouts of diagrams *B  X ! Y . We give a more concrete
description of the tensors and cotensors in K =B and KB given by Proposition 1.*
*2.8
and Proposition 1.2.9. For a space X over B, we let Xb denote the fiber p1(b).*
* If
X is an exspace, then Xb has the basepoint s(b).
Definition 1.3.5. Let X be a space over B and K be a space. Define X xB K
to be the space X x K with projection the product of the projections X ! B and
K ! *. Define Map B(K, X) to be the subspace of Map (K, X) consisting of those
maps f :K ! X that factor through some fiber Xb; the projection sends such a
map f to b.
Definition 1.3.6. Let X be an exspace over B and K be a based space.
Define X ^B K to be the quotient of X xB K obtained by taking fiberwise smash
products, so that (X^B K)b = Xb^K; the basepoints of fibers prescribe the secti*
*on.
Define FB (K, X) to be the subspace of Map B(K, X) consisting of the based maps
K ! Xb X for some b 2 B, so that FB (K, X)b = F (K, Xb); the section sends
b to the constant map at s(b).
Remark 1.3.7. As observed by Lewis [57, p. 85], if p is an open map, then so
are the projections of X ^B K and FB (K, Y ). Therefore O*(B) is tensored and
cotensored over T .
The category K =B is closed cartesian monoidal under the fiberwise cartesian
product X xB Y and the function space Map B(X, Y ) over B. The category KB
is closed symmetric monoidal under the fiberwise smash product X ^B Y and the
function exspace FB (X, Y ). We recall the definitions.
Definition 1.3.8. For spaces X and Y over B, X xB Y is the pullback of the
projections p: X ! B and q :Y ! B, with the evident projection XxB Y ! B.
When X and Y have sections s and t, their pushout X _B Y specifies the coproduc*
*t,
or wedge, of X and Y in KB , and s and t induce a map X _B Y ! X xB Y over
B that sends x and y to (x, tp(x)) and (sq(y), y). Then X ^B Y is the pushout in
K =B displayed in the diagram
X _B Y _____//X xB Y
 
 
fflffl fflffl
*B _______//X ^B Y.
This arranges that (X ^B Y )b = Xb^Yb, and the section and projection are evide*
*nt.
The following result is [11, 8.3].
16 1. THE POINTSET TOPOLOGY OF PARAMETRIZED SPACES
Proposition 1.3.9. If X and Y are weak Hausdorff exspaces over B, then so
is X ^B Y . That is, UB is closed under ^B .
Function objects are considerably more subtle, and we need a preliminary def
inition in order to give the cleanest description.
Definition 1.3.10. For a space Y 2 K , define the partial map classifier "Yto
be the union of Y and a disjoint point !, with the topology whose closed subspa*
*ces
are "Yand the closed subspaces of Y . The point ! is not a closed subset, and "*
*Yis
not weak Hausdorff. The name "partial map classifier" comes from the observation
that, for any space X, pairs (A, f) consisting of a closed subset A of X and a
continuous map f :A ! Y are in bijective correspondence with continuous maps
"f:X ! "Y. Given (A, f), "frestricts to f on A and sends X  A to !; given "f,
(A, f) is "f1(Y ) and the restriction of "f.
Definition 1.3.11. Let p: X ! B and q :Y ! B be spaces over B. Define
Map B(X, Y ) to be the pullback displayed in the diagram
Map B(X, Y )____//Map(X, "Y)
 Map(id,"q)
 
fflffl fflffl
B ____~____//_Map(X, "B).
Here ~ is the adjoint of the map X x B ! "Bthat corresponds to the composite
of the inclusion Graph (p) X x B and the projection X x B  ! B to the
second coordinate. The graph of p is the inverse image of the diagonal under
p x id:X x B ! B x B, and the assumption that B is weak Hausdorff ensures
that it is a closed subset of X x B, as is needed for the definition to make se*
*nse.
Explicitly, ~(b) sends Xb to b and sends X  Xb to the point ! 2 "B.
This definition gives one reason that we require the base spaces of exspace*
*s to
be weak Hausdorff. On fibers, Map B(X, Y )b = Map (Xb, Yb). The space of sectio*
*ns
of Map B(X, Y ) is K =B(X, Y ). We have (categorically equivalent) adjunctions
(1.3.12) MapB (X xB Y, Z) ~=Map B(X, MapB (Y, Z)),
(1.3.13) K =B (X xB Y, Z) ~=K =B (X, MapB (Y, Z)).
These results are due to Booth [5, 6, 7]; see also [10, x7], [11, x8], [51, IIx*
*9], [57].
Examples in [10, 5.3] and [57, 1.7] show that Map B(X, Y ) need not be weak
Hausdorff even when X and Y are. The question of when Map B(X, Y ) is Hausdorff
or weak Hausdorff was studied in [10, x5] and later in [51, 52], but the defini*
*tive
criterion was given by Lewis [57, 1.5].
Proposition 1.3.14. Consider a fixed map p: X  ! B and varying maps
q :Y  ! B, where X and the Y are weak Hausdorff. The map p is open if and
only if the space Map B(X, Y ) is weak Hausdorff for all q.
Proposition 1.3.15. If p: X ! B and q :Y ! B are Hurewicz fibrations,
then the projections XxB Y ! B and MapB (X, Y ) ! B are Hurewicz fibrations.
The second statement is false with Hurewicz fibrations replaced by Serre fibrat*
*ions.
Proof. The statement about X xB Y is clear. The statements about
Map B(X, Y ) are due to Booth [5, 6.1] or, in the present formulation [6, 3.4];*
* see
also [51, 23.17].
1.4. CONVENIENT CATEGORIES OF EXGSPACES 17
Definition 1.3.16. For exspaces X and Y over B, define FB (X, Y ) to be the
subspace of Map B(X, Y ) that consists of the points that restrict to based maps
Xb ! Yb for each b 2 B; the section sends b to the constant map from Xb to the
basepoint of Yb. Formally, FB (X, Y ) is the pullback displayed in the diagram
FB (X, Y_)_____//MapB(X, Y )
 Map (s,id)
  B
fflffl fflffl
B ____t___//Y ~=Map B(B, Y ),
where s and t are the sections of X and Y .
The space of maps S0B! FB (X, Y ) is KB (X, Y ), and we have adjunctions
(1.3.17) FB (X ^B Y, Z) ~=FB (X, FB (Y, Z)),
(1.3.18) KB (X ^B Y, Z) ~=KB (X, FB (Y, Z)).
Proposition 1.3.14 implies the following analogue of Proposition 1.3.9.
Proposition 1.3.19. If X and Y are weak Hausdorff exspaces over B and
X ! B is an open map, then FB (X, Y ) is weak Hausdorff.
We record the following analogue of Proposition 1.3.15. The second part is
again due to Booth, who sent us a detailed writeup. The argument is similar to
his proofs in [5, 6.1(i)] or [6, 3.4], but a little more complicated, and a gen*
*eral result
of the same form is given by Morgan [78].
Proposition 1.3.20. If X and Y are exspaces over B whose sections are
Hurewicz cofibrations and whose projections are Hurewicz fibrations, then the p*
*ro
jections of X ^B Y and FB (X, Y ) are Hurewicz fibrations.
1.4. Convenient categories of exGspaces
The discussion just given generalizes readily to the equivariant context. L*
*et
G be a compactly generated topological group. Subgroups of G are understood to
be closed. Let B be a compactly generated Gspace (with G acting from the left).
We consider Gspaces over B and exGspaces (X, p, s). The total space X is a
Gspace in K , and the section and projection are Gmaps. The fiber Xb is a bas*
*ed
Gbspace with Gbfixed basepoint s(b), where Gb is the isotropy group of b.
Recall from [61, IIx1] the distinction between the category KG of Gspaces a*
*nd
nonequivariant maps and the category GK of Gspaces and equivariant maps; the
former is enriched over GK , the latter over K . We have a similar dichotomy on*
* the
exspace level. Here we have a conflict of notation with our notation for categ*
*ories
of exspaces, and we agree to let KG,B denote the category whose objects are the
exGspaces over B and whose morphisms are the maps of underlying exspaces over
B, that is, the maps f :X ! Y such that f O s = t and q O f = p. Henceforward,
we call these maps "arrows" to distinguish them from Gmaps, which we often
abbreviate to maps. For g 2 G, gf is also an arrow of exspaces over B, so that
KG,B(X, Y ) is a Gspace. Moreover, composition is given by Gmaps
KG,B(Y, Z) x KG,B(X, Y ) ! KG,B(X, Z).
We obtain the category GKB by restricting to Gmaps f, and we may view it as
the Gfixed point category of KG,B. Of course, GKB (X, Y ) is a space and not a
18 1. THE POINTSET TOPOLOGY OF PARAMETRIZED SPACES
Gspace. The pair (KG,B, GKB ) is an example of a Gcategory, a structure that
we shall recall formally in x10.2.
Since *B is an initial and terminal object in both KG,B and GKB , their mor
phism spaces are based. Thus KG,B is enriched over the category GK* of based
Gspaces and GKB is enriched over K*. As discussed in [61, II.1.3], if we were *
*to
think exclusively in enriched category terms, we would resolutely ignore the fa*
*ct
that the Gspaces KG,B(X, Y ) have elements (arrows), thinking of these Gspaces
as enriched hom objects. From that point of view, GKB is the "underlying cate
gory" of our enriched Gcategory. While we prefer to think of KG,B as a categor*
*y, it
must be kept in mind that it is not a very wellbehaved one. For example, becau*
*se
its arrows are not equivariant, it fails to have limits or colimits.
In contrast, the category GKB is bicomplete. Its limits and colimits are con
structed in KB and then given induced Gactions. The category KG,B, although
not bicomplete, is tensored and cotensored over KG,*. The tensors X ^B K and
cotensors FB (K, X) are constructed in KB and then given induced Gactions. They
satisfy the adjunctions
(1.4.1) KG,B(X ^B K, Y ) ~=KG,*(K, KG,B(X, Y )) ~=KG,B(X, FB (K, Y ))
and, by passage to fixed points,
(1.4.2) GKB (X ^B K, Y ) ~=GK*(K, KG,B(X, Y )) ~=GKB (X, FB (K, Y )).
It follows that GKB is tensored and cotensored over GK* and, in particular, is
topologically bicomplete.
The category KG,B is closed symmetric monoidal via the fiberwise smash prod
ucts X ^B Y and function objects FB (X, Y ). Again, these are defined in KB and
then given induced Gactions. The unit is the exGspace S0B= B x S0. The cat
egory GKB inherits a structure of closed symmetric monoidal category. We have
homeomorphisms of based Gspaces
(1.4.3) KG,B(X ^B Y, Z) ~=KG,B(X, FB (Y, Z))
and, by passage to Gfixed points, homeomorphisms of based spaces
(1.4.4) GKB (X ^B Y, Z) ~=GKB (X, FB (Y, Z)).
The first of these implies an associated homeomorphism of exGspaces
(1.4.5) FB (X ^B Y, Z) ~=FB (X, FB (Y, Z)).
Nonequivariantly, the functor that sends an exspace X over B to the fiber Xb
has a left adjoint, denoted ()b. It sends a based space K to the wedge Kb =
B _ K, where B is given the basepoint b; the section and projection are evident.
Nonobviously, the same set B _ K admits a quite different topology under which *
*it
gives a right adjoint to the fiber functor X 7! Xb. We shall prove the equivari*
*ant
analogue conceptually in Example 2.3.12, but we describe the left adjoint to the
fiber functor explicitly here.
Construction 1.4.6. Let b 2 B. Then the functor GKB  ! GbK* that
sends Y to Yb has a left adjoint. It sends a based Gbspace K to the exGspace
Kb given by the pushout
Kb = (G xGb K) [G B.
1.5. APPENDIX: NONASSOCIATIVITY OF SMASH PRODUCTS IN T op* 19
Here G xGb K is the (left) Gspace (G x K)= ~, where (gh, k) ~ (g, hk) for g 2 *
*G,
h 2 Gb, and k 2 K. The pushout is defined with respect to the map G ! B that
sends g to gb and the map G ! G xGb K that sends g to (g, k0), where k0 is the
(Gbfixed) basepoint of K. The section is given by the evident inclusion of B a*
*nd
the projection is obtained by passage to pushouts from the identity map of B and
the Gmap ssb:G xGb K ! B given by ssb(g, k) = gb. Thus we first extend the
group action on K from Gb to G and then glue the orbit of the basepoint of K to
the orbit of b. If K is an unbased Gbspace, then (K+ )b = (G xGb K) q B.
Remark 1.4.7. There is an alternative parametrized view of equivariance that
is important in torsor theory but that we shall not study. It focuses on "topol*
*og
ical groups GB over B" and "GB spaces E over B", where GB is a space over a
nonequivariant space B with a product GB xB GB ! GB that restricts on fibers
to the products of topological groups Gb and E is a space over B with an action
GB xB E ! E that restricts on fibers to actions Gb x Eb ! Eb. That theory
intersects ours in the special case GB = G x B for a topological group G. Since*
*, at
least implicitly, all of our homotopy theory is done fiberwise, our work adapts*
* with
out essential difficulty to give a development of parametrized equivariant homo*
*topy
theory in that context.
1.5.Appendix: nonassociativity of smash products in T op*
In a 1958 paper [82], Puppe asserted the following result, but he did not gi*
*ve a
proof. It was the subject of a series of emails among Mike Cole, Tony Elmendor*
*f,
Gaunce Lewis and the first author. Since we know of no published source that gi*
*ves
the details of this or any other counterexample to the associativity of the sma*
*sh
product in T op*, we include the following proof. It is due to Kathleen Lewis.
Let Q and N be the rational numbers and the nonnegative integers, topologized
as subspaces of R and given the basepoint zero. Consider smash products as quo
tient spaces, without applying the kification functor. Then we have the follow*
*ing
counterexample to associativity.
Theorem 1.5.1. (Q ^ Q) ^ N is not homeomorphic to Q ^ (Q ^ N).
Proof. Consider the following diagram.
Q x Q xON
idxp0ooooo  OOOOpxidOO
oooo  OO
wwooo  OOO''
Q x (Q ^ N) q (Q ^ Q) x N

  
s  r
fflffl t fflffl~= fflffl
Q ^ (Q ^ N)oo___Q ^ Q ^ N_____//(Q ^ Q) ^ N
Here Q ^ Q ^ N denotes the evident quotient space of Q x Q x N. The maps p, p0,
q, r, and s are quotient maps. Since N is locally compact, p x idis also a quot*
*ient
map, hence so is r O (p x id). The universal property of quotient spaces then g*
*ives
the bottom right homeomorphism. Since Q is not locally compact, idx p0 need
not be a quotient map, and in fact it is not. The map t is a continuous bijecti*
*on
given by the universal property of the quotient map q, and we claim that t is n*
*ot
a homeomorphism. To show this, we display an open subset of Q ^ Q ^ N whose
image under t is not open.
20 1. THE POINTSET TOPOLOGY OF PARAMETRIZED SPACES
Let fi be an irrational number, 0 < fi < 1, and let fl = (1  fi)=2. Define *
*V 0(fi)
to be the open subset of R x R that is the union of the following four sets.
(1)The open ball of radius fi about the origin
(2)The tubes [1, 1)x(fl, fl), (1, 1]x(fl, fl), (fl, fl)x[1, 1), and (fl, *
*fl)x
(1, 1] of width 2fl about the axes.
(3)The open balls of radius fl about the four points ( 1, 0), (0, 1).
(4)For each n 1, the open ballPof radius fl=2n about the four points ( fln, 0*
*),
(0, fln), where fln = 1  k=n1k=0fl=2k.
To visualize this set, it is best to draw a picture. It is symmetric with re*
*spect
to 90 degree rotation. Consider the part lying along the positive xaxis. A tube
of width 2fl covers the part of the xaxis to the right of (1, 0). A ball of ra*
*dius fi
centers at the origin. A ball of radius fl centers at (1, 0). Its vertical diag*
*onal is
the edge of the tube going off to the right. On the left, by the choice of fl, *
*this ball
reaches halfway from its center (1, 0) to the point (fi, 0) at the right edge o*
*f the ball
centered at the origin. The point (1  fl, 0) at the left edge of the ball cent*
*ered at
(1, 0) is the center of another ball, which reaches half the distance from (1 *
* fl, 0) to
(fi, 0). And so on: the point where the left edge of the nth ball crosses the x*
*axis
is the center point of the (n + 1)st ball, which reaches half the distance from*
* its
center to the edge of the ball centered at the origin.
Define V (fi) = V 0(fi)\(QxQ). Note that the only points of the coordinate a*
*xes
of R x R that are not in V 0(fi) are ( fi, 0) and (0, fi). Since fi is irratio*
*nal, V (fi)
contains the coordinate axes of QxQ. Because the radii of the balls in the sequ*
*ence
are decreasing, for each " > fi, there is no ffi > 0 such that ((", ")x(ffi, *
*ffi))\(QxQ)
is contained in V (fi).
Now let ff be an irrational number, 0 < ff < 1. Let o be the basepoint of Q *
*^ N
and * be the basepoint of Q ^ Q ^ N. Let U be the union of {*} and the image
under q of [n 1V (ff=n) x {n}. This is an open subspace of Q ^ Q ^ N since
q1(U) = Q x Q x {0} [ ([n 1V (ff=n) x {n})
is an open subset of Q x Q x N. We claim that t(U) is not open in Q ^ (Q ^ N).
Assume that t(U) is open. Then
s1(t(U)) = (idx p0)(q1(U))
is an open subset of Q x (Q ^ N), hence it contains an open neighborhood V of
(0, o). Now V must contain ((", ") \ Q) x W for some " > 0 and some open
neighborhood W of o in Q ^ N. Since Q ^ N is homeomorphic to the wedge over
n 1 of the spaces Q x {n}, W must contain the wedge over n 1 of subsets
((ffin, ffin) \ Q) x {n}, where ffin > 0. By the definition of U, this implies*
* that
((", ") x (ffin, ffin)) \ (Q x Q) V (ff=n).
However, for n large enough that " > ff=n, there is no ffin for which this hold*
*s.
CHAPTER 2
Change functors and compatibility relations
Introduction
In the previous chapter, we developed the internal properties of the category
GKB of exGspaces over B. As B and G vary, these categories are related by
various functors, such as base change functors, change of groups functors, orbi*
*t and
fixed point functors, external smash product and function space functors, and so
forth. We define these "change functors" and discuss various compatibility rela*
*tions
among them in this chapter.
We particularly emphasize base change functors. We give a general categori
cal discussion of such functors in x2.1, illustrating the general constructions*
* with
topological examples. In x2.2, we discuss various compatibility relations that *
*relate
these functors to smash products and function objects.
In x2.3 and x2.4 we turn to equivariant phenomena and study restriction of
group actions along homomorphisms. As usual, we break this into the study of
restriction along inclusions and pullback along quotient homomorphisms.
In x2.3, we discuss restrictions of group actions to subgroups, together with
the associated induction and coinduction functors. We also consider their compa*
*t
ibilities with base change functors. In particular, this gives us a convenient*
* way
of thinking about passage to fibers and allows us to reinterpret restriction to*
* sub
groups in terms of base change and coinduction. That is the starting point of o*
*ur
generalization of the Wirthm"uller isomorphism in Part IV.
In x2.4, we consider pullbacks of group actions from a quotient group G=N to
G, together with the associated quotient and fixed point functors. Again, we al*
*so
consider compatibilities with base change functors. For an Nfree base space E,
we find a relation between the quotient functor ()=N and the fixed point funct*
*or
()N that involves base change along the quotient map E ! E=N. The good
properties of the bundle construction in Part IV can be traced back to this rel*
*ation,
and it is at the heart of the Adams isomorphism in equivariant stable homotopy
theory.
In x2.5, we describe a different categorical framework, one appropriate to e*
*x
spaces with varying base spaces. We show that the relevant category of retracts*
* over
varying base spaces is closed symmetric monoidal under external smash product
and function exspace functors. The internal smash product and function exspace
functors are obtained from these by use of base change along diagonal maps. The
external smash products are much better behaved homotopically than the internal
ones, and homotopical analysis of base change functors will therefore play a ce*
*ntral
role in the homotopical analysis of smash products.
21
22 2. CHANGE FUNCTORS AND COMPATIBILITY RELATIONS
In much of this chapter, we work in a general categorical framework. In some
places where we restrict to spaces, more general categorical formulations are u*
*n
doubtedly possible. When we talk about group actions, all groups are assumed to
be compactly generated spaces but are otherwise unrestricted.
2.1. The base change functors f!, f*, and f*
Let f :A ! B be a map in a bicomplete subcategory B of a bicomplete
category C . We are thinking of U K or GU GK . We wish to define
functors
f!:CA ! CB , f* :CB ! CA , f*: CA ! CB ,
such that f!is left adjoint and f* is right adjoint to f*. The definitions of f*
** and
f!are dual and require no further hypotheses. The definition of f* does not work
in full generality, but it only requires the further hypothesis that C =B be ca*
*rtesian
closed. Thus we assume given internal hom objects Map B(Y, Z) in C =B that
satisfy the usual adjunction, as in (1.3.13). One reason to work in this genera*
*lity
is to emphasize that no further pointset topology is needed to construct these
base change functors in the context of exspaces. This point is not clear from *
*the
literature, where the functor f* is often given an apparently different, but na*
*turally
isomorphic, description. We work with generic exobjects
p t q
A __s__//X_____//A and B ____//_Y____//B
in this section.
Definition 2.1.1. Define f!X and its structure maps q and t by means of the
map of retracts in the following diagram on the left, where the top square is a
pushout and the bottom square is defined by the universal property of pushouts
and the requirement that q O t = id. Define f*Y and its structure maps p and s *
*by
means of the map of retracts in the following middle diagram, where the bottom
square is a pullback and the top square is defined by the universal property of
pullbacks and the requirement that p O s = id.
f f '
A ______//B A _____//_B B ______//MapB(A, A)
s  t s t t  Map(id,s)
fflffl fflffl fflffl fflffl fflffl fflffl
X _____//f!X f*Y ____//_Y f*X _____//MapB(A, X)
p  q p q q  Map(id,p)
fflffl fflffl fflffl fflffl fflffl fflffl
A __f___//B A __f__//_B B ______//Map(A, A)
' B
Thinking of X and A as objects over B via f O p and f and observing that the
adjoint of the identity map of A gives a map ': B ! Map B(A, A), define f*X and
its structure maps q and t by means of the map of retracts in the above diagram
on the right, where the bottom square is a pullback and the top square is defin*
*ed
by the universal property of pullbacks and the requirement that q O t = id.
Proposition 2.1.2. (f!, f*) is an adjoint pair of functors:
CB (f!X, Y ) ~=CA (X, f*Y ).
2.2. COMPATIBILITY RELATIONS 23
Proof. Maps in both hom sets are specified by maps k :X ! Y in C such
that q O k = f O p and k O s = t O f.
Proposition 2.1.3. (f*, f*) is an adjoint pair of functors:
CA (f*Y, X) ~=CB (Y, f*X).
Proof. A map k :f*Y = Y xB A ! X such that p O k = p and k O s = s
has adjoint "k:Y ! Map B(A, X) such that Map (id, p) O "k= ' O q and "kO t =
Map (id, s) O '. The conclusion follows directly.
Remark 2.1.4. Writing these proofs diagrammatically, we see that the ad
junction isomorphisms are given by homeomorphisms in our context of topological
categories.
We specialize to exspaces (or exGspaces), in the rest of the section. Obs*
*erve
that the fiber (f*X)b is the space of sections Ab ! Xb of p: Xb ! Ab.
Remark 2.1.5. If f :A ! B is an open map and X is in U , then f*X is in
U and UA (f*Y, X) ~=UB (Y, f*X) for Y 2 U , by [57, 1.5].
Example 2.1.6. Let f :A ! B be an inclusion. Then f*Y is the restriction
of Y to A and f!X = B [A X. The exspace f*X over B is analogous to the
prolongation by zero of a sheaf over A. The fiber (f*X)b is Xa if a 2 A and a p*
*oint
{b} otherwise. To see this from the definition, recall that Map (;, K) is a poi*
*nt for
any space K and that Map B(A, X)b = Map (Ab, Xb). As a set, f*X ~=B [A X,
but the topology is quite different. It is devised so that the map Y ! f*f*Y t*
*hat
restricts to the identity on Ya for a 2 A but sends Yb to {b} for b =2A is cont*
*inuous.
Example 2.1.7. Let r :B ! * be the unique map. For a based space X and
an exspace E = (E, p, s) over B, we have
r*X = B x X, r!E = E=s(B), and r*E = Sec(B, E),
where Sec(B, E) is the space of maps t: B  ! E such that p O t = id, with
basepoint the section s. These elementary base change functors are the key to u*
*sing
parametrized homotopy theory to obtain information in ordinary homotopy theory.
Let ": r!r* ! id and j :id! r*r! be the counit and unit of the adjunction
(r!, r*). Then r!r*X ~=B+ ^X and " is r+ ^id. Similarly, r!r*r!E ~=B+ ^E=B, and
r!j :r!E ! r!r*r!E is the "Thom diagonal" E=B ! B+ ^ E=B. If p: E ! B is
a spherical fibration with section, such as the fiberwise onepoint compactific*
*ation
of a vector bundle, then r!E is the Thom complex of p.
2.2. Compatibility relations
The term "compatibility relation" has been used in algebraic geometry in the
context of Grothendieck's six functor formalism that relates base change functo*
*rs
to tensor product and internal hom functors in sheaf theory. We describe how the
analogous, but simpler, formalism appears in our categories of exobjects.
We recall some language. We are especially interested in the behavior of base
change functors with respect to closed symmetric monoidal structures that, in o*
*ur
topological context, are given by smash products and function objects. Relevant
categorical observations are given in [40]. We say that a functor T :B ! A
between closed symmetric monoidal categories is closed symmetric monoidal if
T SB ~=SA , T (X ^B Y ) ~=T X ^A T Y, and T FB (X, Y ) ~=FA (T X, T Y ),
24 2. CHANGE FUNCTORS AND COMPATIBILITY RELATIONS
where SB , ^B and FB denote the unit object, product, and internal hom of B, and
similarly for A . These isomorphisms must satisfy appropriate coherence conditi*
*ons.
In the language of [40], the following result states that any map f of base spa*
*ces
gives rise to a "Wirthm"uller context", which means that the functor f* is clos*
*ed
symmetric monoidal and has both a left adjoint and a right adjoint.
Proposition 2.2.1. If f :A ! B is a map of base Gspaces, then the functor
f* :GKB  ! GKA is closed symmetric monoidal. Therefore, by definition and
implication, f*S0B~=S0Aand there are natural isomorphisms
(2.2.2) f*(Y ^B Z) ~=f*Y ^A f*Z,
(2.2.3) FB (Y, f*X) ~=f*FA (f*Y, X),
(2.2.4) f*FB (Y, Z) ~=FA (f*Y, f*Z),
(2.2.5) f!(f*Y ^A X) ~=Y ^B f!X,
(2.2.6) FB (f!X, Y ) ~=f*FA (X, f*Y ),
where X is an exGspace over A and Y and Z are exGspaces over B.
Proof. The isomorphism f*S0B~=S0Ais evident since f*(B x K) ~=A x K for
based Gspaces K. The isomorphism (2.2.2) is obtained by passage to quotients
from the evident homeomorphism
(Y xB A) xA (Z xB A) ~=(Y xB Z) xB A
As explained in [40, xx2, 3], the isomorphism (2.2.2) is equivalent to the isom*
*or
phism (2.2.3), and it determines natural maps from left to right in (2.2.4), (2*
*.2.5),
and (2.2.6) such that all three are isomorphisms if any one is. By a comparison*
* of
definitions, we see that the categorically defined map in (2.2.4), which is den*
*oted
ff in [40, 3.3], coincides in the present situation with the map, also denoted *
*ff, on
[11, p. 167]. As explained on [11, p. 178], in the pointset topological frame*
*work
that we have adopted, that map ff is a homeomorphism.
Remark 2.2.7. Only the very last statement refers to topology. The categori
cally defined map ff should quite generally be an isomorphism in analogous cont*
*exts,
but we have not pursued this question in detail. An alternative selfcontained *
*proof
of the previous proposition is given in Remark 2.5.6 below by using Proposition*
* 2.2.9
to prove (2.2.5) instead of (2.2.4). In that argument, the only nonformal ingr*
*edient
is the fact that the functor D xB () commutes with pushouts.
We shall later need a purely categorical coherence observation about the cat
egorically defined map ff of (2.2.4). In fact, it will play a key role in the p*
*roof of
the fiberwise duality theorem of x15.1. It is convenient to insert it here.
Remark 2.2.8. Let T :B ! A be a symmetric monoidal functor. We are
thinking of T as, for example, a base change functor f*. The map
ff: T FB (X, Y ) ! FA (T X, T Y )
is defined to be the adjoint of
T FB (X, Y ) ^A T X ~=T (FB (X, Y ) ^BTX)ev//_T Y.
2.3. CHANGE OF GROUP AND RESTRICTION TO FIBERS 25
The dual of X is DB X = FB (X, SB ), where SB is the unit of B. Taking Y = SB ,
the definition of ff implies that the top triangle commutes in the diagram
~= Tev
T DB X ^A T X ________//_T (DB X ^B X)___//_T2SB2ffffff
ffffff
ff^Aid fffffevffffff ~=
fflfflffffff fflffl
FA (T X, T SB ) ^A T_X~=//_DA f*X ^A f*X_ev__//_SA .
The bottom triangle is a naturality diagram. The outer rectangle is [40, 3.7], *
*but
its commutativity in general was not observed there. However, it was observed in
[40, 3.8] that its commutativity implies the commutativity of the diagram
~= T
T DB X ^A T Y ____//_T (DB X ^B Y_)___//T FB (X, Y )
ff^ATY  ff
fflffl fflffl
DA T X ^A T Y _______________________//_FA (T X, T Y ),
where :DB X ^B Y ! FB (X, Y ) is the adjoint of
DB X ^B Y ^B X ~=DB X ^B X ^B Y ev^id//_SB ^B Y ~=Y.
In other contexts, the analogue of (2.2.5) is called the "projection formula*
*",
and we shall also use that term. The following base change commutation relations
with respect to pullbacks are also familiar from other contexts. We state the r*
*esult
for spaces but, apart from use of the fact that the functor D xB () commutes w*
*ith
pushouts, the proof is formal.
Proposition 2.2.9. Suppose given a pullback diagram of base spaces
g
C _____//D
i j
fflfflfflffl
A __f__//B.
Then there are natural isomorphisms of functors
(2.2.10) j*f!~=g!i*, f*j* ~=i*g*, f*j!~=i!g*, j*f* ~=g*i*.
Proof. The first isomorphism is one of left adjoints, and the second is the
corresponding "conjugate" isomorphism of right adjoints. Similarly for the thi*
*rd
and fourth isomorphisms. By symmetry, it suffices to prove the first isomorphis*
*m.
The functor j* = D xB () commutes with pushouts. For a space X over A
regarded by composition with f as a space over B, C xA X ~=D xB X. This gives
j*f!X = D xB (B [A X) ~=D [C (C xA X) = g!i*X.
2.3. Change of group and restriction to fibers
This section begins the study of equivariant phenomena that have no non
equivariant counterparts. In particular, using a conceptual reinterpretation of*
* the
adjoints of the fiber functors ()b, we relate restriction to subgroups to rest*
*riction to
fibers. Recall that subgroups of G are understood to be closed and fix an inclu*
*sion
': H G throughout this section. Parametrized theory gives a convenient way of
studying restriction along ' without changing the ambient group from G to H.
26 2. CHANGE FUNCTORS AND COMPATIBILITY RELATIONS
Proposition 2.3.1. The category GKG=H of exGspaces over G=H is equiv
alent to the category HK* of based Hspaces.
Proof. The equivalence sends an exGspace (Y, p, s) over G=H to the H
space p1(eH) with basepoint the Hfixed point s(eH). Its inverse sends a based
Hspace X to the induced Gspace G xH X, with the evident structure maps.
More formally, recall that there are "induction" and "coinduction" functors *
*'!
and '* from Hspaces to Gspaces that are left and right adjoint to the forgetf*
*ul
functor '* that sends a Gspace Y to Y regarded as an Hspace. Explicitly, for *
*an
Hspace X,
(2.3.2) '!X = G xH X and '*X = Map H(G, X).
The latter is the space of maps of (left) Hspaces, with (left) action of G ind*
*uced
by the right action of G on itself. Similarly, when X is a based Hspace, we ha*
*ve
the based analogues
(2.3.3) '!X = G+ ^H X and '*X = FH (G, X).
With this notation, some familiar natural isomorphisms take the forms
(2.3.4) '!('*Y x X) ~=Y x '!X and '*Map ('*Y, X) ~=Map (Y, '*X)
and, in the based case,
(2.3.5) '!('*Y ^ X) ~=Y ^ '!X and '*F ('*Y, X) ~=F (Y, '*X).
By the uniqueness of adjoints, or inspection of definitions, we see that these *
*familiar
change of groups functors are change of base functors along r :G=H ! *.
Corollary 2.3.6. The change of group and change of base functors associated
to ' and r agree under the equivalence of categories between HK* and GKG=H :
'* ~=r*, '!~=r!, and '* ~=r*.
We can generalize this equivalence of categories, using the following defini*
*tions.
We have a forgetful functor '*: GKB ! HK'*B. It doesn't have an obvious left
or right adjoint, but we have obvious analogues of induction and coinduction th*
*at
involve changes of base spaces. The first will lead to a description of '* as a*
* base
change functor and thus as a functor with a left and right adjoint.
Definition 2.3.7. Let A be an Hspace and X be an Hspace over A. Define
'!:HKA ! GK'!Aby letting '!X be the Gspace G xH X over '!A = G xH A.
Define '*: HKA ! GK'*A by letting '*X be the Gspace Map H(G, X) over
'*A = Map H(G, A).
For an Hspace A and a Gspace B, let
(2.3.8) ~: G xH '*B = '!'*B ! B and :A ! '*'!A = '*(G xH A)
be the counit and unit of the ('!, '*) adjunction. The following result says t*
*hat
exHspaces over an Hspace A are equivalent to exGspaces over the Gspace '!*
*A.
Proposition 2.3.9. The functor '!:HKA ! GK'!Ais a closed symmetric
monoidal equivalence of categories with inverse the composite
* *
GK'!A'! HK'*'!A! HKA .
2.3. CHANGE OF GROUP AND RESTRICTION TO FIBERS 27
Applied to A = '*B, this equivalence leads to the promised description of
'*: GKB ! HK'*B as a base change functor.
Proposition 2.3.10. The functor '*: GKB ! HK'*B is the composite
*
GKB _~__//_GK'!'*B~=HK'*B
Change of base and change of groups are related by various further consisten*
*cy
relations. The following result gives two of them.
Proposition 2.3.11. Let f :A ! '*B be a map of Hspaces and "f:'!A ! B
be its adjoint map of Gspaces. Then the following diagrams commute up to natur*
*al
isomorphism.
"f! f"*
GK'!AO_____//_GKBOOO GKB ______//GK'!A
'! ~!O'! '* *O'*
  fflffl fflffl
HKA __f!_//_HK'*B HK'*B __f*_//_HKA
Proof. Since "f= ~ O '!f, we have
f"!O '!~=(~ O '!f)!O '!~=~!O ('!f)!O '!~=~!O '!O f!,
where the last isomorphism holds because GxH () commutes with pushouts. Since
f = '*f"O , we have
f* O '* ~=('*f"O )* O '* ~= * O ('*f")* O '* ~= * O '* O "f*,
where the last isomorphism holds because pulling the G action back to an Hacti*
*on
commutes with pullbacks.
The reader may find it illuminating to work out these isomorphisms in the
context of Proposition 2.3.1. That result leads to the promised conceptual rein*
*ter
pretation of Construction 1.4.6.
Example 2.3.12. For b 2 B, we also write b: * ! B for the map that sends
* to b, and we write "b:G=Gb ! B for the induced inclusion of orbits. Thus
b is a Gbmap and "bis a Gmap. Under the equivalence GKG=Gb ~= GbK* of
Proposition 2.3.1, "b*may be interpreted as the fiber functor GKB ! GbK* that
sends X to Xb, "b!may be interpreted as the left adjoint of Construction 1.4.6 *
*that
sends K to Kb, and "b*specifies a right adjoint to the fiber functor, which we *
*denote
by bK. With these notations, the isomorphisms of Proposition 2.2.1 specialize to
the following natural isomorphisms, where Y and Z are in GKB and K is in GbK*.
(Y ^B Z)b ~=Yb^ Zb,
FB (Y, bK) ~=bF (Yb, K),
FB (Y, Z)b ~=F (Yb, Zb),
(Yb^ K)b ~=Y ^B Kb,
FB (Kb, Y ) ~=bF (K, Yb).
28 2. CHANGE FUNCTORS AND COMPATIBILITY RELATIONS
Example 2.3.13. Several earlier results come together in the following situa
tion. Let f :A ! B be a Gmap. For b 2 B, let b: {b} ! B and ib:Ab ! A
denote the evident inclusions of Gbspaces. We have the following compatible pu*
*ll
back squares, the first of Gbspaces and the second of Gspaces.
fb GxGbfb
Ab_____//{b} G xGb Ab _____//G=Gb
ib b "b "b
fflffl fflffl fflffl fflffl
A __f___//B A ____f____//_B
Applying Proposition 2.2.9 to the righthand square and interpreting the conclu*
*sion
in terms of fibers by Definition 2.3.7, we obtain canonical isomorphisms of Gb*
*spaces
(f!X)b ~=fb!i*bX and (f*X)b ~=fb*i*bX,
where X is an exGspace over A, regarded on the righthand sides as an exGb
space over A by pullback along ': Gb ! G.
2.4. Normal subgroups and quotient groups
Observe that any homomorphism ` :G ! G0 factors as the composite of a
quotient homomorphism ", an isomorphism, and an inclusion '. We studied change
of groups along inclusions in the previous section. Here we consider a quotient
homomorphism ffl: G ! J of G by a normal subgroup N. We still have a restricti*
*on
functor
ffl*: JKA ! GKffl*A,
and we also have the functors
()=N :GKB ! JKB=N and ()N :GKB ! JKBN
obtained by passing to orbits over N and to Nfixed points. When B is a point,
these last two functors are left and right adjoint to ffl*, but in general chan*
*ge of base
must enter in order to obtain such adjunctions. The following observation follo*
*ws
directly by inspection of the definitions.
Proposition 2.4.1. Let j :BN  ! B be the inclusion and p: B ! B=N be
the quotient map. Then the following factorization diagrams commute.
()=N ()N
GKB _____//_JKB=N99 and GKB _____//_JKBN
ss ttt99
p! ssss j* tttt N
fflffl()=Nsss fflffl()tt
GKB=N GKBN
It follows that (()=N, p*ffl*) and (j!ffl*, ()N ) are adjoint pairs.
We have the following analogue of Proposition 2.3.11.
2.4. NORMAL SUBGROUPS AND QUOTIENT GROUPS 29
Proposition 2.4.2. Let f :A ! B be a map of Gspaces. Then the following
diagrams commute up to natural isomorphisms.
f! f* f!
GKA _______//GKB GKB ______//GKA GKA _____//_GKB
()=N  ()=N ()N ()N ()N ()N
fflffl fflffl fflffl fflffl fflffl fflffl
JKA=N (f=N)//_JKB=N JKBN _____//JKAN JKAN _____//JKBN
! (fN )* (fN )!
Proof. For exGspaces X over A and Y over B, these isomorphisms are given
by the homeomorphisms
(X [A B)=N ~=X=N [A=N B=N,
(Y xB A)N ~=Y N xBN AN ,
and
(X [A B)N ~=XN [AN BN .
As a quibble, the last requires A ! X to be a closed inclusion, but this will *
*hold
for the sections of compactly generated exGspaces over A by Lemma 1.1.3(i).
Specializing to Nfree Gspaces, we obtain a factorization result that is an*
*al
ogous to those in Proposition 2.4.1, but is less obvious. It is a precursor of*
* the
Adams isomorphism, which we will derive in x16.4.
Proposition 2.4.3. Let E be an Nfree Gspace, let B = E=N, and let
p: E ! B be the quotient map. Then the diagram
()=N
GKE _____//JKB;;
vv
p* vvvv
fflffl()Nvvv
GKB
commutes up to natural isomorphism. Therefore the left adjoint ()=N of the
functor p*"* is also its right adjoint.
Proof. Let X be an exGspace over E with projection q. Comparing the
pullbacks that are used to define the functors p* and Map Bin Definitions 2.1.1*
* and
1.3.11, we find that p*X fits into a pullback diagram
p*X _____//Map(E, "X)

 "q
 
fflffl fflffl
B ______//Map(E, "E).
Here (b), b = Ne, corresponds as in Definition 1.3.10 to the inclusion of the *
*closed
subset Ne in E. Passing to Nfixed points, we see that it suffices to prove tha*
*t the
following commutative diagram is a pullback.
X=N ___~___//MapN(E, "X)
q=N "q
fflffl fflffl
E=N = B _____//MapN(E, "E)
30 2. CHANGE FUNCTORS AND COMPATIBILITY RELATIONS
Here ~ is induced from the adjoint of the map X x E ! "Xthat sends (x, e) to nx
if e = nq(x) and sends (x, e) to ! otherwise. With this description, ~ is well*
*defined
since E is Nfree. It suffices to give a continuous inverse to the induced map
OE: X=N ! Map N(E, "X) xMapN(E,E")E=N.
If (f, Ne) is a point in the pullback, then f corresponds to a map Ne ! X, and
OE1(f, Ne) = Nf(e) in X=N. For continuity, note that OE1 is obtained from the
evaluation map Map (E, "X) x E ! "Xby passage to subquotient spaces.
Remark 2.4.4. This leads to a useful alternative description of the functor
'!:HKA  ! GK'!A, where A is an Hspace and '!A = G xH A. We have the
projection ss :G x A ! A of (G x H)spaces, where the G x H actions on the
source and target are given by
(g, h)(g0, a) = (gg0h1, ha) and (g, h)a = ha.
Consider exHspaces X over A as (G x H)spaces with G acting trivially and
let ffl: G x H ! H be the projection. We see from the definition that '!X =
(ss*"*X)=H. Since GxA is an Hfree (GxH)space, we conclude from the previous
result that '!X ~=(p*ss*"*X)H , where p: G x A ! G xH A = '!A is the quotient
map.
2.5. The closed symmetric monoidal category of retracts
Let B be a topologically bicomplete full subcategory of a topologically bico*
*m
plete category C . We are thinking of U K or GU GK . We have the
category of retracts CB . The objects of CB are the retractions B s! X p! B
with B 2 B and X 2 C , abbreviated (X, p, s) or just X. The morphisms of CB are
the evident commutative diagrams. When B = C , this is just a diagram category
for the evident two object domain category.
The importance of the category CB is apparent from its role in Definition 2.*
*1.1:
focus on this category is natural when we consider base change functors. In our
examples, B and C are enriched and topologically bicomplete over the appropriate
category of spaces, U for B and K for C . For a space K 2 K , the tensors  x K
and cotensors Map (K, ) applied to retractions give retractions, and we have t*
*he
adjunction homeomorphisms
(2.5.1) CB (X x K, Y ) ~=K (K, CB (X, Y )) ~=CB (X, Map(K, Y )).
The category GKGU is closed symmetric monoidal under an external smash
product functor, denoted X ZY , and an external function exspace functor, deno*
*ted
~F(Y, Z). If X, Y , and Z are exspaces over A, B, and A x B, respectively, then
X Z Y is an exspace over A x B and ~F(Y, Z) is an exspace over A. We have
(2.5.2) GKAxB (X Z Y, Z) ~=GKA (X, ~F(Y, Z)),
which gives the required adjunction in GKGU . It specializes to parts of (1.4.2*
*) when
A or B is a point. The exspace X Z Y is the evident fiberwise smash product, w*
*ith
(X Z Y )(a,b)= Xa ^ Yb. The fiber ~F(Y, Z)a is FB (Y, Za), where Za is the exs*
*pace
over B whose fiber Za,bover b is the inverse image of (a, b) under the projecti*
*on
Z ! A x B. Rather than describe the topology of the exspace ~F(Y, Z) directly,
we give alternative descriptions of X Z Y and ~F(Y, Z) in terms of internal sma*
*sh
products and internal function exspaces. Let ssA and ssB be the projections of
A x B on A and B and observe that ss*AX ~=X x B and ss*BY ~=A x Y . If one like*
*s,
2.5. THE CLOSED SYMMETRIC MONOIDAL CATEGORY OF RETRACTS 31
the following results can be taken as a definition of the external operations a*
*nd a
characterization of the internal operations, or vice versa.
Lemma 2.5.3. The external smash product and function exspace functors are
determined by the internal functors via natural isomorphisms
X Z Y ~=ss*AX ^AxB ss*BY and ~F(Y, Z) ~=ssA*FAxB (ss*BY, Z),
where X, Y , and Z are exspaces over A, B, and A x B, respectively.
With these isomorphisms taken as definitions, the adjunction (2.5.2) follows
from the adjunctions (ss*A, ssA*), (ss*B, ssB*), and (^AxB , FAxB ).
Lemma 2.5.4. The internal smash product and function exspace functors are
determined by the external functors via natural isomorphisms
X ^B Y ~= *(X Z Y ) and FB (X, Y ) ~=~F(X, *Y ),
where X and Y are exspaces over B and : B ! B x B is the diagonal map.
With these isomorphisms taken as definitions, the adjunction (^B , FB ) foll*
*ows
from the adjunctions ( *, *) and (2.5.2). Since * is symmetric monoidal and
the composite of either projection ssi:B x B ! B with is the identity map of
B, we see that, if we have constructed both internal and external smash product*
*s,
then they must be related by natural isomorphisms as in Lemmas 2.5.3 and 2.5.4.
Remark 2.5.5. The first referee suggests that we point out another consisten*
*cy
check. The fiber ( *Y )(b,c)is a point if b 6= c and is Yb if b = c. Therefor*
*e the
fiber over b of the restriction ( *Y )b of *Y to {b} x B is Yb[ (B  {b}), sui*
*tably
topologized, and
~F(X, *Y )b = FB (X, ( *Y )b)b ~=F (Xb, Yb) = FB (X, Y )b.
Remark 2.5.6. The description of the internal smash product in terms of the
external smash product sheds light on the basic compatibility isomorphisms (2.2*
*.2)
and (2.2.5). For maps f :A ! B and g :A0 ! B0 and for exspaces X over B
and Y over B0, it is easily checked that
(2.5.7) f*Y Z g*Z ~=(f x g)*(Y Z Z).
Similarly, for exspaces W over A and X over A0,
(2.5.8) f!W Z g!X ~=(f x g)!(W Z X).
Now take A = A0, B = B0 and f = g. For exspaces Y and Z over B,
f*(Y ^B Z) ~=f* *B(Y Z Z) ~=( B O f)*(Y Z Z).
On the other hand, using (2.5.7),
f*Y ^A f*Z ~= *A(f x f)*(Y Z Z) ~=((f x f) O A )*(Y Z Z).
The right sides are the same since B O f = (f x f) O A . Similarly,
f!(f*Y ^A X) ~=f! *A(f x id)*(Y Z X) ~=f!((f x id) O A )*(Y Z X),
while
Y ^B f!X ~= *B(idx f)!(Y Z X).
32 2. CHANGE FUNCTORS AND COMPATIBILITY RELATIONS
Since the diagram
A fxid
A _____//A x A____//_B x A
f idxf
fflffl fflffl
B _________B______//B x B
is a pullback, the right sides are isomorphic by Proposition 2.2.9.
It is illuminating conceptually to go further and consider group actions from
an external point of view. For groups H and G, an Hspace A, and a Gspace B,
we have an evident external smash product
(2.5.9) Z: HKA x GKB ! (H x G)KAxB .
For an exHspace X over A and an exGspace Y over B, X ZY is just the internal
smash product over the (H x G)space A x B of ss*Hss*AX and ss*Gss*BY , where t*
*he
ss0s are the projections from H x G and A x B to their coordinates. It is easil*
*y seen
that this definition leads to another (Z, ~F) adjunction.
When H = G, the diagonal : G ! G x G is a closed inclusion since G
is compactly generated. We can pull back along , and then our earlier external
smash product X Z Y over the Gspace *(A x B) is given in terms of (2.5.9) as
the pullback *(X Z Y ). Note that, by Proposition 2.3.10, * here can be viewed
as a base change functor.
CHAPTER 3
Proper actions, equivariant bundles, and fibrations
Introduction
Much of the work in equivariant homotopy theory has focused on compact Lie
groups. However, as was already observed by Palais [81], many results can be
generalized to arbitrary Lie groups provided that one restricts to proper actio*
*ns.
These are wellbehaved actions whose isotropy groups are compact, and all actio*
*ns
by compact Lie groups are proper. The classical definition of a Lie group [17, *
*p.
129] includes all discrete groups (even though they need not be second countabl*
*e)
and, for discrete groups, the proper actions are the properly discontinuous one*
*s.
In the parametrized world, the homotopy theory is captured on fibers. When
we restrict to proper actions on base spaces, the fibers have actions by the co*
*mpact
isotropy groups of the base space. So even though our primary interest is still*
* in
compact Lie groups of equivariance, proper actions on the base space provide the
right natural level of generality. We set the stage for such a theory in this c*
*hapter
by generalizing various classical results about equivariant bundles and fibrati*
*ons to
a setting focused on proper actions by Lie groups. The reader interested primar*
*ily
in the nonequivariant theory should skip this chapter since only some very stan*
*dard
material in it is relevant nonequivariantly.
In x3.1, we recall some basic results about proper actions of locally compact
groups. We use this discussion to generalize some results about equivariant bun*
*dles
in x3.2. We generalize Waner's equivariant versions of Milnor's results on spac*
*es
of the homotopy types of CW complexes in x3.3. In x3.4, we recall and generalize
classical theorems of Dold and Stasheff about Hurewicz fibrations. We also reca*
*ll
an important but little known result of Steinberger and West that relates Serre*
* and
Hurewicz fibrations. We recall the definition of equivariant quasifibrations in*
* x3.5.
3.1. Proper actions of locally compact groups
We recall relevant definitions and basic results about proper actions in this
section. For appropriate generality and technical convenience, we let G be a lo*
*cally
compact topological group whose underlying topological space is compactly gen
erated. Local compactness means that the identity element, hence any point, has
a compact neighborhood. We see from Proposition 1.1.2 that G is Hausdorff and,
since all compact subsets are closed, it follows that each neighborhood of any *
*point
contains a compact neighborhood.
Remark 3.1.1. We comment on the assumptions we make for G. If G is any
topological group whose underlying space is in K , then an action of G on X in
K may not come from an action in T op. The point is that the product G x X in
K is defined by applying the kification functor to the product G xc X in T op,
and not every action G x X ! X need be continuous when viewed as a function
33
34 3. PROPER ACTIONS, EQUIVARIANT BUNDLES, AND FIBRATIONS
G xcX ! X. However, when G is locally compact, G xcX is already in K by
Proposition 1.1.2, and kification is not needed. There is then no ambiguity ab*
*out
what we mean by a Gspace, and we need not worry about refining the topology
on products with G.
Another reason for restricting to locally compact groups is that many useful
properties of proper actions only hold in that case. In the literature, such re*
*sults
are usually derived for actions on Hausdorff spaces, but we shall see that weak
Hausdorff generally suffices.
We begin with some standard equivariant terminology.
Definition 3.1.2. Let X be a Gspace and let H G.
(i)An Htube U in X is an open Ginvariant subset of X together with a Gmap
ss :U ! G=H. If x 2 U and H = Gx, then U is a tube around x. A tube is
contractible if ss is a Ghomotopy equivalence.
(ii)An Hslice S in X is an Hinvariant subset such that the canonical Gmap
G xH S ! GS X is an embedding onto an open subset. Then GS is
an Htube with S = ss1(eH). Conversely, if (U, ss) is an Htube in X, then
S = ss1(eH) is an Hslice and U = GS. On isotropy subgroups, we then
have Gy = Hy H for all y 2 S, but equality need not hold. If x 2 S and
H = Gx, then S is a slice through x.
(iii)We say that X has enough slices if every point x 2 X is contained in an
Hslice for some compact subgroup H. This implies that every point x has
compact isotropy group, but in general it does not imply that there must be
a slice through every point x.
(iv)A Gnumerable cover of X is a cover {Uj} by tubes such that there exists a
locally finite partition of unity by Gmaps ~j:X ! [0, 1] with support Uj.
The following is the equivariant generalization of [33, 6.7].
Proposition 3.1.3. Any GCW complex admits a Gnumerable cover by con
tractible tubes.
Proof. The proof given by Dold [33] in the nonequivariant case goes through
with only a minor change in the initial construction, which we sketch. From the*
*re,
the technical details are unchanged. Let Xn be the nth skeletal filtration of*
* a
GCW complex X. Let X`ndenote the subspace obtained by deleting the centers
G=H x 0 of all ncells in Xn and let rn :`Xn! Xn1 denote the obvious retract.
Starting from the interior en = G=H x (Dn  Sn1) of an ncell cn, define Vnm
inductivelySfor m n by setting Vnn= en and Vnm+1 = r1m+1(Vnm). Then the union
Vn1 = m n Vnm is a contractible tube, where the projection to G=H is induced
by the projection of en to G=H x 0.
We now give the definition of a proper group action in K . We shall see that
the definition could equivalently be made in U . For further details, but in T *
*op,
see for example [13, 32]. Recall that a continuous map is proper if it is a clo*
*sed
map with compact fibers.
Definition 3.1.4. A Gspace X in GK is proper (or Gproper) if the map
` :G x X ! X x X
specified by `(g, x) = (x, gx) is proper.
3.1. PROPER ACTIONS OF LOCALLY COMPACT GROUPS 35
We warn the reader that the definition is not quite the standard one. We are
working in the category K , and the product X x X on the right hand side is
the kspace obtained by kifying the standard product topology on X xc X. In
T op there are various other notions of a proper group action; see [4] for a ca*
*reful
discussion. They all agree for actions of locally compact groups on completely
regular spaces. If X is proper, then the isotropy groups Gx are compact since t*
*hey
are the fibers `1(x, x). Moreover, since points are closed subsets of G, the d*
*iagonal
X = `({e} x X) must be a closed subset of X x X and thus X must be weak
Hausdorff. This means that proper Gspaces must be in U . Since G is locally
compact, we have the following useful characterizations.
Proposition 3.1.5. For a Gspace X in GK the following are equivalent.
(i)The action of G on X is proper.
(ii)The isotropy groups Gx are compact and for all (x, y) 2 X x X and all neig*
*h
borhood U of `1(x, y) in GxX, there is a neighborhood V of (x, y) in X xX
such that `1(V ) U.
(iii)The isotropy groups Gx are compact and for all (x, y) 2 X x X and all neig*
*h
borhoods U of {g  gx = y} in G, there is a neighborhood V of (x, y) such
that
{g 2 G  ga = b for some (a, b)}2 VU.
(iv)The space X is weak Hausdorff and every point (x, y) 2 X x X has a neigh
borhood V such that
{g 2 G  ga = b for some (a, b)}2 V
has compact closure in G.
Proof. This holds by essentially the same proof as [4, 1.6(b)]. One must on*
*ly
keep in mind that we are now working in K rather than in T op and adjust the
argument accordingly.
Corollary 3.1.6. If G is discrete, then a Gspace X is proper if and only if
any point (x, y) 2 X x X has a neighborhood V such that
{g 2 G  ga = b for some (a, b)}2 V
is finite.
Corollary 3.1.7. If G is compact, then any Gspace in GU is proper.
Remark 3.1.8. There is an alternative description of the set displayed in
Proposition 3.1.5 that may clarify the characterization. Define
OE: G x X x X ! X x X
by OE(g, x, y) = (gx, y). For V X x X, let OEV be the restriction of OE to G *
*x V
and let ss :G x V ! G be the projection, which is an open map since G x V has
the product topology. Then the displayed set is ssOE1V( X ). If X x X = X xcX,
then the condition in Proposition 3.1.5 is equivalent to the more familiar one *
*that
any two points x and y in X have neighborhoods Vx and Vy such that
{g 2 G  gVx \ Vy 6= ;}
has compact closure in G.
Proposition 3.1.9. Proper actions satisfy the following closure properties.
36 3. PROPER ACTIONS, EQUIVARIANT BUNDLES, AND FIBRATIONS
(i)The restriction of a proper action to a closed subgroup is proper.
(ii)An invariant subspace of a proper Gspace is also proper.
(iii)Products of proper Gspaces are proper.
(iv)If X is a proper Hausdorff Gspace in GK and C is a compact Hausdorff
Gspace, then the Gspace Map (C, X) is proper.
(v)An Hspace S is Hproper if and only if G xH S is Gproper.
Proof. The first three are standard and elementary; see for example [32,
I.5.10]. The fifth is [4, 2.3]. We prove (iv). We must show that the map
` :G x Map(C, X) ! Map (C, X) x Map(C, X)
is proper, which amounts to showing that it is closed and that the isotropy gro*
*ups
Gf are compact for f 2 Map (C, X). For the latter, let {gi} be a net in Gf and
fix c 2 C. Note that f(gic) = gif(c). Since C is compact, we can assume by
passing to a subnet that {gic} converges to some ~c2 C. Let V be a neighborhood
of (f(c), f(~c)) such that
B = {g 2 G  ga = b for some (a, b)}2 V
has compact closure. Since C is compact, C xC xMap (C, X) has the usual product
topology. Since the map
C x C x Map(C, X) ! X x X
that sends (c, d, f) to (f(c), f(d)) is continuous and the net {c, gic, f} conv*
*erges to
(c, ~c, f), the net {(f(c), f(gic))} = {(f(c), gif(c))} must converge to (f(c),*
* f(~c)). It
follows that a subnet of {gi} lies in B and therefore has a converging subsubn*
*et.
To show that ` is closed, let A be a closed subset of G x Map (C, X) and let
{(fi, gifi)} be a net in `(A) that converges to (f, F ). We must show that (f, *
*F ) is
in `(A). For c 2 C, the net {g1ic} has a subnet that converges to some ~c, by *
*the
compactness of C, so we may as well assume that the original net converges to ~*
*c.
Let V be a neighborhood of (f(~c), F (c)) such that
B0= {g 2 G  ga = b for some (a, b)}2 V
has compact closure. By continuity and the compactness of C, there is a compact
neighborhood K1xK2 of (~c, c) that (f, F ) maps into V . Since {(fi, gifi)} con*
*verges
to (f, F ), there is an h such that (fi, gifi)(K1 x K2) V for i h. It follo*
*ws that
there is a k h such that (fi(g1ic), gifi(g1ic)) 2 V for all i k. Then t*
*he
subnet {gi}i k is contained in B0 and therefore has a subsubnet that converges
to some g 2 G. We have now seen that our original net {(gi, fi)} in A has a
subnet {(gij, fij)} that converges to (g, f), and (g, f) 2 A since A is closed.*
* By the
continuity of `, {`(gi, fi)} must converge to (f, F ) = `(g, f) 2 `(A). In this*
* last
statement, we are using the uniqueness of limits, which we ensure by requiring X
and C to be Hausdorff.
The following theorem of Palais [81], as generalized by Biller [4], is funda*
*men
tal. Those sources work in T op, but the arguments work just as well in U .
Theorem 3.1.10 (Palais). Let X be a Gspace in GU .
(i)If X has enough slices, then it is proper.
(ii)Conversely, if X is completely regular and proper, then it has enough slic*
*es.
(iii)If G is a Lie group and X is completely regular and proper, then there is a
slice through each point of X.
3.3. SPACES OF THE HOMOTOPY TYPES OF GCW COMPLEXES 37
Proof. Part (i) is given by [4, 2.4]. Part (iii) is given by [81, 2.3.3]. P*
*art (ii)
is deduced from part (iii) in [4, 2.5].
3.2. Proper actions and equivariant bundles
We introduce here the equivariant bundles to which we will apply our basic
foundational results in Part IV. As we explain, Theorem 3.1.10 allows us to gen*
*er
alize some basic results about such bundles from actions of compact Lie groups *
*to
proper actions of Lie groups.
Let be a normal subgroup of a Lie group such that = = G and let
q : ! G be the quotient homomorphism. By a principal ( ; )bundle we mean
the quotient map p: P  ! P= where P is a free space such that acts
properly on P . It follows that the induced Gaction on B = P= is proper. If F
is a space, then we have the associated Gmap E = P x F ! P x * ~=P= ,
which we say is a bundle with structure group and fiber F . For compact Lie
groups, bundles of this general form are studied in [55], which generalizes the*
* study
of the classical case = G x given in [54]. A summary and further references
are given in [68, Chapter VII]. We recall an observation about such bundles.
Lemma 3.2.1. For b 2 B, the action of on F induces an action of the isotro*
*py
group Gb on the fiber Eb through a homomorphism aeb:Gb ! such that q O aeb is
the inclusion Gb ! G and Eb ~=ae*bF .
Proof. Choose z 2 P such that ss(z) = b. The isotropy group z intersects
in the trivial group, and q maps z isomorphically onto Gb. Let aeb be the comp*
*osite
of q1 :Gb ! z and the inclusion z ! . Since the subspace {z}xF of P xF
is zinvariant and maps homeomorphically onto Eb on passage to orbits over ,
the conclusion follows. Note that changing the choice of z changes aebby conjug*
*ation
by an element of and changes the identification of Eb with F correspondingly.
Bundles should be locally trivial. When P is completely regular, local trivi*
*ality
is a consequence of Theorem 3.1.10(iii), just as in the case when is a compact
Lie group [55, Lemma 3], and this justifies our bundletheoretic terminology. N*
*ote
that if P is completely regular, then so is B = P= .
Lemma 3.2.2. A completely regular principal ( ; )bundle P is locally trivi*
*al.
That is, for each b 2 B, there is a slice Sb through b and a homeomorphism
~=
x Sb _____//_p1(GSb)
qx1  p
fflffl~= fflffl
G xGb Sb_______//GSb
where only intersects in the identity element and is mapped isomorphica*
*lly
to Gb by q. The action on Sb is given by pulling back the Gbaction along q.
3.3.Spaces of the homotopy types of GCW complexes
In this section, we recall and generalize the equivariant version of Milnor'*
*s re
sults [76] about spaces of the homotopy types of CW complexes. For compact Lie
groups, Waner formulated and proved such results in [95, x4]. With a few obser
vations, his proofs generalize to deal with proper actions by general Lie group*
*s.
38 3. PROPER ACTIONS, EQUIVARIANT BUNDLES, AND FIBRATIONS
We first note the following immediate consequence of Proposition 3.1.3 and Theo
rem 3.1.10.
Theorem 3.3.1. For any locally compact group G, a GCW complex is proper
if and only if it is constructed from cells of the form G=KxDn, where K is comp*
*act.
We also note the following recent "triangulation theorem" of Illman [48, The
orem II]. It is this result that led us to try to generalize some of our result*
*s from
compact Lie groups to general Lie groups.
Theorem 3.3.2 (Illman). If G is a Lie group that acts smoothly and properly
on a smooth manifold M, then M has a GCW structure.
Many of our applications of this result are based on the following observati*
*on.
Lemma 3.3.3. If H and K are closed subgroups of a topological group G and
K is compact, then the diagonal action of G on G=H x G=K is proper.
Proof. The proof given in [32, I.5.16] that G acts properly on G=K generali*
*zes
directly. Set X = G=H x G. Let G act diagonally from the left and let K act on
the second factor from the right. Note that these actions commute. It suffices *
*to
show that ` :GxX ! X xX is proper. Indeed, consider the commutative square
G x X ____`____//X x X
 
 
fflffl~` fflffl
G x X=K ____//_X=K x X=K.
The right vertical map is proper and the left vertical map is surjective. There*
*fore,
by [32, VI.2.13], the bottom horizontal map is proper if the top horizontal map*
* is
proper. Since X is a free Gspace, ` is proper if and only if the image Im(`) i*
*s a
closed subspace of X x X and the map OE: Im(`) ! G specified by OE(x, gx) = g
is continuous. The diagonal subspace of G=H x G=H is closed, and its preimage
under the map i :X x X ! G=H x G=H specified by
i((xH, y), (~xH, ~y)) = (~yy1xH, ~xH)
is precisely Im(`), which is therefore closed. The function OE is the restrict*
*ion to
Im(`) of the continuous map : X x X ! G specified by
((xH, y), (~xH, ~y)) = ~yy1
and is therefore continuous.
We shall also make essential use of the following corollary of Theorem 3.3.2.
Corollary 3.3.4. If X is a proper GCW complex, then, viewed as an Hspace
for any closed subgroup H of G, X has the structure of an Hcell complex.
Proof. Each cell G=K x Dn has K compact. Since G acts smoothly and
properly on the smooth manifold G=K, the closed subgroup H also acts smoothly
and properly. We use the resulting HCW structure on all of the cells to obtain
an Hcell structure. It is homotopy equivalent to an HCW complex obtained by
"sliding down" cells that are attached to higher dimensional ones, but we shall*
* not
need to use that.
3.4. SOME CLASSICAL THEOREMS ABOUT FIBRATIONS 39
Theorem 3.3.5 (Milnor, Waner). Let G be a Lie group and (X; Xi) be an nad
of closed subGspaces of a proper Gspace X. If (X; Xi) has the homotopy type *
*of
a GCW nad and (C; Ci) is an nad of compact Gspaces, then (X; Xi)(C;Ci)has
the homotopy type of a GCW nad.
Proof. We only remark how the proof of Waner for the case of actions by a
compact Lie group generalizes to the case of proper actions by a Lie group. Def*
*ine
a Gsimplicial complex to be a GCW complex such that X=G with the induced cell
structure is a simplicial complex. In [95, x5], Waner proves that any GCW comp*
*lex
is Ghomotopy equivalent to a colimit of finite dimensional Gsimplicial comple*
*xes
and cellular inclusions and that a Gspace dominated by a GCW complex is G
homotopy equivalent to a GCW complex. The arguments apply verbatim to any
topological group G.
The rest of the argument requires two key lemmas. In [95, 4.2], Waner defines
the notion of a Gequilocally convex, or GELC, Gspace. The first lemma says
that every finite dimensional Gsimplicial complex is GELC. The essential star*
*ting
point is that orbits are GELC, the proof of which uses the Lie group structure
just as in [95, p.358] in the compact case. From there, Waner's proof [95, x6] *
*goes
through unchanged. The second says that any completely regular, Gparacompact,
GELC, proper Gspace is dominated by a GCW complex. When G is compact Lie,
this is proven in [95, x7]. However, the hypothesis on G is only used to guaran*
*tee
the existence of enough slices, hence the proof holds without change for proper
actions of Lie groups, indeed of locally compact groups.
The rest of the proof goes as in [76, Theorem 3]. One only needs to make two
small additional observations. First, if a Gsimplicial complex K has the homot*
*opy
type of a proper Gspace X, then it is proper. This holds since if f :K ! X is*
* a
homotopy equivalence, then Gk Gf(k)is compact. Second, for an nad (K; Ki)
of Gsimplicial complexes and a compact nad (C; Ci), (X; Xi)(C;Ci)is proper si*
*nce
it is a subspace of the proper Gspace XC ; see (i) and (iv) of Proposition 3.1*
*.9.
Since it is also completely regular, Gparacompact, and GELC, it is dominated *
*by
a GCW complex, and the result follows from the steps above.
3.4.Some classical theorems about fibrations
A basic principle of parametrized homotopy theory is that homotopical infor
mation is given on fibers. We recall two relevant classical theorems about Hure*
*wicz
fibrations and a comparison theorem relating Serre and Hurewicz fibrations. We
begin with Dold's theorem [33, 6.3]. The nonequivariant proof in [64, 2.6] is g*
*ener
alized to the equivariant case in Waner [96, 1.11]. Waner assumes throughout [9*
*6]
that G is a compact Lie group, but that assumption is not used in the cited pro*
*of.
Theorem 3.4.1 (Dold). Let G be any topological group and let B be a Gspace
that has a Gnumerable cover by contractible tubes. Let X ! B and Y ! B be
Hurewicz fibrations. Then a map X ! Y over B is a fiberwise Ghomotopy equiv
alence if and only if each fiber restriction Xb ! Yb is a Gbhomotopy equivale*
*nce.
We next recall and generalize a classical result that relates the homotopy t*
*ypes
of fibers to the homotopy types of total spaces. Nonequivariantly, it is due to
Stasheff [89] and, with a much simpler proof, Sch"on [84]. The generalization *
*to
the equivariant case, for compact Lie groups, is given by Waner [96, 6.1]. With
Theorems 3.4.1, 3.3.5 and 3.3.2 in place, Sch"on's argument generalizes directl*
*y to
40 3. PROPER ACTIONS, EQUIVARIANT BUNDLES, AND FIBRATIONS
give the following version. Since the result plays an important role in our wor*
*k and
the argument is so pretty, we can't resist repeating it in full.
Theorem 3.4.2 (Stasheff, Sch"on). Let G be a Lie group and B be a proper
Gspace that has the homotopy type of a GCW complex. Let p: X ! B be a
Hurewicz fibration. Then X has the homotopy type of a GCW complex if and only
if each fiber Xb has the homotopy type of a GbCW complex.
Proof. First assume that X has the homotopy type of a GCW complex. For
b 2 B, let ': Gb ! G be the inclusion and consider the Gbmap '*p: '*X ! '*B
of Gbspaces. It is still a Hurewicz fibration, as we see by using the left ad*
*joint
G xGb () of '*. By Corollary 3.3.4, '*X and '*B have the homotopy types of
GbCW complexes. Factor '*p through the inclusion into its mapping cylinder
i: '*X ! M'*p. Since Gb is compact, it follows from Theorem 3.3.5 that the
homotopy fiber Fbi = (M'*p; {b}, '*X)(I;0,1)has the homotopy type of a GbCW
complex. Since Fbi is homotopy equivalent to Fb'*p, by the gluing lemma, and
Fb'*p is homotopy equivalent to the fiber Xb, this proves the forward implicati*
*on.
For the converse, assume that each fiber Xb has the homotopy type of a Gb
CW complex. Let fl : X ! X be a GCW approximation of X. The mapping
path fibration of fl gives us a factorization of fl as the composite of a Ghom*
*otopy
equivalence : X ! Nfl and a Hurewicz fibration q :Nfl ! X. We may view
q as a map of fibrations over B.
Nfl______q_____//BX
BBB """"
pOqBBB__B""p""""
B
The fibers of p O q have the homotopy types of GbCW complexes by the first part
of the proof, since X is a GCW complex, and the fibers of p have the homotopy
types of GbCW complexes by hypothesis. Comparison of the long exact sequences
associated to p O q and p gives that q restricts to a Gbhomotopy equivalence on
each fiber. Noting that we can pull back a numerable cover by contractible tubes
along a homotopy equivalence B ! B0, where B0 is a GCW complex, it follows
from Theorem 3.4.1 that q is a homotopy equivalence.
Although it no longer plays a role in our theory, the following little known
result played a central role in our thinking. It shows that the dichotomy betwe*
*en
Serre and Hurewicz fibrations diminishes greatly over CW base spaces. It is due*
* to
Steinberger and West [90], with a correction by Cauty [16].
Theorem 3.4.3 (Steinberger and West; Cauty). A Serre fibration whose base
and total spaces are CW complexes is a Hurewicz fibration.
We believe that this remains true equivariantly for compact Lie groups, and *
*it
certainly remains true for finite groups. Before we understood the limitations*
* of
the qmodel structure, we planned to use this result to relate our model theore*
*tic
homotopy category of exspaces over a CW complex B to a classical homotopy
category defined in terms of Hurewicz fibrations and thereby overcome the probl*
*ems
illustrated in Counterexample 0.0.1. Such a comparison is still central to our *
*theory,
and it is this result that convinced us that such a comparison must hold.
3.5. QUASIFIBRATIONS 41
3.5. Quasifibrations
For later reference, we recall the definition of quasifibrations. Here G can*
* be
any topological group.
Definition 3.5.1. A map p: E ! Y in K is a quasifibration if the map of
pairs p: (E, Ey) ! (Y, y) is a weak equivalence for all y in Y . A map p: E !*
* Y
in K =B or KB is a quasifibration if it is a quasifibration on total spaces. A *
*Gmap
p: E ! Y is a quasifibration if each of its fixed point maps pH :EH  ! Y H is*
* a
nonequivariant quasifibration.
The condition that p: (E, Ey) ! (Y, y) is a weak equivalence means that for
all e 2 Ey the following two conditions hold.
(i)p*: ssn(E, Ey, e) ! ssn(Y, y) is an isomorphism for all n 1.
(ii)For any x 2 E, p(x) is in the path component of y precisely when the path
component of x in E intersects Ey. In other words, the sequence
ss0(Ey, e) ! ss0(E, e) ! ss0(Y, y)
of pointed sets is exact.
Warning 3.5.2. In contrast to the usual treatments in the literature, we do *
*not
require p to be surjective and therefore ss0(E, e) ! ss0(Y, y) need not be sur*
*jective.
Hurewicz and, more generally, Serre fibrations are examples of quasifibrations,*
* and
they are not always surjective, as the trivial example {0} ! {0, 1} illustrate*
*s.
Model categorically, one point is that the initial map ; ! Y is always a Serre
fibration since the empty lifting problem always has a solution.
The definition of a quasifibration is arranged so that the long exact sequen*
*ce
of homotopy groups associated to the triple (E, Ey, e) is isomorphic to a long *
*exact
sequence
. ..! ssn+1(Y, y) ! ssn(Ey, e) ! ssn(E, e) ! ssn(Y, y) ! . ..! ss0(Y, y).
Part II
Model categories and parametrized
spaces
Introduction
In Part III, we shall develop foundations for parametrized equivariant stable
homotopy theory. In making that theory rigorous, it became apparent to us that
substantial foundational work was already needed on the level of exspaces. That
work is of considerable interest for its own sake, and it involves general poin*
*ts about
the use of model categories that should be of independent interest. Therefore, *
*rather
than rush through the space level theory as just a precursor of the spectrum le*
*vel
theory, we have separated it out in this more leisurely and discursive expositi*
*on.
In Chapter 4, which is entirely independent of our parametrized theory, we g*
*ive
general model theoretic background, philosophy, and results. In contrast to the
simplicial world, we often have both a classical htype and a derived qtype mo*
*del
structure in topologically enriched categories, with respective weak equivalenc*
*es the
homotopy equivalences and the weak homotopy equivalences. We describe what is
involved in verifying the model axioms for these two types of model structures.
In Chapter 5, we describe how the parametrized world fits into this general
framework. There are several different htype model structures on our categories
of parametrized Gspaces, with different homotopy equivalences based on differe*
*nt
choices of cylinders. These mesh in unexpected ways. Understanding of this part*
*ic
ular case leads us to a conceptual axiomatic description of how the classical h*
*type
homotopy theory and the qtype model structure must be related in order to be
able to do homotopy theory satisfactorily in a topologically enriched category.
In Chapter 6, we work nonequivariantly and develop our preferred "qtype"
model category structure, the "qfmodel structure", on the categories K =B and
KB . This chapter is taken directly from the second author's thesis [88].
In Chapter 7, we give the equivariant generalization of the qfmodel struc
ture and begin the study of the resulting homotopy categories by discussing tho*
*se
adjunctions that are given by Quillen pairs. There is another new twist here in
that we need to use many Quillen equivalent qftype model structures. In fact,
this is already needed nonequivariantly in the study of base change along bundl*
*es
f :A ! B.
In Chapter 8, we discuss exfibrations and an exfibrant approximation func
tor that better serves our purposes than model theoretic fibrant approximation *
*in
studying those adjunctions that are not given by Quillen pairs. In Chapter 9, we
describe our parametrized homotopy categories in terms of classical homotopy ca*
*t
egories of exfibrations and use this description to resolve the issues concern*
*ing base
change functors and smash products that are discussed in the Prologue.
44
CHAPTER 4
Topologically bicomplete model categories
Introduction
In x4.1, we describe a general philosophy about the role of different model
structures on a given category C . It is natural and important in many contexts,
and it helps to clarify our thinking about topological categories of parametriz*
*ed
objects. In particular, we advertise a remarkable unpublished insight of Mike C*
*ole.
It is a pleasure to thank him for keeping us informed of his ideas. We describe*
* how
a classical "htype" model structure and a suitably related Quillen "qtype" mo*
*del
structure, can be mixed together to give an "mtype" model structure such that
the mequivalences are the qequivalences and the mfibrations are the hfibrat*
*ions.
This is a completely general phenomenon, not restricted to topological contexts.
In xx4.2 and 4.3, we describe classical structure that is present in any top*
*o
logically bicomplete category C . Here we follow up a very illuminating paper *
*of
Schw"anzl and Vogt [85]. There are two classes of (Hurewicz) hfibrations and t*
*wo
classes of hcofibrations, ordinary and strong. Taking weak equivalences to be *
*ho
motopy equivalences, the ordinary hfibrations pair with the strong hcofibrati*
*ons
and the strong hfibrations pair with the ordinary hcofibrations to give two i*
*n
terrelated model like structures. For each choice, all of the axioms for a pro*
*per
topological model category are satisfied except for the factorization axioms, w*
*hich
hold in a weakened form. To prove that C is a model category, it suffices to pr*
*ove
one of the factorization axioms since the other will follow. Again, the theory *
*can
easily be adapted to other contexts than our topological one.
We signal an ambiguity of nomenclature. In the model category literature, the
term "simplicial model structure" is clear and unambiguous, since there is only
one model structure on simplicial sets in common use. In the topological contex*
*t,
we understand "topological model structures" to refer implicitly to the hmodel
structure on spaces for model structures of htype and to the qmodel structure*
* on
spaces for model structures of qtype. The meaning should always be clear from
context.
In x4.4, we give another insight of Cole, which gains power from the work
of Schw"anzl and Vogt. Cole provides a simple hypothesis that implies the miss
ing factorization axioms for an hmodel structure of either type on a topologic*
*ally
bicomplete category C . When we restrict to compactly generated spaces, the hy
pothesis applies to give an hmodel structure on U . In K , this seems to fail,*
* and
we give a streamlined version of Strom's original proof [93], together with his*
* proof
that the strong hcofibrations in K are just the closed ordinary hcofibration*
*s.
This works in exactly the same way for the categories GK and GU , where G is
any (compactly generated) topological group.
45
46 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES
In x4.5, we describe how to construct compactly generated qtype model struc
tures, giving a slight variant of standard treatments. In particular, GK and GU
have the usual qmodel structures in which the qequivalences are the weak equi*
*va
lences and the qfibrations are the Serre fibrations. Again, G can be any topol*
*ogical
group. However we only know that the model structure is Gtopological when G is
a compact Lie group.
4.1. Model theoretic philosophy: h, q, and mmodel structures
The point of model categories is to systematize "homotopy theory". The ho
motopy theory present in many categories of interest comes in two flavors. Ther*
*e is
a "classical" homotopy theory based on homotopy equivalences, and there is a mo*
*re
fundamental "derived" homotopy theory based on a weaker notion of equivalence
than that of homotopy equivalence. This dichotomy pervades the applications, re
gardless of field. It is perhaps well understood that both homotopy theories ca*
*n be
expressed in terms of model structures on the underlying category, but this asp*
*ect
of the classical homotopy theory has usually been ignored in the model theoreti*
*cal
literature, a tradition that goes back to Quillen's original paper [83]. The "c*
*lassi
cal" model structure on spaces was introduced by Strom [93], well after Quillen*
*'s
paper, and the "classical" model structure on chain complexes was only introduc*
*ed
explicitly quite recently, by Cole [21] and Schw"anzl and Vogt [85].
Perhaps for this historical reason, it may not be widely understood that the*
*se
two model structures can profitably be used in tandem, with the hmodel structu*
*re
used as a tool for proving things about the qmodel structure. This point of vi*
*ew is
implicit in [39, 61, 62], and a variant of this point of view will be essential*
* to our
work. In the cited papers, the terms "qfibration" and "qcofibration" were used
for the fibrations and cofibrations in the Quillen model structures, and the te*
*rm
"hcofibration" was used for the classical notion of a Hurewicz cofibration spe*
*cified
in terms of the homotopy extension property (HEP). The corresponding notion
of an "hfibration" defined in terms of the covering homotopy property (CHP) is
fortuitously appropriate1. Just as the "q" is meant to suggest Quillen, the "h"
is meant to suggest Hurewicz, as well as homotopy. It is logical to follow this
idea further (as was not done in [39, 61, 62]) by writing qfibrant, qcofibran*
*t,
hfibrant, and hcofibrant for clarity. Following this still further, we shoul*
*d also
write "hequivalence" for homotopy equivalence and "qequivalence" for (Quillen)
weak equivalence. The relations among these notions are as follows in all of t*
*he
relevant categories C :
________________________________
 hequivalence=) qequivalence 
 hcofibration(= qcofibration 
 hcofibrant(= qcofibrant 
 hfibration=) qfibration 
_____hfibrant=)__qfibrant____ 
Therefore, the identity functor is the right adjoint of a Quillen adjoint pa*
*ir
from C with its hmodel structure to C with its qmodel structure. It follows t*
*hat
we have an adjoint pair relating the classical homotopy category, hC say, to the
____________
1However, the notation conflicts with the notation often used for Dold's no*
*tion of a weak or
"halb"fibration. We shall make no use of that notion, despite its real importa*
*nce in the theory
of fibrations. We do not know whether or not it has a model theoretic role to p*
*lay.
4.2. STRONG HUREWICZ COFIBRATIONS AND FIBRATIONS 47
derived homotopy category qC = Ho C . This formulation packages standard in
formation. For example, the Whitehead theorem that a weak equivalence between
cell complexes is a homotopy equivalence, or its analogue that a quasiisomorph*
*ism
between projective complexes is a homotopy equivalence, is a formal consequence
of this adjunction between homotopy categories.
Recently, Cole [23] discovered a profound new way of thinking about the di
chotomy between the kinds of model structures that we have been discussing. He
proved the following formal model theoretic result.
Theorem 4.1.1 (Cole). Let (Wh, Fibh, Cofh) and (Wq, Fibq, Cofq) be two mo
del structures on the same category C . Suppose that Wh Wq and Fibh Fibq.
Then there is a mixed model structure (Wq, Fibh, Cofm ) on C . The mixed cofi
brations Cofm are the maps in Cofh that factor as the composite of a map in Wh
and a map in Cofq. An object is mcofibrant if and only it is hcofibrant and o*
*f the
hhomotopy type of a qcofibrant object. If the h and qmodel structures are le*
*ft or
right proper, then so is the mmodel structure.
By duality, the analogue with the inclusion Fibh Fibq replaced by an inclu
sion Cofh Cofq also holds. In the category of spaces with the h and qmodel
structures discussed above, the theorem gives a mixed model structure whose m
cofibrant spaces are the spaces of the homotopy types of CWcomplexes. This
mmodel structure combines weak equivalences with Hurewicz fibrations, and it
might conceivably turn out to be as important and convenient as the Quillen mod*
*el
structure. It is startling that this model structure was not discovered earlier.
The pragmatic point is twofold. On the onehand, there are many basic resul*
*ts
that apply to hcofibrations and not just qcofibrations. Use of hcofibrations*
* limits
the need for qcofibrant approximation and often clarifies proofs by focusing a*
*tten
tion on what is relevant. Many examples appear in [39, 62, 61], where properties
of hcofibrations serve as scaffolding in the proof that qmodel structures are*
* in fact
model structures. We shall formalize and generalize this idea in the next chapt*
*er.
On the other hand, there are many vital results that apply only to hfibrati*
*ons
(Hurewicz fibrations), not to qfibrations (Serre fibrations). For example, a l*
*ocal
Hurewicz fibration is a Hurewicz fibration, but that is not true for Serre fibr*
*ations.
The mixed model structure provides a natural framework in which to make use of
Hurewicz fibrations in conjunction with weak equivalences. While we shall make *
*no
formal use of this model structure, it has provided a helpful guide to our thin*
*king.
The philosophy here applies in algebraic as well as topological contexts, but we
shall focus on the latter.
4.2. Strong Hurewicz cofibrations and fibrations
Fix a topologically bicomplete category C throughout this section and the ne*
*xt.
With no further hypotheses on C , we show that it satisfies most of the axioms *
*for
not one but two generally different proper topological htype model structures.*
* We
alert the reader to the fact that we are here using the term "hmodel structure"
in a generic sense. When we restrict attention to parametrized spaces, we will
use the term in a different specific sense derived from the hmodel structure on
underlying total spaces. The material of these sections follows and extends mat*
*erial
in Schw"anzl and Vogt [85].
We have cylinders X x I and cocylinders Map (I, X). When C is based, we
focus on the based cylinders X ^ I+ and cocylinders F (I+ , X). In either case,*
* these
48 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES
define equivalent notions of homotopy, which we shall sometimes call hhomotopy.
We will later use these and cognate notations, but, for the moment, it is conve*
*nient
to introduce the common notations Cyl(X) and Cocyl(X) for these objects. There
are obvious classes of maps that one might hope would specify a model structure.
Definition 4.2.1. Let f be a map in C .
(i)f is an hequivalence if it is a homotopy equivalence in C .
(ii)f is a Hurewicz fibration, abbreviated hfibration, if it satisfies the CH*
*P in
C , that is, if it has the right lifting property (RLP) with respect to the*
* maps
i0 : X ! Cyl(X) for X 2 C .
(iii)f is a Hurewicz cofibration, abbreviated hcofibration, if it satisfies th*
*e HEP
in C , that is, if it has the left lifting property (LLP) with respect to t*
*he maps
p0: Cocyl(X) ! X.
These sometimes do give a model structure, but then the hcofibrations must
be exactly the maps that satisfy the LLP with respect to the hacyclic hfibrat*
*ions,
and dually. In general, that does not hold. We shall characterize the maps in C
that do satisfy the LLP with respect to the hacyclic hfibrations and, dually,*
* the
maps that satify the RLP with respect to the hacyclic hfibrations. For this, *
*we
need the following relative version of the above notions.
Definition 4.2.2. We define strong Hurewicz fibrations and cofibrations.
(i)A map p: E ! Y is a strong Hurewicz fibration, abbreviated ~hfibration, if
it satisfies the relative CHP with respect to all hcofibrations i : A ! X*
*, in
the sense that a lift exists in any diagram
A ____i____//_X______//E44jjjjj;;
jjjj ww
i0 jjjj ww p
fflffljjjjjfflfflw fflffl
Cyl(A)_____//Cyl(X)____//Y.
(ii)A map i : A ! X is a strong Hurewicz cofibration, abbreviated ~hcofibrati*
*on,
if it satisfies the relative HEP with respect to all hfibrations p : E ! Y*
* , in
the sense that a lift exists in any diagram
A _____//Cocyl(E)____//Cocyl(Y:):44
u iiiiii
i uuiii iiii p0
fflffluuiifflffliiiii fflffl
X ________//E_____p_____//Y.
We recall the standard criteria for maps to be hfibrations or hcofibration*
*s.
Define the mapping cylinder Mf and mapping path fibration Nf by the usual
pushout and pullback diagrams
f
X _______//_Y and Nf _____//Cocyl(Y )
i0   p0
fflffl fflffl fflffl fflffl
Cyl(X) ____//_Mf X ____f____//Y.
Lemma 4.2.3. Let f be a map in C .
4.2. STRONG HUREWICZ COFIBRATIONS AND FIBRATIONS 49
(i)f is an hfibration if and only if it has the RLP with respect to the map
i0 : Nf ! Cyl(Nf).
(ii)f is an hcofibration if and only if it has the LLP with respect to the map
p0 : Cocyl(Mf) ! Mf.
The ~hfibrations and ~hcofibrations admit similar characterizations. These*
* were
taken as definitions in [85, 2.4].
Lemma 4.2.4. (i) A map p: E ! Y is an ~hfibration if and only if it has
the RLP with respect to the canonical map Mi ! Cyl(X) for any hcofibration
i : A ! X; this holds if and only if the canonical map Cocyl(E) ! Np has the RLP
with respect to all hcofibrations.
(ii) A map i : A ! X is an ~hcofibration if and only if it has the LLP with re*
*spect
to the canonical map Cocyl(E) ! Np for any hfibration p : E ! Y ; this holds
if and only if the canonical map Mi ! Cyl(X) has the LLP with respect to all
hfibrations.
Observe that the map i0 : X ! Cyl(X) is an ~hcofibration and the map p0 :
Cocyl(X) ! X is an ~hfibration. Since the cylinder objects associated to init*
*ial
objects are initial objects, ~hfibrations are in particular hfibrations. Simi*
*larly, ~h
cofibrations are hcofibrations. Observe too that every object is both ~hcofib*
*rant
and ~hfibrant, hence both hcofibrant and hfibrant.
We shall see in x4.4 that these distinctions are necessary in K but disappe*
*ar
in U , where the h and ~hnotions coincide. Even there, however, the conceptual
distinction sheds light on classical arguments.
The results of this section and the next are quite formal. Amusingly, the ma*
*in
nonformal ingredient is just the use in the following proof of the standard fa*
*ct that
{0, 1} ! I has the LLP with respect to all hacyclic hfibrations.
Lemma 4.2.5. Let i: A ! X and p: E ! B be maps in C .
(i)If i is an hacyclic hcofibration, then i is the inclusion of a strong def*
*ormation
retraction r :X ! A.
(ii)If i is the inclusion of a strong deformation retraction r : X ! A, then i*
* is a
retract of Mi ! Cyl(X).
(iii)If p is an hacyclic hfibration, then p is a strong deformation retractio*
*n.
(iv)If p is a strong deformation retraction, then p is a retract of Cocyl(E) !*
* Np.
Proof. The last two statements are dual to the first two. For (i), since the
hequivalence i is an hcofibration, application of the HEP shows that i has a
homotopy inverse r : X ! A such that ri = idA. Since {0, 1} ! I has the LLP
with respect to hacyclic hfibrations, an adjunction argument shows that p(0,1*
*)has
the RLP with respect to hcofibrations. Thus a lift exists in the diagram on the
left, which means that r is a strong deformation retraction with inclusion i.
A ____i____//_XGG
Cocyl(i)   GGrGG
A __c__//Cocyl(A)___//_Cocyl(X)44i i0 i0 GGG
i i i1 fflfflpr  G##
i  i i i p(0,1)A _____//Cyl(A)J_______________//A
fflffliiii fi fflffl  JJJJ  
X ________(iOr,idX)__//_X x X i Cyl(i)JJJ  i
fflffl J%%fflffl fflffl
X _________i1_____//_Cyl(X)_fi//_X
50 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES
For (ii), we are given fi in the diagram on the left displaying r as a strong d*
*e
formation retraction with inclusion i. Then the diagram on the right commutes,
where the composites displayed in the lower two rows are identity maps. Using t*
*he
universal property of Mi to factor the crossing arrows i0 and pr through Mi, we
see that i is a retract of the canonical map Mi ! Cyl(X).
4.3. Towards classical model structures in topological categories
We now have two candidates for a classical model structure on C based on
the hequivalences. We can either take the hfibrations and the ~hcofibrations*
* or
the hcofibrations and the ~hfibrations. The following result shows that all o*
*f the
axioms for a proper topological model category are satisfied except that, in ge*
*neral,
only a weakened form of the factorization axioms holds.
Theorem 4.3.1. The following versions of the axioms for a proper topological
model category hold.
(i)The classes of hcofibrations, ~hcofibrations, hfibrations and ~hfibrati*
*ons are
closed under retracts.
(ii)Let i be an hcofibration and p be an hfibration. The pair (i, p) has the*
* lifting
property if i is strong and p is hacyclic or if p is strong and i is hacy*
*clic.
(iii)Any map f : X ! Y factors as
X __i_//_Mf__r_//_Y
where i is an ~hcofibration and r has a section that is an hacyclic ~hco*
*fibration
and as
p
X __s__//Nf____//_Y
where p is an ~hfibration and s has a retraction that is an hacyclic ~hf*
*ibration.
(iv)Let i : A ! X be an hcofibration and p : E ! B be an hfibration, where i
or p is strong. Then the map
C (i, p) : C (X, E) ! C (A, E) xC(A,B)C (X, B)
induced by i and p is an hfibration of spaces. It is hacyclic if i or p i*
*s acyclic
and it is an ~hfibration if both i and p are strong.
(v)The hequivalences are preserved under pushouts along hcofibrations and pu*
*ll
backs along hfibrations.
Proof. Part (i) is clear since all classes are defined in terms of lifting *
*prop
erties. Part (ii) follows directly from Lemma 4.2.4 and Lemma 4.2.5. The factor
izations of part (iii) are the standard ones. We consider the first. The evid*
*ent
section j :Y  ! Mf is an hacyclic ~hcofibration since it is the pushout of o*
*ne.
Consider the lifting problem in the left diagram below, in which the middle ver*
*tical
composite is i. Here p is an hacyclic hfibration, and we choose a section s o*
*f p.
X FF
FF
i1 FffFFF ___sOfiOjOfqff//_55
i0 fflfflF##F~0 X q X 0 l l lE
X ____//_Cyl(X)```//E55llll<
 >>
i0 jn>>>
fflffl~n fflff>>>qnl
Cyl(Nqn) ____//_VZn+1q >>
VVVVV OO n+1>>
VVVVVV OO >>
VVVVV O >>O
VVVVVOEOE>++V''O
Y
The map Cyl(Nqn) ! Y is the adjoint of the projection Nqn ! Cocyl(Y ) given
by the definition of Nqn, and qn+1 is the induced map. The maps jn are hacyclic
~hcofibrations since they are pushouts of such maps. Let Z be the colimit of t*
*he
Zn and j and q be the colimits of the jn and qn. Certainly f = q O j and j is a*
*n h
acyclic ~hcofibration. By Hypothesis 4.4.1, Nq is the colimit of the Nqn. Sinc*
*e the
cylinder functor preserves colimits, we see by Lemma 4.2.3 that q is an hfibra*
*tion
since the ~n give a lift Cyl(Nq) ! Z by passage to colimits.
The dual version of Theorem 4.4.2 admits a dual proof.
54 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES
Theorem 4.4.3 (Cole). If C is a topologically bicomplete category which sat
isfies the dual of Hypothesis 4.4.1, then the hequivalences, ~hfibrations, an*
*d h
cofibrations specify a proper topological hmodel structure on C .
From now on, we break the symmetry by focusing on hfibrations and ~h
cofibrations. These give model structures in K and U . Everything in the rest
of the section works equally in GK and GU . The following theorem combines
several results of Strom [91, 92, 93].
Theorem 4.4.4 (Strom). The following statements hold.
(i)The hequivalences, hfibrations, and ~hcofibrations give K a proper topo*
*log
ical hmodel structure. Moreover, a map in K is an ~hcofibration if and on*
*ly
if it is a closed hcofibration.
(ii)The hequivalences, hfibrations, and ~hcofibrations give U a proper topo*
*logi
cal hmodel structure. Moreover, a map in U is an ~hcofibration if and only
if it is an hcofibration.
Proof. Theorem 4.4.2 applies to prove the first statement in (ii), but it d*
*oes
not seem to apply to prove the first statement in (i). The reasons are explaine*
*d in
Remark 1.1.4. Taking Z = Y Iand p = p0 there, the comparison map ff specializes
to the map colimNfn ! Nf of Hypothesis 4.4.1. It may be that ff is a homeo
morphism in this special case, but we do not have a proof. It is a homeomorphism
when we work in U . The characterization of the ~hcofibrations in U follows fr*
*om
Lemma 1.1.3 and their characterization in K .
For (i), we give a streamlined version of Strom's original arguments that us*
*es
the material of the previous section to prove both statements together. We proc*
*eed
in four steps. The first step is Strom's key observation, the second and third *
*steps
give the second statement, and the fourth step proves the needed factorization
axiom. Consider an inclusion i: A ! X.
Step 1. By Strom's [91, Thm. 3], if i is the inclusion of a strong deformati*
*on
retract and there is a map _ :X ! I such that _1(0) = A, then i has the LLP
with respect to all hfibrations. By Proposition 4.3.3(i), this means that i i*
*s an
hacyclic ~hcofibration.
Step 2. If i is an hcofibration, then the canonical map j :Mi ! X x I is
an hacyclic hcofibration and therefore, by Lemma 4.2.5, the inclusion of a st*
*rong
deformation retract. If i is closed, then (X, A) is an NDRpair and there exis*
*ts
OE: X ! I such that OE1(0) = A. Define _ :X x I ! I by _(x, t) = tOE(x). Then
_1(0) = Mi. Applying Step 1, we conclude that j has the LLP with respect to
all hfibrations. By Lemma 4.2.4, this means that i is an ~hcofibration.
Step 3. We can factor any inclusion i as the composite
A _i0_//_Ess_//X,
where E is the subspace X x (0, 1] [ A x I of X x I and ss is the projection. N*
*ote
that A = _1(0), where _ :E ! I is the projection on the second coordinate.
By direct verification of the CHP [93, p. 436], ss is an hfibration. If i is *
*an ~h
cofibration, then it has the LLP with respect to ss, hence we can lift the iden*
*tity
map of X to a map ~: X ! E such that ~ O i = i0. It follows that i(A) is closed
in X since i0(A) is closed in E.
Step 4. Let f :X ! Y be a map. Use Theorem 4.3.1(ii) to factor f as p O s,
where s: X ! Nf is the inclusion of a strong deformation retract and p is an
4.5. COMPACTLY GENERATED qTYPE MODEL STRUCTURES 55
~hfibration. Use Step 3 to factor s as
X _i0_//_Nf x (0, 1] [ X_xsIs//_Nf.
Here i0 is the inclusion of a strong deformation retract and X = _1(0), as in *
*Step
3. By Step 1, i0 is an hacyclic ~hcofibration. By Step 3, p O ss is an hfibr*
*ation.
There are several further results of Strom about hcofibrations that deserve*
* to
be highlighted. In order, the following results are [92, Theorem 12], [93, Lemma
5], and [92, Corollary 5].
Proposition 4.4.5. If p: E ! Y is an hfibration and the inclusion X Y
is an ~hcofibration, then the induced map p1(X) ! E is an ~hcofibration.
Proposition 4.4.6. If i: A ! B and j :B ! X are maps in K such that
j and j O i are hcofibrations, then i is an hcofibration.
Proposition 4.4.7. If an inclusion A X is an hcofibration, then so is the
induced inclusion ~A X.
In view of the characterization of ~hcofibrations in Theorem 4.4.4, it is n*
*atural
to ask if there is an analogous characterization of ~hfibrations. Only the fol*
*lowing
sufficient condition is known. It is stated without proof in [85, 4.1.1], and i*
*t gives
another reason for requiring the base spaces of exspaces to be in U .
Proposition 4.4.8. An hfibration p: E ! Y with Y 2 U is an ~hfibration.
__
Proof. Let k :A ! X be an hacyclic hcofibration and let j :A ! X
be the induced_inclusion. By Propositions 4.4.7 and 4.4.6, j and the inclusion
i: A A are hcofibrations. By Lemma 4.2.5(i), k is the inclusion of a deforma*
*tion
retraction r :X ! A_and the deformation restricts to a homotopy from (i O r) O*
* j
to the identity on A. It follows that j and hence also i are hacyclic. Since*
* j is
an hacyclic ~hcofibration, it has the LLP with respect to p, and we see by a *
*little
diagram chase that it suffices to verify that i has the LLP with respect to p. *
*Factor
p as the composite of s: E ! Np and q :Np:  ! Y , as usual. Since q is an
~hfibration, (i, q) has the lifting property, and it suffices to show that (i,*
* s) has the
__
lifting property. Suppose given a lifting problem f :A ! E and g :A ! Np
such that s O f = g O i. Note that s(e) = (e, cp(e)) for e 2 E, where cy denote*
*s the
constant path at y. Since Y is weak Hausdorff, the constant paths give a closed
subset of Y Iand Np = Y IxY E is a closed subset of Y Ix E. Therefore s(E) is
closed in Np. We conclude that
__ ____ ______ ____
g(A ) g(A)= s(f(A) s(E)= s(E),
__
which means that there is a lift A ! E.
4.5. Compactly generated qtype model structures
We give a variant of the standard procedure for constructing qtype model
structures. The exposition prepares the way for a new variant that we will expl*
*ain
in x5.4 and which is crucial to our work. Although our discussion is adapted to
topological examples, C need not be topological until otherwise specified. We
first recall the small object argument in settings where compactness allows use*
* of
sequential colimits.
Definition 4.5.1. Let I be a set of maps in C .
56 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES
(i)A relative Icell complex is a map Z0 ! Z, where Z is the colimit of a
sequence of maps Zn ! Zn+1 such that Zn+1 is the pushout Y [X Zn of a
coproduct X ! Y of maps in I along a map X ! Zn.
(ii)I is compact if for every domain object X of a map in I and every relative
Icomplex Z0 ! Z, the map colimC (X, Zn) ! C (X, Z) is a bijection.
(iii)An Icofibration is a map that satisfies the LLP with respect to any map t*
*hat
satisfies the RLP with respect to I.
Lemma 4.5.2 (Small object argument). Let I be a compact set of maps in C ,
where C is cocomplete. Then any map f : X ! Y in C factors functorially as a
composite
p
X ___i_//W_____//Y
such that p satisfies the RLP with respect to I and i is a relative Icell comp*
*lex and
therefore an Icofibration.
Definition 4.5.3. A model structure on C is compactly generated if there are
compact sets I and J of maps in C such that the following characterizations hol*
*d.
(i)The fibrations are the maps that satisfy the RLP with respect to J, or equi*
*v
alently, with respect to retracts of relative Jcell complexes.
(ii)The acyclic fibrations are the maps that satisfy the RLP with respect to I*
*, or
equivalently, with respect to retracts of relative Icell complexes.
(iii)The cofibrations are the retracts of relative Icell complexes.
(iv)The acyclic cofibrations are the retracts of relative Jcell complexes.
The maps in I are called the generating cofibrations and the maps in J are call*
*ed
the generating acyclic cofibrations.
We find it convenient to separate out properties of classes of maps in a mod*
*el
category, starting with the weak equivalences.
Definition 4.5.4. A subcategory of C is a subcategory of weak equivalences if
it satisfies the following closure properties.
(i)All isomorphisms in C are weak equivalences.
(ii)A retract of a weak equivalence is a weak equivalence.
(iii)If two out of three maps f, g, g O f are weak equivalences, so is the thir*
*d.
Theorem 4.5.5. Let C be a bicomplete category with a subcategory of weak
equivalences. Let I and J be compact sets of maps in C . Then C is a compactly
generated model category with generating cofibrations I and generating acyclic *
*cofi
brations J if the following two conditions hold:
(i)(Acyclicity condition) Every relative Jcell complex is a weak equivalence.
(ii)(Compatibility condition) A map has the RLP with respect to I if and only *
*if
it is a weak equivalence and has the RLP with respect to J.
Proof. This is the formal part of Quillen's original proof of the qmodel s*
*truc
ture on topological spaces and is a variant of [44, 2.1.19] or [43, 11.3.1]. Th*
*e fi
brations are defined to be the maps that satisfy the RLP with respect to J. The
cofibrations are defined to be the Icofibrations and turn out to be the retrac*
*ts of
relative Icell complexes. The retract axioms clearly hold and, by (ii), the co*
*fibra
tions are the maps that satisfy the LLP with respect to the acyclic fibrations,*
* which
gives one of the lifting axioms. The maps in J satisfy the LLP with respect to *
*the
4.5. COMPACTLY GENERATED qTYPE MODEL STRUCTURES 57
fibrations and are therefore cofibrations, which verifies something that is tak*
*en as
a hypothesis in the versions in the cited sources. Applying the small object ar*
*gu
ment to I, we factor a map f as a composite of an Icofibration followed by a m*
*ap
that satisfies the RLP with respect to I; by (ii), the latter is an acyclic fib*
*ration.
Applying the small object argument to J, we factor f as a composite of a relati*
*ve
Jcell complex that is a Jcofibration followed by a fibration. By (i), the fir*
*st map
is acyclic, and it is a cofibration because it satisfies the LLP with respect t*
*o all
fibrations, in particular the acyclic ones. Finally, for the second lifting axi*
*om, if we
are given a lifting problem with an acyclic cofibration f and a fibration p, th*
*en a
standard retract argument shows that f is a retract of an acyclic cofibration t*
*hat
satisfies the LLP with respect to all fibrations.
Using the following companion to Definition 4.5.4, we codify the usual patte*
*rn
for verifying the acyclicity condition.
Definition 4.5.6. A subcategory of a cocomplete category C is a subcategory
of cofibrations if it satisfies the following closure properties.
(i)All isomorphisms in C are cofibrations.
(ii)All coproducts of cofibrations are cofibrations.
(iii)If i: X ! Y is a cofibration and f :X ! Z is any map, then the pushout
j :Y ! Y [X Z of f along i is a cofibration.
(iv)If X is the colimit of a sequence of cofibrations in :Xn ! Xn+1, then the
induced map i: X0 ! X is a cofibration.
(v)A retract of a cofibration is a cofibration.
In more general contexts, (iv) should be given a transfinite generalization,*
* but
we shall not have need of that. Note that if a subcategory of cofibrations is d*
*efined
in terms of a left lifting property, then all of the conditions hold automatica*
*lly.
Lemma 4.5.7. Let C be a cocomplete category together with a subcategory of
cofibrations, denoted gcofibrations, and a subcategory of weak equivalences, s*
*atis
fying the following properties.
(i)A coproduct of weak equivalences is a weak equivalence.
(ii)If i: X ! Y is an acyclic gcofibration and f :X ! Z is any map, then
the pushout j :Y ! Y [X Z of f along i is a weak equivalence.
(iii)If X is the colimit of a sequence of acyclic gcofibrations in :Xn ! Xn+1,
then the induced map i: X0 ! X is a weak equivalence.
If every map in a set J is an acyclic gcofibration, then every relative Jcell*
* complex
is a weak equivalence.
We emphasize that the gcofibrations are not the model category cofibrations
and may or may not be the intrinsic hcofibrations or ~hcofibrations. They ser*
*ve as
a convenient scaffolding for proving the model axioms.
Remark 4.5.8. The properties listed in Lemma 4.5.7 include some of the ax
ioms for a "cofibration category" given by Baues [1, pp 6, 182]. However, our
purpose is to describe features of categories that are more richly structured t*
*han
model categories, often with several relevant subcategories of cofibrations, ra*
*ther
than to describe deductions from axiom systems for less richly structured categ*
*ories,
which is his focus. The gcofibrations in Lemma 4.5.7 need not be the cofibrati*
*ons
of any cofibration category or model category.
58 4. TOPOLOGICALLY BICOMPLETE MODEL CATEGORIES
The qmodel structures on K and U are obtained by Theorem 4.5.5, taking
the qequivalences to be the weak equivalences, that is, the maps that induce i*
*so
morphisms on all homotopy groups, and the qfibrations to be the Serre fibratio*
*ns.
We also have the equivariant generalization, which applies to any topological g*
*roup
G. We introduce the following notations, which will be used throughout.
Definition 4.5.9. Nonequivariantly, let I and J denote the set of inclusions
i: Sn1 ! Dn (where S1 is empty) and the set of maps i0: Dn ! Dn x I.
Equivariantly, let I and J denote the set of all maps of the form G=H x i, where
H is a (closed) subgroup of G and i runs through the maps in the nonequivariant
sets I and J. In the based categories K* and GK* we continue to write I and J
for the sets obtained by adjoining disjoint base points to the specified maps.
A map f :X ! Y of Gspaces is said to be a weak equivalence or Serre fibra
tion if all fixed point maps fH :XH  ! Y H are weak equivalences or Serre fibr*
*a
tions. Just as nonequivariantly, we also call these qequivalences and qfibrat*
*ions.
Observe that qequivalences are defined in terms of the equivariant homotopy gr*
*oups
ssHn(X, x) = ssn(XH , x) for H G and x 2 XH and that qfibrations are defined*
* in
terms of the RLP with respect to the cells in J.
If X0 ! X is a relative I or Jcell complex, then X=X0 is in GU and
Lemma 1.1.5 gives all that is needed to verify the compactness hypothesis in De*
*f
inition 4.5.1(ii). Taking the gcofibrations to be the hcofibrations, Lemma 4*
*.5.7
applies to verify the acyclicity condition of Theorem 4.5.5. With considerable *
*sim
plification, our verification of the compatibility condition for the qfmodel s*
*tructure
in Chapter 6 specializes to verify it here. Nonequivariantly, the qmodel struc*
*ture
is discussed in [37, x8] and, with somewhat different details, in [44, 2.4] (wh*
*ere the
details on transfinite sequences are unnecessary).
Equivariantly, a detailed proof of the following result is given in [61, III*
*x1].
The argument there is given for based Gspaces, in GT , but it works equally we*
*ll
for unbased Gspaces, in GK .
Theorem 4.5.10. For any G, GK is a compactly generated proper model cate
gory whose qequivalences, qfibrations, and qcofibrations are the weak equiva*
*lences,
the Serre fibrations, and the retracts of relative Gcell complexes. The sets I*
* and
J are the generating qcofibrations and the generating acyclic qcofibrations, *
*and
all qcofibrations are ~hcofibrations. If G is a compact Lie group, then the m*
*odel
structure is Gtopological.
The notion of a Gtopological model category is defined in the same way as
the notion of a simplicial or topological model category and is discussed forma*
*lly in
x10.3 below. The point of the last statement is that if H and K are subgroups o*
*f a
compact Lie group G, then G=H x G=K has the structure of a GCW complex. By
Theorem 3.3.2, this remains true when G is a Lie group and H and K are compact
subgroups. We shall see how to use this fact model theoretically in Chapter 7.
CHAPTER 5
Wellgrounded topological model categories
Introduction
It is essential to our theory to understand the interrelationships among the
various model structures that appear naturally in the parametrized context, both
in topology and in general. This understanding leads us more generally to an
axiomatization of the properties that are required of a good qtype model struc*
*ture
in order that it relate well to the classical homotopy theory on a topological *
*category.
The obvious qmodel structure on exspaces over B does not satisfy the axioms, *
*and
in the next chapter we will introduce a new model structure, the qfmodel struc*
*ture,
that does satisfy the axioms.
As we recall in x5.1, any model structure on a category C induces a model
structure on the category of objects over, under, or over and under a given obj*
*ect
B. When C is topologically bicomplete, so are these over and under categories.
They then have their own intrinsic htype model structures, which differ from t*
*he
one inherited from C . This leads to quite a few different model structures on
the category CB of objects over and under B, each with its own advantages and
disadvantages. Letting B vary, we also obtain a model structure on the category
of retracts. We shall only be using most of these structures informally, but t*
*he
plethora of model structures is eye opening.
In x5.2, we focus on spaces and compare the various classical notions of fib*
*ra
tions and cofibrations that are present in our over and under categories. Altho*
*ugh
elementary, this material is subtle, and it is nowhere presented accurately in *
*the
literature. In particular, we discuss htype, ftype and fptype model structur*
*es,
where f and fp stand for "fiberwise" and "fiberwise pointed". For simplicity, we
discuss this material nonequivariantly, but it applies verbatim equivariantly.
The comparisons among the q, h, f, and fp classes of maps and model struc
tures guide our development of parametrized homotopy theory. We think of the
fnotions as playing a transitional role, connecting the fp and hnotions. In t*
*he
rest of the chapter, we work in a general topologically bicomplete category C ,*
* and
we sort out this structure and its relationship to a desired qtype model struc*
*ture
axiomatically.
Here we shift our point of view. We focus on three basic types of cofibratio*
*ns
that are in play in the general context, namely the Hurewicz cofibrations deter*
*mined
by the cylinders in C , the ground cofibrations that come in practice from a gi*
*ven
forgetful functor to underlying spaces, and the qtype model cofibrations. The *
*first
two are intrinsic, but we think of the qtype cofibrations as subject to negoti*
*ation. In
KB , the Hurewicz cofibrations are the fpcofibrations and the ground cofibrati*
*ons
are the hcofibrations, which is in notational conflict with the point of view *
*taken
in the previous chapter.
59
60 5. WELLGROUNDED TOPOLOGICAL MODEL CATEGORIES
In xx5.3 and 5.4, we ignore model theoretic considerations entirely. We desc*
*ribe
how the two intrinsic types of cofibrations relate to each other and to colimit*
*s and
tensors, and we explain how this structure relates to weak equivalences.
We define the notion of a "wellgrounded model structure" in x5.5. We believe
that this notion captures exactly the right blend of classical and model catego*
*rical
homotopical structure in topological situations. It describes what is needed f*
*or
a qtype model structure in a topologically bicomplete category to be compatible
with its intrinsic htype model structure and its ground structure. Crucially, *
*the
qtype cofibrations should be "bicofibrations", meaning that they are both Hure*
*wicz
cofibrations and ground cofibrations. To illustrate the usefulness of the axiom*
*ati
zation, and for later reference, we derive the long exact sequences associated *
*to
cofiber sequences and the lim1exact sequences associated to colimits in x5.6.
A clear understanding of the desiderata for a good qtype model structure
reveals that the obvious over and under qmodel structure is essentially worthl*
*ess
for serious work in parametrized homotopy theory. This will lead us to introduce
the new qfmodel structure, with better behaved qtype cofibrations, in the next
chapter. The formalization given in xx5.35.6 might seem overly pedantic were it
only to serve as motivation for the definition of the qfmodel structure. Howev*
*er, we
will encounter exactly the same structure in Part III when we construct the lev*
*el and
stable model structures on parametrized spectra. We hope that the formalization
will help guide the reader through the rougher terrain there.
We note parenthetically that there is still another interesting model struct*
*ure
on the category of exspaces over B, one based on local considerations. It is d*
*ue
to Michelle Intermont and Mark Johnson [49]. We shall not discuss their model
structure here, but we are indebted to them for illuminating discussions. It is
conceivable that their model structure could be used in an alternative developm*
*ent
of the stable theory, but that has not been worked out. Their structure suffers*
* the
defects that it is not known to be left proper and that, with their definition *
*of weak
equivalences, homotopy equivalences of base spaces need not induce equivalences*
* of
homotopy categories.
We focus mainly on the nonequivariant context in this chapter, but G can be
any topological group in all places where equivariance is considered.
5.1. Over and under model structures
Recall from x1.2 that, for any category C and object B in C , we let C =B and
CB denote the categories of objects over B and of exobjects over B. We also ha*
*ve
the category B\C of objects under B. If C is bicomplete, then so are C =B, B\C
and CB . We begin with some general observations about over and under model
categories before returning to topological categories.
We have forgetful functors U :C =B ! C and V :CB ! C =B. The first is
left adjoint to the functor that sends an object Y to the object B x Y over B:
(5.1.1) C (UX, Y ) ~=C =B (X, B x Y ).
The second is right adjoint to the functor that sends an object X over B to the
object X q B over and under B:
(5.1.2) CB (X q B, Y ) ~=C =B (X, V Y ).
As a composite of a left and a right adjoint, the total object functor UV :CB *
*! C
does not enjoy good formal properties. This obvious fact plays a significant ro*
*le in
5.1. OVER AND UNDER MODEL STRUCTURES 61
our work. For example, it limits the value of the model structures on CB that a*
*re
given by the following result.
Proposition 5.1.3. Let C be a model category. Then C =B, B\C , and CB
are model categories in which the weak equivalences, cofibrations, and fibratio*
*ns
are the maps over B, under B, or over and under B which are weak equivalences,
fibrations, or cofibrations in C . If C is left or right proper, then so are C *
*=B, B\C ,
and CB .
Proof. As observed in [44, p. 5] and [37, 3.10], the statement about C =B
is a direct verification from the definition of a model category. By the self*
*dual
nature of the axioms, the statement about B\C is equivalent. The statement
about CB follows since it is the category of objects under (B, id) in C =B. The*
* last
statement holds since pushouts and pullbacks in these over and under categories
are constructed in C .
When considering qtype model structures, we start with a compactly generated
model category C . Using the adjunctions (5.1.1) and (5.1.2), we then obtain the
following addendum to Proposition 5.1.3.
Proposition 5.1.4. If C is a compactly generated model category, then C =B
and CB are compactly generated. The generating (acyclic) cofibrations in C =B a*
*re
the maps i such that Ui is a generating (acyclic) cofibration in C . The genera*
*ting
(acyclic) cofibrations in CB are the maps i q B where i is a generating (acycli*
*c)
cofibration in C =B.
We now return to the case when C is topologically bicomplete. Then it has the
resulting "classical", or htype, structure that was discussed in x4.3 and x4.4*
*. If our
philosophy in x4.1 applies to C , then it also has q and mstructures and the c*
*ate
gories C =B and CB both inherit over and under model structures that are related
as we discussed there. However, since C is topologically bicomplete, so is C =B*
* by
Proposition 1.2.8, and CB is based topologically bicomplete by Proposition 1.2.*
*9.
These categories therefore have classical htype structures when they are regar*
*ded
in their own right as topologically bicomplete categories. To fix notation and *
*avoid
confusion we give an overview of all of these structures.
We start with the hclasses of maps in C that are given in Definition 4.2.1 *
*and
Lemma 4.2.4. As in our discussion of spaces, we work assymmetrically, ignoring *
*the
~hfibrations and focusing on the candidates for htype model structures given *
*by
the hfibrations and ~hcofibrations. We agree to use the letter h for the inhe*
*rited
classes of maps in C =B and CB , although that contradicts our previous use of h
for the classical classes of maps in an arbitrary topologically bicomplete cate*
*gory,
such as C =B or CB . We shall resolve that ambiguity shortly by introducing new
names for the classes of "classical" maps in those categories.
Definition 5.1.5. A map g in C =B is an hequivalence, hfibration, hco
fibration, or ~hcofibration if Ug is such a map in C . A map g in CB is an h
equivalence, hfibration, hcofibration, or ~hcofibration if V g is such a map*
* in C =B
or, equivalently, UV g is such a map in C .
The ~hcofibrations are hcofibrations, but not conversely in general. Since*
* the
object *B = (B, id, id) is initial and terminal in CB , an object of CB is hco*
*fibrant
(or ~hcofibrant) if its section is an hcofibration (or ~hcofibration) in C .*
* It is h
fibrant if its projection is an hfibration in C .
62 5. WELLGROUNDED TOPOLOGICAL MODEL CATEGORIES
In C =B, we have the notion of a homotopy over B, defined in terms of X xB I
or, equivalently, Map B(I, X). The adjective "fiberwise" is generally used in *
*the
literature to describe these homotopies. See, for example, the books [29, 51] on
fiberwise homotopy theory. To distinguish from the hmodel structure, we agree
to write f rather than h for the fiberwise specializations of Definition 4.2.1 *
*and
Lemma 4.2.4. To avoid any possible confusion, we formalize this, making use of
Proposition 4.3.3.
Definition 5.1.6. Let g be a map in C =B.
(i)g is an fequivalence if it is a fiberwise homotopy equivalence.
(ii)g is an ffibration if it satisfies the fiberwise CHP, that is, if it has *
*the RLP
with respect to the maps i0: X ! X xB I for X 2 C =B.
(iii)g is an fcofibration if it satisfies the fiberwise HEP, that is, if it ha*
*s the LLP
with respect to the maps p0: MapB (I, X) ! X.
(iv)g is an ~fcofibration if it has the LLP with respect to the facyclic ffi*
*brations.
A map g in CB is an fequivalence, ffibration, fcofibration, or ~fcofibratio*
*n if
V g is one in C =B.
Again, f~cofibrations are fcofibrations, but not conversely in general. T*
*he
orem 4.4.2 often applies to show that the ffibrations and f~cofibrations defi*
*ne
an fmodel structure on C =B and therefore, by Proposition 5.1.3, on CB . As is
always the case for an intrinsic classical model structure, every object of C =*
*B is
both fcofibrant and ~fcofibrant as well as ffibrant. While this is obvious f*
*rom
the definitions, it may seem counterintuitive. It does not follow that every ob*
*ject
of CB is fcofibrant since the two categories have different initial objects.
In CB , we also have the notion of a homotopy over and under B, defined in
terms of X ^B I+ or, equivalently, FB (I+ , X). The adjective "fiberwise pointe*
*d" is
used in [29, 51] to describe these homotopies. Again, for notational clarity, w*
*e agree
to write fp rather than h for the fiberwise pointed specializations of Definiti*
*on 4.2.1
and Lemma 4.2.4, and we formalize this to avoid any possible confusion.
Definition 5.1.7. Let g be a map in CB .
(i)g is an fpequivalence if it is a fiberwise pointed homotopy equivalence.
(ii)g is an fpfibration if it satisfies the fiberwise pointed CHP, that is, i*
*f it has
the RLP with respect to the maps i0: X ! X ^B I+ .
(iii)g is a fpcofibration if it satisfies the fiberwise pointed HEP, that is, *
*if it has
the LLP with_respect to the maps p0: FB (I+ , X) ! X.
(iv)g is an fpcofibration if it has the LLP with respect to the fpacyclic fp
fibrations.
___
Again, fpcofibrations are fpcofibrations, but not conversely_in general, a*
*nd
Theorem 4.4.2 often applies to show that the fpfibrations and fpcofibrations *
*de
fine an fpmodel structure on CB . We summarize some general formal implications
relating our classes of maps.
Proposition 5.1.8. Let C , C =B and CB be topologically bicomplete categories
with h, f, and fpclasses of maps defined as above. Then the following implicat*
*ions
hold for maps in CB .
5.2. THE SPECIALIZATION TO OVER AND UNDER CATEGORIES OF SPACES 63
___________________________________________________
 fpequivalence=) fequivalence=) hequivalence 
 fpcofibration(= fcofibration=) hcofibration 
 ___ * * * 
 fpcofibration(= ~fcofibration=) ~hcofibration 
____fpfibration=)__ffibration_(=___hfibration__ 
Moreover, every object of CB is both fpfibrant and fpcofibrant.
Proof. Trivial inspections of lifting diagrams show that an hfibration is *
*an__
ffibration, an fcofibration is an fpcofibration, and an f~cofibration is an*
* fp
cofibration. Use of the adjunctions (5.1.1) and (5.1.2) shows that an fcofibra*
*tion
is an hcofibration, an ~fcofibration is an ~hcofibration, and an fpfibratio*
*n is an f
fibration. The last statement holds since fiberwise pointed homotopies with dom*
*ain
or target B are constant at the section or projection of the target or source.
Remark 5.1.9. Assume that these classes of maps define model structures.
Then the implications in Proposition 5.1.8 lead via Theorem 4.1.1 and its dual
version to two new mixed model structures on CB , one with weak equivalences the
fequivalences and fibrations the fpfibrations and one with weak equivalences *
*the
hequivalences and cofibrations the ~fcofibrations.
The category CB of retracts introduced in x2.5 suggests an alternative model
theoretic point of view. We give the basic definitions, but we shall not pursue*
* this
idea in any detail. Again, Theorem 4.4.2 often applies to verify the model cate*
*gory
axioms. Note that the intrinsic homotopies are given by homotopies of total obj*
*ects
over and under homotopies of base objects.
Definition 5.1.10. Assume that CB is topologically bicomplete and let g be
a map in CB .
(i)g is an requivalence if it is a homotopy equivalence of retractions.
(ii)g is an rfibration if it satisfies the retraction CHP, that is, if it has*
* the RLP
with respect to the maps i0: X ! X x I for X 2 CB .
(iii)g is an rcofibration if it satisfies the retraction HEP, that is, if it h*
*as the LLP
with respect to the maps p0: Map(I, X) ! X.
(iv)g is an ~rcofibration if it has the LLP with respect to the racyclic rfi*
*brations.
Remark 5.1.11. The initial and terminal object of CB are the identity retrac
tions of the initial and terminal objects of B and every object is both rcofib*
*rant
and rfibrant. It might be of interest to characterize the retractions for whi*
*ch
the map *B  ! (X, p, s) induced by s is an rcofibration or for which the map
(X, p, s) ! *B induced by p is an rfibration. By specialization of the liftin*
*g prop
erties, an exmap over B that is an rcofibration or rfibration is an fpcofib*
*ration
or fpfibration in CB , but we have not pursued this question further.
5.2.The specialization to over and under categories of spaces
Now we take C to be K or U . We discuss the relationships among our various
classes of fibrations and cofibrations in this special case, and we consider wh*
*en the
f and fp classes of maps give model structures. Everything in this section appl*
*ies
equally well equivariantly.
We first say a bit about based spaces, which are exspaces over B = {*}.
Here the fact that * is a terminal object greatly simplifies matters. All of t*
*he
64 5. WELLGROUNDED TOPOLOGICAL MODEL CATEGORIES
fnotions coincide with the corresponding hnotions, and our trichotomy reduces
to the familiar dichotomy between free (or h) notions and based (or fp) notions.
Recall that a based space is wellbased, or nondegenerately based, if the inclu*
*sion
of the basepoint is an hcofibration. Every based space is fpcofibrant, and an
fpcofibration between wellbased spaces is an hcofibration [93, Prop. 9]. Ev*
*ery
based space is fpfibrant, and an hfibration of based spaces satisfies the bas*
*ed CHP
with respect to wellbased source spaces. Of course, the over and under hmodel
structure differs from the intrinsic fpmodel structure.
None of the reverse implications in Proposition 5.1.8 holds in general. We g*
*ave
details of that result since it is easy to get confused and think that more is *
*true
than we stated.
Scholium 5.2.1. On [29, p. 66], it is stated that a fiberwise pointed cofibr*
*ation
which is a closed inclusion is a fiberwise cofibration. That is false even when*
* B is
a point, since it would imply that every point of a T1space is a nondegenerate
basepoint. On [29, p. 69], it is stated that a fiberwise pointed map (= exmap)*
* is
a fiberwise pointed fibration if and only if it is a fiberwise fibration. That *
*is also
false when B is a point, since the unbased CHP does not imply the based CHP.
However, as for based spaces, the reverse implications in parts of Proposi
tion 5.1.8 often do hold under appropriate additional hypotheses.
Proposition 5.2.2. The following implications hold for an arbitrary topologi
cally bicomplete category C .
(i)A map in C =B between hfibrant objects over B is an hequivalence if and
only if it is an fequivalence.
(ii)An exmap between fcofibrant exobjects over B is an fequivalence if and
only if it is an fpequivalence.
Proof. The first part follows from Proposition 4.3.5(ii) since an fequival*
*ence
in C =B is the same as an hequivalence over B in C . The second part follows
similarly from Proposition 4.3.5(i) since an fpequivalence in CB is the same a*
*s an
fequivalence under B in C =B.
The following results hold for spaces. We are doubtful that they hold in gen*
*eral.
Proposition 5.2.3. The following implications hold in both GK and GU .
(i)An exmap between ~fcofibrant exspaces is an fcofibration if and only if*
* it
is an fpcofibration.
(ii)An exmap whose source is ~fcofibrant is an ffibration if and only if it*
* is an
fpfibration.
Proof. Part (ii) is [29, 16.3]. Part (i) is stated on [93, p. 441] and the *
*proof
given there for based spaces generalizes using the following lemma.
It is easy to detect fcofibrations by means of the following result, whose *
*proof
is the same as that of the standard characterization of Hurewicz cofibrations (*
*e.g.
[71, p. 43], see also [91, Thm. 2], [92, Lem. 4] and [29, 4.3]).
Lemma 5.2.4. An inclusion i: X ! Y in K =B is an fcofibration if and
only if (Y, X) is a fiberwise NDRpair in the sense that there is a map u: Y !*
* I
such that X u1(0) and a homotopy h: Y xB I ! Y over B such that h0 = id,
ht = id on X for 0 t 1, and h1(y) 2 X if u(y) < 1. A closed inclusion
5.2. THE SPECIALIZATION TO OVER AND UNDER CATEGORIES OF SPACES 65
i : X ! Y in K =B is an f~cofibration if and only if the map u above can be
chosen so that X = u1(0).
We introduce the following names here, but we defer a full discussion to x8.*
*1.
Definition 5.2.5. An exspace is said to be wellsectioned if it is ~fcofib*
*rant.
An exspace is said to be exfibrant or, synonomously, to be an exfibration if*
* it is
both ~fcofibrant and hfibrant. Thus an exfibration is a wellsectioned exs*
*pace
whose projection is an hfibration.
The term exfibrant is more logical than exfibration, since we are defining*
* a
type of object rather than a type of morphism of KB , but the term exfibration
goes better with Serre and Hurewicz fibration and is standard in the literature*
*. We
have the following implication of Propositions 5.1.8 and 5.2.2. It helps explai*
*n the
usefulness of exfibrations.
Corollary 5.2.6. Let g be an exmap between exfibrations over B.
(i)g is an hequivalence if and only if g is an fequivalence, and this hold i*
*f and
only if g is an fpequivalence.
(ii)g is an fcofibration if and only if g is an fpcofibration, and then g is*
* an
hcofibration.
(iii)g is an ffibration if and only if g is an fpfibration, and this holds if*
* g is an
hfibration.
Remark 5.2.7. The model theoretic significance of exfibrations over B is un
clear. They are fibrant and cofibrant objects in the mixed model structure on
exspaces over B whose weak equivalences are the hequivalences and whose cofi
brations are the ~fcofibrations. However, the converse fails since there are *
*well
sectioned ffibrant exspaces that are fequivalent to hfibrant exspaces, hen*
*ce
are mixed fibrant, but are not themselves hfibrant.
The previous remark anticipated the following result on over and under model
structures in the categories of spaces and exspaces over B. Note that Lemma 1.*
*1.3
applies to K =B and KB as well as to K to show that both fcofibrations and
fpcofibrations are inclusions which are closed when the total spaces are in U .
Theorem 5.2.8. The following statements hold.
(i)The fequivalences, ffibrations, and ~fcofibrations give K =B a proper to*
*po
logical model structure. Moreover, a map in K =B is an ~fcofibration if and
only if it is a closed fcofibration.
(ii)The fequivalences, ffibrations, and ~fcofibrations give U =B a proper t*
*opo
logical model structure. Moreover, a map in U =B is an ~fcofibration if and
only if it is an fcofibration. ___
(iii)The fpequivalences, fpfibrations, and fpcofibrations give UB an fpmodel
structure.
(iv)The rclasses of maps give the category UU of retracts a proper topological
rmodel structure.
Proof. Apart from the factorization axioms, the model structures follow from
the discussion in 4.3. In particular, the lifting axioms, the properness, and *
*the
topological property of all of these model structures are given by Theorem 4.3.*
*1.
In (ii), (iii), and (iv), the factorization axioms follow from Theorem 4.4.2 si*
*nce
66 5. WELLGROUNDED TOPOLOGICAL MODEL CATEGORIES
the argument in Remark 1.1.4 verifies Hypothesis 4.4.1. The rest of (i) can be
proven by direct mimicry of the proof of Theorem 4.4.4, using Lemma 5.2.4, and
the characterization of the ~fcofibrations in (ii) follows.
Remark 5.2.9._We do not know whether or not KB is an fpmodel category or
whether the fpcofibrations in KB are characterized as the closed fpcofibratio*
*ns.
We also do not know whether or not KU is an rmodel category. The problem here
is related to the fact that, while the sections of exspaces are always inclusi*
*ons,
they need not be closed inclusions unless the total spaces are in U . Steps 1 a*
*nd 3
of the proof of Theorem 4.4.4 fail in KB , and we also do not see how to carry *
*over
Strom's original proofs in [92, 93]. Theorem 4.3.1 still applies, giving much o*
*f the
information carried by a model structure. Observe too that if i: A ! X is a map
of wellsectioned exspaces over B, then i is an fpcofibration if and only if *
*it is an
fcofibration, by Proposition 5.2.2(iii). For exspaces that are not wellsecti*
*oned,
we have little understanding_of fpcofibrations, even_when_B is a point. We have
little understanding of fpcofibrations that are not fcofibrations in any case.
There is a certain tension between the fp and hnotions, with the fnotions
serving as a bridge between the two. Fiberwise pointed homotopy is the intrinsi*
*cally
right notion of homotopy in KB , hence the fpstructure is the philosopically r*
*ight
classical htype model structure on KB , or at least on UB . It is the one tha*
*t is
naturally related to fiber and cofiber sequences, the theory of which works for*
*mally
in any based topologically bicomplete category in exactly the same way as for b*
*ased
spaces, as we will recall in x5.6. A detailed exposition in the case of exspac*
*es is
given in [29, 51, 52].
However, with h replaced by fp, we do not have the implications that we em
phasized in the general philosophy of x4.1. In particular, with the over and un*
*der
qmodel structure, qcofibrations need not be fpcofibrations and fpfibrations*
* need
not be qfibrations, let alone hfibrations. The qmodel structure is still rel*
*ated to
the hmodel structure as in x4.1, but this does not serve to relate the qmodel*
* struc
ture to parametrized fiber and cofiber sequences in the way that we are familiar
with in the nonparametrized context. This already suggests that the qmodel str*
*uc
ture might not be appropriate in parametrized homotopy theory. In the following
four sections, we explore conceptually what is required of a qtype model struc*
*ture
to connect it up with the intrinsic homotopy theory in a topologically bicomple*
*te
category.
5.3.Wellgrounded topologically bicomplete categories
Let C be a topologically bicomplete category in either the based or the unba*
*sed
sense; we use the notations of the based context. In our work here, and in other
topological contexts, C is topologically concrete in the sense that there is a *
*faithful
and continuous forgetful functor from C to spaces. In practice, appropriate "gr*
*ound
cofibrations" can then be specified in terms of underlying spaces. These cofibr*
*ations
should be thought of as helpful background structure in our category C .
To avoid ambiguity, we use the term "Hurewicz cofibration", abbreviated no
tationally to cylcofibration, for the maps that satisfy the HEP with respect t*
*o the
cylinders in C . We also have_the notion of a strong Hurewicz cofibration, whic*
*h we
abbreviate notationally to cylcofibration. For example, the cylcofibrations i*
*n K ,
K =B, and KB are the hcofibrations, the fcofibrations, and the fpcofibration*
*s,
5.3. WELLGROUNDED TOPOLOGICALLY BICOMPLETE CATEGORIES 67
___
respectively, and similarly for cylcofibrations. As we have seen, it often hap*
*pens
that cylcofibrations between suitably nice objects of C , which we shall call *
*"well
grounded", are also ground cofibrations. We introduce language to describe this
situation. The following definitions codify the behavior of the wellgrounded o*
*bjects
with respect to the cylcofibrations, colimits, and tensors in C . It is conven*
*ient to
build in the appropriate equivariant generalizations of our notions, although we
defer a formal discussion of Gtopologically bicomplete Gcategories to x10.2; *
*see
Definition 10.2.1. The examples in x1.4 give the idea.
Definition 5.3.1. An unbased space is wellgrounded if it is compactly gener
ated. A based space is wellgrounded if it is compactly generated and wellbase*
*d.
The same definitions apply to Gspaces for a topological group G.
Let C be a topologically bicomplete category.
Definition 5.3.2. A full subcategory of C is said to be a subcategory of wel*
*l
grounded objects if the following properties hold.
(i)The initial object of C is wellgrounded.
(ii)All coproducts of wellgrounded objects are wellgrounded.
(iii)If i: X ! Y is a cylcofibration and f :X ! Z is any map, where X, Y ,
and Z are wellgrounded, then the pushout Y [X Z is wellgrounded.
(iv)The colimit of a sequence of cylcofibrations between wellgrounded objects*
* is
wellgrounded.
(v)A retract of a wellgrounded object is wellgrounded.
(vi)If X is a wellgrounded object and K is a wellgrounded space, then X ^ K
(X x K in the unbased context) is wellgrounded.
When C is Gtopologically bicomplete, we replace spaces by Gspaces in (vi).
Definition 5.3.3. A ground structure on C is a (full) subcategory of well
grounded objects together with a subcategory of cofibrations, called the ground
cofibrations and denoted gcofibrations, such that every cylcofibration between
wellgrounded objects is a gcofibration. A map that is both a gcofibration an*
*d a
cylcofibration is called a bicofibration.
Thus a cylcofibration between wellgrounded objects is a bicofibration. The
need for focusing on bicofibrations and the force of the definition come from t*
*he
following fact.
Warning 5.3.4. In practice, (iii) often fails if i is a gcofibration betwee*
*n well
grounded objects that is not a cylcofibration, as we shall illustrate in x6.1.*
* In
particular, in GKB with the canonical ground structure described below, it can
already fail for an inclusion i of Icell complexes, where I is the standard se*
*t of
generators for the qcofibrations.
In the next chapter, we will construct a qtype model structure for GKB with*
* a
set of generating cofibrations to which the following implication of Definition*
*s 4.5.6
and 5.3.2 applies.
Lemma 5.3.5. Let I be a set of cylcofibrations between wellgrounded objects
and let f :X ! Y be a retract of a relative Icell complex W ! Z. Then f is a
bicofibration. If W is wellgrounded, then so are X, Y , and Z.
Our categories of equivariant parametrized spaces have canonical ground stru*
*c
tures. Recall that the classes of f and ~fcofibrations in GU =B and GUB coinci*
*de.
68 5. WELLGROUNDED TOPOLOGICAL MODEL CATEGORIES
Definition 5.3.6. A space over B is wellgrounded if its total space is com
pactly generated. An exspace over B is wellgrounded if it is wellsectioned a*
*nd
its total space is compactly generated. In both GK =B and GKB , define the g
cofibrations to be the hcofibrations.
Note that the only distinction between wellsectioned and wellgrounded ex
spaces is the condition on total spaces. The distinction is relevant when we co*
*nsider
relative Icell complexes X0 ! X in GKB . If X0 is wellsectioned, then so is *
*X,
whereas X=X0 is an Icell complex and is therefore wellgrounded for any X0.
Proposition 5.3.7. These definitions specify ground structures on GK =B and
on GKB .
Proof. For GK =B, the Hurewicz cofibrations are the fcofibrations, and
these are hcofibrations. It is standard that GU =B has the closure properties
specified in Definition 5.3.2. For GKB , the Hurewicz cofibrations are the fp
cofibrations. Between wellsectioned exspaces, these are fcofibrations and th*
*ere
fore hcofibrations by Proposition 5.2.3(i). Parts (i)(v) of Definition 5.3.2*
* are
clear since wellsectioned means ~fcofibrant, which is a lifting property. Fin*
*ally we
consider part (vi). Recall that X ^B K can be constructed as the pushout of
*B oo___X q (B x K) _____//X x K
in the category of spaces over B. By the equivariant version of the NDRpair
characterization of fcofibrations in Lemma 5.2.4, these spaces are fcofibrant*
* and
the map on the right is an fcofibration. This implies that X ^B K is fcofibra*
*nt.
5.4.Wellgrounded categories of weak equivalences
The following definition describes how the weak equivalences and the ground
structure are related in practice.
Definition 5.4.1. Let C be a topologically bicomplete category with a given
ground structure. A subcategory of weak equivalences in C is wellgrounded if t*
*he
following properties hold (where acyclicity refers to the weak equivalences).
(i)A homotopy equivalence is a weak equivalence.
(ii)A coproduct of weak equivalences between wellgrounded objects is a weak
equivalence.
(iii)(Gluing lemma) Assume that the maps i and i0 are bicofibrations and the
vertical arrows are weak equivalences in the following diagram.
f
Y ooi___X _____//_Z
  
  
fflffl fflfflfflffl
Y 0oi0oX0___f0_//Z0
Then the induced map of pushouts is a weak equivalence. In particular,
pushouts of weak equivalences along bicofibrations are weak equivalences.
(iv)(Colimit lemma) Let X and Y be the colimits of sequences of bicofibrations
in :Xn ! Xn+1 and jn :Yn ! Yn+1 such that both X=X0 and Y=Y0 are
wellgrounded. If f :X ! Y is the colimit of a sequence of compatible weak
5.4. WELLGROUNDED CATEGORIES OF WEAK EQUIVALENCES 69
equivalences fn :Xn ! Yn, then f is a weak equivalence. In particular, if
each in is a weak equivalence, then the induced map i: X0 ! X is a weak
equivalence.
(v)For a map i: X ! Y of wellgrounded objects in C and a map j :K ! L
of wellgrounded spaces, i j is a weak equivalence if i is a weak equivalen*
*ce
or j is a qequivalence.
Here, in the based context, i j is the evident induced map
(X ^ L) [X^K (Y ^ K) ! Y ^ L.
The gluing lemma implies that acyclic bicofibrations are preserved under push
outs, as of course holds for pushouts of acyclic cofibrations in model categori*
*es.
The special case mentioned in (iii) corresponds to the left proper axiom in mod*
*el
categories. As there, it can be used to prove the general case of the gluing le*
*mma
provided that we have suitable factorizations.
Lemma 5.4.2. Assume the following hypotheses.
(i)Weak equivalences are preserved under pushouts along bicofibrations.
(ii)Every map factors as the composite of a bicofibration and a weak equivalen*
*ce.
Then the gluing lemma holds.
Proof. We use the notations of Definition 5.4.1(iii) and proceed in three c*
*ases.
If f and f0are both weak equivalences, then, by (i), so are the horizontal a*
*rrows
in the commutative diagram
Y ______//Y [X Z
 
 
fflffl fflffl
Y 0_____//Y 0[X0 Z0.
Since Y ! Y 0is a weak equivalence, the right arrow is a weak equivalence by t*
*he
two out of three property of weak equivalences.
If f and f0 are both bicofibrations, consider the commutative diagram
pX _________i_________//Y9n
fpppp  nnnnn  99
pp  wwnnnn  99
xxppp_________//_Y [ Z  999
Z   ?X?  99
   ???  99
 fflffl  ?? fflffl 9oo
 X0 ____________???_//_Y [X_X0_//Y 0
 f0 qqq  ?? sss
 qqqq  oooo??? sss
fflfflxxqqq fflfflwwooOO yyss
Z0 ______________//_Y [X Z0__//_Y 0[X0 Z0.
The back, front, top, and two bottom squares are pushouts, and the middle com
posite X0 ! Y 0is i0. Since f and f0 are bicofibrations, so are the remaining
three arrows from the back to the front. Similarly, i and its pushouts are bico*
*fi
brations. Since X ! X0, Y ! Y 0, and Z ! Z0 are weak equivalences, (i)
and the two out of three property imply that Y  ! Y [X X0, Y [X X0 ! Y 0,
Y [X Z ! Y [X Z0, and Y [X Z0 ! Y 0[X0Z0are weak equivalences. Composing
the last two, Y [X Z ! Y 0[X0 Z0 is a weak equivalence.
70 5. WELLGROUNDED TOPOLOGICAL MODEL CATEGORIES
To prove the general case, construct the following commutative diagram.
f
Y oo__i____X ________________//_LLZ99ss
  LLLL sss 
  LL%% sss~f 
  W 
  
   
fflffli0 fflfflf0 fflffl
Y 0oo______X0 _______________//_Z0
KKK  tt99
KKK  ttt
K%%fflffl~f0tt
X0[X W
Here we first factor f as the composite of a bicofibration and a weak equivalen*
*ce
~fand then define a map ~f0by the universal property of pushouts. By hypothesis
(i), W ! X0[X W is a weak equivalence, and by the two out of three property,
so is ~f0. By the second case,
Y [X W ! Y 0[X0 (X0[X W ) ~=Y 0[X W
is a weak equivalence and by the first case, so is
Y [X Z ~=(Y [X W ) [W Z ! (Y 0[X W ) [(X0[XW) Z0~=Y 0[X0 Z0.
Remark 5.4.3. Clearly the previous result applies to any categories of weak
equivalences and cofibrations that satisfy (i) and (ii). The essential point is*
* that,
in practice, we often need bicofibrations in order to verify (i).
Similarly, but more simply, the following observation reduces the verificati*
*on
of Definition 5.4.1(v) to special cases. Here we assume that C is based.
Lemma 5.4.4. Let i: X ! Y be a map in C and j :K ! L be a map of
based spaces. Display i j in the diagram
id^j
X ^ K ___________________________//_X ^ L
 k jjjjj 
 jjjjj 
 uuj 
i^id (X ^ L) [X^K (Y ^ K) i^id
 jj55j TTTT 
 jjjjj TTTTT 
fflffljjj i j T))T fflffl
Y ^ K ____________id^j___________//_Y ^ L.
If the maps i ^ idand the pushout k of i ^ idalong id^ j are weak equivalences,
then so is i j, and similarly with the roles of i and j reversed.
Together with Lemma 5.3.5, the notion of a wellgrounded category of weak
equivalences encodes a variant of Lemma 4.5.7 that often applies when the latter
does not.
Lemma 5.4.5. If J is a set of acyclic cylcofibrations between wellgrounded
objects, then all relative Jcell complexes are weak equivalences.
Proof. This follows from (ii), (iii), and (iv) of Definition 5.4.1, togethe*
*r with
the observation that if X0 ! X is a relative Jcell complex, then X=X0 is a J*
*cell
complex and is therefore wellgrounded, so that (iv) applies.
There is an analogous reduction of the problem of determining when a functor
preserves weak equivalences.
5.4. WELLGROUNDED CATEGORIES OF WEAK EQUIVALENCES 71
Lemma 5.4.6. Let F :C ! D be a functor between topologically bicomplete
categories that come equipped with subcategories of wellgrounded weak equivale*
*nces
with respect to given ground structures. Let J be a set of acyclic cylcofibra*
*tions
between wellgrounded objects in C . Assume that F has a continuous right adjoi*
*nt
and that F takes maps in J to weak equivalences between wellgrounded objects.
Then F takes a retract of a relative Jcell complex to an acyclic map in D.
Proof. The functor F preserves cylcofibrations since it has a continuous r*
*ight
adjoint and hence F J consists of acyclic cylcofibrations between wellgrounded
objects. The conclusion follows from Lemma 5.4.5 and the fact that left adjoints
commute with colimits and therefore the construction of cell complexes.
Similarly, cell complexes are relevant to the verification of Definition 5.4*
*.1(v).
Recall that the cylcofibrations in K* are the fpcofibrations, that is, the ba*
*sed
cofibrations.
Lemma 5.4.7. Let I be a set of cylcofibrations between wellgrounded objects
of C and let J be a set of fpcofibrations between wellbased spaces. If i is a*
* retract
of a relative Icell complex, j is a retract of a relative Jcell complex, and *
*either I
or J consists of weak equivalences, then i j is a weak equivalence.
Proof. Assume that I consists of weak equivalences; the proof of the other
case is symmetric. Since the functor  ^ K commutes with coproducts, pushouts,
sequential colimits, and retracts, we can construct j ^ K by first applying  ^*
* K to
the generators, then construct the cell complex, and finally pass to retracts. *
*Since
^K preserves cylcofibrations and wellgrounded objects by Definition 5.3.2(vi*
*), it
takes maps in I to cylcofibrations between wellgrounded objects. By Lemma 5.4*
*.5,
the resulting cell complex is acyclic and therefore so also is any retract of i*
*t. Thus
j ^ K is an acyclic bicofibration. Since such maps are preserved under pushouts,
Lemma 5.4.4 applies to give the conclusion.
The following classical example is implicit in the literature.
Proposition 5.4.8. The qequivalences in GK are wellgrounded with respect
to the ground structure whose wellgrounded objects are the compactly generated
spaces and whose gcofibrations are the hcofibrations.
Proof. Parts (i), (ii), and, here in the unbased case, (v) of Definition 5.*
*4.1
are clear, and (iv) follows easily from Lemma 1.1.5. The essential point is the
gluing lemma of (iii). By passage to fixed point spaces, it suffices to prove *
*this
nonequivariantly. Using the gluing lemma for the proper hmodel structure on K ,
we see that f and f0 can be replaced by their mapping cylinders. Then the induc*
*ed
map of pushouts is the map of double mapping cylinders induced by the original
diagram. This map is equivalent to a map of excisive triads, and in that case t*
*he
result is [67, 1.3], whose proof is corrected in [98].
Proposition 5.4.9. The qequivalences in GK =B and GKB are wellgrounded
with respect to the ground structures of Proposition 5.3.7. In these cases, one*
* need
only assume that the relevant maps in the gluing and colimit lemmas are ground
cofibrations (= hcofibrations), not both ground and Hurewicz cofibrations.
Proof. We verify this for GKB . Part (i) of Definition 5.4.1 holds since an*
*y fp
equivalence is a qequivalence and part (iii) follows directly from the gluing *
*lemma
72 5. WELLGROUNDED TOPOLOGICAL MODEL CATEGORIES
in GK . For part (ii), the total space of _B Xi is the pushout in GK of
*B oo___q*B _____//qXi.
Since the Xi are wellgrounded, the map on the right is an hcofibration, hence
(ii) also follows from the gluing lemma in GK . In part (iv), the relevant quot*
*ient
in GKB is given by the pushout, X=BX0, of the diagram *B  X0 ! X.
Since X=BX0 is wellgrounded, the quotient total space is in U and one can apply
Lemma 1.1.5 just as on the space level. Finally consider (v). As in the proof of
Proposition 5.3.7(vi), X ^B K can be constructed as the pushout of the following
diagram of fcofibrant spaces over B.
*B oo___X q (B x K) _____//X x K
The map on the right is an fcofibration. By the gluing lemma in GK , it suffic*
*es
to observe that X x K preserves qequivalences in both variables since homotopy
groups commute with products.
5.5. Wellgrounded compactly generated model structures
Let C be a topologically bicomplete category or, equivariantly, a Gtopologi*
*cally
bicomplete Gcategory. In the notion of a "wellgrounded model structure", we f*
*or
mulate the properties that a compactly generated model structure on C should
have in order to mesh well with the intrinsic hstructure on C described in x4.*
*3.
When C has such a model structure, and when the classical hstructure actually
is a model structure, the identity functor on C is a Quillen left adjoint from *
*the
wellgrounded model structure to the hmodel structure. Thus this notion gives a
precise axiomatization for the implementaton of the philosophy that we advertis*
*ed
in x4.1. We begin with a variant of Theorem 4.5.5.
Theorem 5.5.1. Let C be a topologically bicomplete category with a ground
structure, a subcategory of wellgrounded weak equivalences, and compact sets I
and J of maps that satisfy the following conditions.
(i)(Acyclicity condition) Every map in J is a weak equivalence.
(ii)(Compatibility condition) A map has the RLP with respect to I if and only *
*if
it is a weak equivalence and_has_the RLP with respect to J.
(iii)Every map in I and J is a cylcofibration between wellgrounded objects.
Then C is a compactly generated model category with generating sets I and J of
cofibrations and acyclic cofibrations. Every cofibration is a bicofibration and*
* every
cofibrant object is wellgrounded. A pushout of a weak equivalence along a bico*
*fi
bration is a weak equivalence and, in particular, the model structure is left p*
*roper.
The model structure is topological or, equivariantly, Gtopological if the foll*
*owing
condition holds.
(iv)i j is an Icell complex if i: X ! Y is a map in I and j :K ! L is
a map of spaces (or Gspaces) in I.
Proof. By Lemma 5.4.5, Theorem 4.5.5 applies to verify the model axioms.
Condition (iii) implies the statements about cofibrations and cofibrant objects*
* by
Lemma 5.3.5, and the gluing lemma implies the statement about pushouts of weak
equivalences. In the last statement, the set I of generating cofibrations in t*
*he
relevant category of (based or unbased) spaces is as specified in Definition 4.*
*5.9.
By passage to coproducts, pushouts, sequential colimits, and retracts, (iv) imp*
*lies
5.6. PROPERTIES OF WELLGROUNDED MODEL CATEGORIES 73
that i j is a cofibration if i: X ! Y is a cofibration in C and j :K ! L is a
qcofibration of spaces (or Gspaces). Together with Lemma 5.4.7, this implies *
*that
the model structure is topological.
We emphasize the difference between the acyclicity conditions stated in Theo
rem 4.5.5 and in Theorem 5.5.1. In the applications of the former, it is the ve*
*rifi
cation of the acyclicity of Jcell complexes that is problemmatic, but in the l*
*atter
our axiomatization has built in that verification. Similarly, our axiomatizatio*
*n has
built in the verification of the acyclicity condition required for the model st*
*ructure
to be topological.
Definition 5.5.2. A compactly generated model structure on C is said to be
wellgrounded if it is right proper and satisfies all of the hypotheses of the *
*preceding
theorem. It is therefore proper and topological or, equivariantly, Gtopologica*
*l.
5.6. Properties of wellgrounded model categories
Assume that C is a wellgrounded model category. To derive properties of its
homotopy category HoC , we must sort out the relationship between homotopies
defined in terms of cylinders and homotopies in the model theoretic sense, which
we call "model homotopies". Of course, the cylinder objects Cyl(X) in C have
maps i0, i1: X ! Cyl(X) and p: Cyl(X) ! X, and i0 (or i1) and p are inverse
homotopy equivalences since tensors with spaces preserve homotopies in the space
variable. Definition 5.4.1(i) ensures that p is therefore a weak equivalence. *
* This
means that Cyl(X) is a model theoretic cylinder object in C , provided that we
adopt the nonstandard definition of [37]. With the language there, it need not*
* be
a good cylinder object since i0 q i1: X q X ! Cyl(X) need not be a cofibration.
As pointed out in [37, p. 90], this already fails for spaces, where the inclus*
*ion
X q X ! X x I is not a qcofibration unless X is qcofibrant. With the standard
definition given in [43, 44, 83], cylinder objects are required to have this co*
*fibration
property. Under that interpretation, the cylinder objects Cyl(X) would not qual*
*ify
as model theoretic cylinder objects in general. (We note parenthetically that "*
*good
cylinders" are defined in [85] in such a way as to include all standard cylinde*
*rs in
the category of spaces). We record the following observations.
Lemma 5.6.1. Consider maps f, g :X ! Y in C .
(i)If f is homotopic to g, then f is left model homotopic to g.
(ii)If X is cofibrant, then Cyl(X) is a good cylinder object.
(iii)If X is cofibrant and Y is fibrant, then f is homotopic to g if and only i*
*f f is
left and right model homotopic to g.
Proof. Part (i) is [37, 4.6], part (ii) follows from Definition 5.3.2(iii),*
* and part
(iii) follows from [37, 4.23].
Let [X, Y ] denote the set of morphisms X  ! Y in Ho C and let ss(X, Y )
denote the set of homotopy classes of maps X ! Y . We shall only use the latter
notation when homotopy and model homotopy coincide.
Lemma 5.6.2 (Cofiber sequence lemma). Assume that C is based. Consider
the cofiber sequence
X ! Y ! Cf ! X ! Y ! Cf ! 2X ! . . .
74 5. WELLGROUNDED TOPOLOGICAL MODEL CATEGORIES
of a wellgrounded map f :X ! Y . For any object Z, the induced sequence
. ..! [ n+1X, Z] ! [ nCf, Z] ! [ nY, Z] ! [ nX, Z] ! . ..! [X, Z]
of pointed sets (groups left of [ X, Z], Abelian groups left of [ 2X, Z]) is ex*
*act.
Proof. As usual, giving I the basepoint 1, we define
CX = X ^ I, X = X ^ S1, and Cf = Y [f CX.
If X is cofibrant, then X is wellgrounded and X ! CX is a cofibration and
therefore a bicofibration. If X and Y are cofibrant, then so is Cf, as one sees*
* by
solving the relevant lifting problem by first using that Y is cofibrant, then u*
*sing
that X ! CX is a cofibration, and finally using that Cf is a pushout. Thus,
taking Z to be fibrant, the conclusion follows in this case from the sequence of
homotopy classes of maps
. ..! ss( X, Z) ! ss(Cf, Z) ! ss(Y, Z) ! ss(X, Z),
which is proven to be exact in the same way as on the space level. If X and
Y are not cofibrant, let Qf :QX  ! QY be a cofibrant approximation to f.
The gluing lemma applies to give that the canonical map CQf ! Cf is a weak
equivalence. Therefore the conclusion follows in general from the special case*
* of
cofibrant objects.
Warning 5.6.3. While the proof just given is very simple, it hides substanti*
*al
subtleties. It is crucial that cofibrant objects X be wellgrounded, so that t*
*he
cylcofibration X ! CX is a bicofibration and the gluing lemma applies.
Of course, the group structures are defined just as classically. The pinch m*
*aps
S1 ~=I={0, 1} ! I={0, 1_2, 1} ~=S1 _ S1 and I ! I={1_2, 1} ~=I _ S1
induce pinch maps
X ! X _ X and Cf ! Cf _ X
that give X the structure of a cogroup object in Ho C and Cf a coaction by
X; 2X is an abelian cogroup object for the same reason that higher homotopy
groups are abelian. Therefore [ X, Z] is a group, [Cf, Z] is a [ X, Z]set, and
[ X, Z] ! [Cf, Z] is a [ X, Z]map.
Lemma 5.6.4 (Wedge lemma). For any Xi and Y in C , [qXi, Y ] ~= [Xi, Y ].
Proof. This is standard, using that a coproduct of cofibrant approximations
is a cofibrant approximation.
Lemma 5.6.5 (Lim 1lemma). Assume that C is based. Let X be the colimit
of a sequence of wellgrounded cylcofibrations in :Xn ! Xn+1. Then, for any
object Y , there is a lim1exact sequence of pointed sets
* ! lim1[ Xn, Y ] ! [X, Y ] ! lim[Xn, Y ] ! *.
Proof. The telescope TelXn is defined to be colimTn, where the Tn and a
ladder of weak equivalences jn :Xn ! Tn and rn :Tn ! Xn are constructed
5.6. PROPERTIES OF WELLGROUNDED MODEL CATEGORIES 75
inductively by setting T0 = X0 and letting jn+1 and rn+1 be the maps of pushouts
induced by the following diagram.
Xn __________Xn ____in___//_Xn+1
i1 2 2
fflffli(0,1)fflffljnqin fflffl
CylXn oo___Xn q Xn _____//Tn q Xn+1
p  rnqid
fflffl  fflffl
Xn oo_r___Xn q Xn idqin//_Xn q Xn+1
Since jn+1 is a pushout of the bicofibration i1: Xn ! Cyl(Xn), the gluing lemma
and colimit lemma specified in Definition 5.4.1(iii) and (iv) apply to show tha*
*t the
induced maps TelXn ! colimXn = X are weak equivalences.
As in the cofiber sequence lemma, we can use cofibrant approximation to redu*
*ce
to a question about ss(, ). Then the telescope admits an alternative descript*
*ion
from which the lim1exact sequence is immediate. It would take us too far afield
to go into full details of what should be a standard argument, but we give a sk*
*etch
since we cannot find our preferred argument in the literature.
Recall that the classical homotopy pushout, or double mapping cylinder, of
f f0
Y oo___X ____//_Y 0
is the ordinary pushout M(f, f0) of
i0,1 fqf0
CylX oo___X q X _____//Y q Y 0.
It fits into a cofiber sequence
Y q Y 0! M(f, g) ! X.
There results a surjection from ss(M(f, g), Z) to the evident pullback, the ker*
*nel of
which is the set of orbits of the right action of ss( Y, Z) x ss( Y 0, Z) on ss*
*( X, Z)
given by x(y, y0) = ( f)*(y)1x( f0)*(y0).
The classical homotopy coequalizer C(f, g) of parallel maps f, g :X ! Y is
the homotopy pushout of the coproduct f qg :X qX ! Y qY and the codiagonal
r: X qX ! X. Using a little algebra, we see that ss(C(f, g), Z) maps surjectiv*
*ely
to the equalizer of f* and g* with kernel isomorphic to the set of orbits of ss*
*( X, Z)
under the right action of ss( Y, Z) specified by xy = ( f)*(y)1x( g)*(y).
In this language, TelXn is the classical homotopy coequalizer of the identity
and the coproduct of the in, both being self maps of the coproduct of the Xn.
By algebraic inspection, the lim1exact sequence follows directly. A quicker, l*
*ess
conceptual, argument is possible, as in [71, p. 146] for example.
Remark 5.6.6. Let C be an arbitrary pointed model category with (for sim
plicity) a functorial cylinder construction Cyl. If X is cofibrant, let X deno*
*te the
quotient Cyl(X)=(X _ X). Quillen [83] constructed a natural cogroup structure
on X in Ho C . For a cofibration X ! Y between cofibrant objects, he also
constructed a natural coaction of X on the quotient Y=X. One can then define
cofiber sequences in HoC just as in the homotopy category of a topological model
category, and one can define fiber sequences dually.
76 5. WELLGROUNDED TOPOLOGICAL MODEL CATEGORIES
The cofiber sequences and fiber sequences each give HoC a suitably weakened
form of the notion of a triangulation, called a "pretriangulation" [44, 83], an*
*d they
are suitably compatible. If HoC is closed symmetric monoidal one can take this a
step further and formulate what it means for the pretriangulation to be compati*
*ble
with that structure, as was done in [74] for triangulated categories. However,
proving the compatibility axioms from this general point of view would at best *
*be
exceedingly laborious, if it could be done at all.
These purely model theoretic constructions of the suspension and looping fun*
*c
tors and are more closely related to the familiar topological constructions*
* than
might appear. The homotopy category of any model category is enriched and biten
sored over the homotopy category of spaces (obtained from the qmodel structure)
[36, 44], and the suspension and loop functors are given by the (derived) tensor
and cotensor with the unit circle. That is, X ' X ^ S1 and X ' F (S1, X).
This general point of view is not one that we wish to emphasize. For topolog*
*ical
model categories, the structure described in this section is far easier to defi*
*ne and
work with directly, as in classical homotopy theory, and we have axiomatized wh*
*at
is required of a model structure in order to allow the use of such standard and
elementary classical methods. In our topological context, the homotopy category
HoC is automatically enriched over HoK* and ( , ) is a Quillen adjoint pair th*
*at
descends to an adjoint pair on homotopy categories that agrees with the purely
model theoretic adjoint pair just described.
The crucial point for our stable work is that a large part of this structure*
* exists
before one constructs the desired model structure. It can therefore be used as*
* a
tool for carrying out that construction. This is in fact how stable model categ*
*ories
were constructed in [39, 61, 62], but there the compatibility between qtype and
htype structures was too evident to require much comment. The key step in our
construction of the stable model structure on parametrized spectra in Chapter 12
is to show that cofiber sequences induce long exact sequences on stable homotopy
groups. That will allow us to verify that the stable equivalences are suitably *
*well
grounded, and from there the model axioms follow as in the earlier work just ci*
*ted.
CHAPTER 6
The qf model structure on KB
Introduction
In this chapter, we introduce and develop our preferred qtype model structu*
*re,
namely the qfmodel structure. It is a Quillen equivalent variant of the qmodel
structure that has fewer, and better structured, cofibrations. For clarity of e*
*xposi
tion, we work nonequivariantly in this chapter, which is taken from [88].
We begin by comparing the homotopy theory of spaces and the homotopy
theory of exspaces over B, starting with a comparison of the qmodel structures
that we have on both. In the category K of spaces, we have the familiar situati*
*on
described in x4.1. The homotopy category HoK that we care about is defined in
terms of qequivalences, the intrinsic notion of homotopy is given by the class*
*ical
cylinders, and, since all spaces are qfibrant, the category HoK is equivalent*
* to
the classical homotopy category hKc of qcofibrant spaces (or CW complexes).
Since the qcofibrations are hcofibrations, the qmodel structure and the hmo*
*del
structure on K mesh smoothly. Indeed, the classical and model theoretic homotopy
theory have been used in tandem for so long that this meshing of structures goes
without notice. In particular, although cofiber and fiber sequences are defined*
* in
terms of the hmodel structure while the homotopy category is defined in terms *
*of
the qmodel structure, the compatibility seems automatic.
Now consider the category KB . The homotopy category HoKB that we care
about is defined in terms of qequivalences of total spaces, but we need some j*
*ustifi
cation for making that statement. A map of qfibrant exspaces is a qequivalen*
*ce of
total spaces if and only if all of its maps on fibers are qequivalences. This *
*reformula
tion captures the idea that the homotopical information in parametrized homotopy
theory should be encoded on the fibers, and it is such fiberwise qequivalences*
* that
we really care about. It is only for qfibrant exspaces, or exspaces whose pr*
*ojec
tions are at least quasifibrations, that the homotopy groups of total spaces gi*
*ve the
"right answer". There are three notions of homotopy in sight, h, f, and fp. The
last of these is the intrinsic one defined in terms of the relevant cylinders i*
*n KB ,
and HoKB is equivalent to the classical homotopy category hKB cf of qcofibrant
and qfibrant objects, defined with respect to fphomotopy. It is still true t*
*hat
qcofibrations are hcofibrations. However, it is not true that qcofibrations *
*are fp
cofibrations, and it is the latter that are intrinsic to cofiber sequences. The*
* classical
and model theoretic homotopy theory no longer mesh.
Succinctly, the problem is that the qmodel structure is not an example of a
wellgrounded compactly generated model category. The task that lies before us
is to find a model structure which does satisfy the axioms that we set out in x*
*5.5
and therefore can be used in tandem with the fpstructure to do parametrized
homotopy theory. Before embarking on this, we point out the limitations of the
77
78 6. THE qfMODEL STRUCTURE ON KB
qmodel structure more explicitly in x6.1. There are two kinds of problems, tho*
*se
that we are focusing on in our development of the model category theory, and the
more intrinsic ones that account for Counterexample 0.0.1 and which cannot be
overcome model theoretically.
Ideally, to define the qfmodel structure, we would like to take the qfcofi*
*bra
tions to be those qcofibrations that are also fcofibrations. However, with t*
*hat
choice, we would not know how to prove the model category axioms. We get closer
if we try to take as generating sets of cofibrations and acyclic cofibrations t*
*hose
generators in the qmodel structure that are fcofibrations, but with that choi*
*ce
we still would not be able to prove the compatibility condition Theorem 5.5.1(i*
*i).
However, using this generating set of cofibrations and a subtler choice of a ge*
*nerat
ing set of acyclic cofibrations, we obtain a precise enough homotopical relatio*
*nship
to the qequivalences that we can prove the cited compatibility. The constructi*
*on of
the qfmodel structure is given in x6.2, but all proofs are deferred to the fol*
*lowing
three sections.
6.1. Some of the dangers in the parametrized world
We introduce notation for the generating (acyclic) cofibrations for the qmo*
*del
structures on K =B and KB . These maps are identified in Proposition 5.1.4, sta*
*rt
ing from the sets I and J in K specified in Definition 4.5.9. We then make some
comments about these maps that help explain the structure of our theory.
Definition 6.1.1. For maps i: C ! D and d: D ! B of (unbased) spaces,
we have the restriction dOi: C ! B and may view i as a map over B. We agree to
write i(d) for either the map i viewed as a map over B or the map iqid:C qB !
DqB of exspaces over B that is obtained by taking the coproduct with B to adjo*
*in
a section. In either K =B or KB , define IB to be the set of all such maps i(d)*
* with
i 2 I, and define JB to be the set of all such maps j(d) with j 2 J. Observe th*
*at
in KB , each map in JB is the inclusion of a deformation retract of spaces unde*
*r,
but not over, B.
Warning 6.1.2. We cannot restrict the maps d to be open here. That is one
of the reasons we chose KB over O*(B) in x1.3.
Warning 6.1.3. The maps in IB and JB are clearly not fcofibrations, only
hcofibrations. Looking at the NDRpair characterization of fcofibrations give*
*n in
Lemma 5.2.4, we see that, with our arbitrary projections d, there is in general*
* no
way to carry out the required deformation over B. Since the maps in IB and JB
are maps between wellsectioned spaces, they cannot be fpcofibrations in gener*
*al,
by Proposition 5.2.3(i).
Remark 6.1.4. Observe that the maps i in IB or JB are closed inclusions
in U , so that those maps in IB or JB which are fcofibrations are necessarily
~fcofibrations and therefore both f~pcofibrations and ~hcofibrations, by Pro*
*posi
tion 5.1.8 and Theorem 5.2.8.
Warning 6.1.3 shows that the qmodel structure is not wellgrounded since
its generating (acyclic) cofibrations are not fpcofibrations. This may sound l*
*ike a
minor technicality, but that is far from the case. We record an elementary exam*
*ple.
Counterexample 6.1.5. Let B = I and define an exmap i: X ! Y over I
by letting X = {0} q I, Y = I q I, and i be the inclusion. The second copies of*
* I
6.2. THE qf MODEL STRUCTURE ON THE CATEGORY K =B 79
give the sections, and the projections are given by the identity map on each co*
*py of
I. This is a typical generating acyclic qcofibration, and it is not an fpcofi*
*bration.
Let Z be the pushout of i and p: X ! I, where the latter is viewed as a map
of exspaces over I. Then Z is the onepoint union I _ I obtained by identifying
the points 0. The section I ! Z is not an fcofibration, so that Z is not well
sectioned. The same is true if we replace Y by Y 0= {1=(n + 1)  n 2 N} q I and
obtain Z0. The map Z0 ! CIZ0 of Z0 into its cone over I is not an hcofibration
(and therefore not a qcofibration).
Thus we cannot apply the classical gluing lemma to develop cofiber sequences,
as we did in x5.6. This and related problems prevent use of the qmodel structu*
*re
in a rigorous development of parametrized stable homotopy theory. For example,
consider qfibrant approximation. If we have a map f :X ! Y with qfibrant
approximation Rf :RX ! RY , there is no reason to believe that CB Rf is q
equivalent to RCB f.
We are about to overcome modeltheoretically the problems pointed out in the
warnings above. Turning to the intrinsic problems that must hold in any qtype
model structure, we explain why the base change functor f* and the internal sma*
*sh
product cannot be Quillen left adjoints.
Warning 6.1.6. If f :A ! B is a map and d: D ! B is a disk over B, we
have no homotopical control over the pullback A xB D ! A in general.
Warning 6.1.7. In sharp contrast to the nonparametrized case, the generating
sets do not behave well with respect to internal smash products, although they *
*do
behave well with respect to external smash products. We have
(D q A) Z (E q B) ~=(D x E) q (A x B).
If the projections of D and E are d and e, then the projection of D x E is d x *
*e.
However, if A = B, then
(D q B) ^B (E q B) ~=(d x e)1( B) q (A x B).
We have no homotopical control over the space (d x e)1( B) in general.
This has the unfortunate consequence that, when we go on to parametrized
spectra in Part III, we will not be able to develop a homotopically wellbehaved
theory of pointset level parametrized ring spectra. However, we will be able *
*to
develop a satisfactory pointset level theory of parametrized module spectra ov*
*er
nonparametrized ring spectra.
6.2. The qf model structure on the category K =B
Rather than start with a model structure on K to obtain model structures
on K =B and KB , we can start with a model structure on K =B and then apply
Proposition 5.1.3 to obtain a model structure on KB . This gives us the opportu*
*nity
to restrict the classes of generating (acyclic) cofibrations present in the qm*
*odel
structure on K =B to ones that are fcofibrations, retaining enough of them that
we do not lose homotopical information. This has the effect that the generating
(acyclic) cofibrations are fcofibrations between wellgrounded spaces over B, *
*as
is required of a wellgrounded model structure. Such maps have closed images,
hence are ~fcofibrations, and therefore all of the cofibrations in the resulti*
*ng model
structure on K =B are ~fcofibrations.
80 6. THE qfMODEL STRUCTURE ON KB
We call the resulting model structure the "qfmodel structure", where f refe*
*rs
to the fiberwise cofibrations that are used and q refers to the weak equivalenc*
*es.
The latter are the same as in the qmodel structure, namely the weak equivalenc*
*es
on total spaces, or qequivalences. This model structure restores us to a situ*
*a
tion in which the philosophy advertised in x4.1 applies, with the q and hmodel
structures on spaces replaced by the qf and fmodel structures on spaces over B.
Since fcofibrations in KB are fpcofibrations, by Proposition 5.1.8, the philo*
*so
phy also applies to the qf and fpmodel structures on KB , or at least on UB (s*
*ee
Theorem 5.2.8 and Remark 5.2.9).
We need some notations and recollections in order to describe the generating
(acyclic) qfcofibrations and the qffibrations.
Notation 6.2.1. For each n 1, embed Rn1 in Rn = Rn1 x R by sending
x to (x, 0). Let en = (0, 1) 2 Rn. For n 0, define the following subspaces of*
* Rn.
Rn+ = {(x, t) 2 Rn  t 0} Rn= {(x, t) 2 Rn  t 0}
Dn = {(x, t) 2 Rn  x2 + t2 S1}n1= {(x, t) 2 Rn  x2 + t2 = 1}
Sn1+= Sn1 \ Rn+ Sn1= Sn1 \ Rn
Here R0 = {0} and S1 = ;. We think of Sn Rn+1 as having equator Sn1,
upper hemisphere Sn+with north pole en+1 and lower hemisphere Sn.
We recall a characterization of Serre fibrations.
Proposition 6.2.2. The following conditions on a map p: E ! Y in K are
equivalent; p is called a Serre fibration, or qfibration, if they are satisfie*
*d.
(i)The map p satisfies the covering homotopy property with respect to disks Dn;
that is, there is a lift in the diagram
Dn ___ff__//E;;w
 w 
 ww p
fflfflw fflffl
Dn x I __h__//Y.
(ii)If h is a homotopy relative to the boundary Sn1 in the diagram above, then
there is a lift that is a homotopy relative to the boundary.
(iii)The map p has the RLP with respect to the inclusion Sn+ ! Dn+1 of the
upper hemisphere into the boundary Sn of Dn+1; that is, there is a lift in *
*the
diagram
Sn+___ff_//E<<
z
 zz p
 z 
fflfflz fflffl
Dn+1 __~h_//Y.
Proof. Serre fibrations p: E  ! Y are usually characterized by the first
condition. Since the pairs (Dn x I, Dn) and (Dn x I, Dn [ (Sn1 x I)) are home
omorphic, one easily obtains that the first condition implies the second. Simil*
*arly
a homeomorphism of the pairs (Dn+1, Sn+) and (Dn x I, Dn) gives that the first
and third conditions are equivalent. A homotopy h: Dn x I ! Y relative to the
boundary Sn1 factors through the quotient map Dn x I ! Dn+1 that sends
6.2. THE qf MODEL STRUCTURE ON THE CATEGORY K =B 81
p _______
(x, t) to (x, (2t  1) 1  x2). Conversely, any map ~h:Dn+1 ! Y gives rise *
*to
a homotopy h: Dn x I ! Y relative to the boundary Sn1. It follows that the
second condition implies the third.
Property (ii) states that Serre fibrations are the maps that satisfy the "di*
*sk
lifting property" and that is the way we shall think about the qffibrations. I*
*n view
of property (iii), we sometimes abuse language by calling a map h: Dn+1 ! Y a
disk homotopy. The restriction to the upper hemisphere Sn+gives the "initial di*
*sk"
and the restriction to the lower hemisphere Sngives the "terminal disk".
Definition 6.2.3. A disk d: Dn ! B in K =B is said to be an fdisk if
i(d): Sn1 ! Dn is an fcofibration. An fdisk d: Dn+1 ! B is said to be a
relative fdisk if the lower hemisphere Snis also an fdisk, so that the restr*
*iction
i(d): Sn1 ! Snis an fcofibration; the upper hemisphere i(d): Sn1 ! Sn+
need not be an fcofibration.
Definition 6.2.4. Define IfBto be the set of inclusions i(d): Sn1 ! Dn
in K =B, where d: Dn ! B is an fdisk. Define JfBto be the set of inclusions
i(d): Sn+! Dn+1 of the upper hemisphere into a relative fdisk d: Dn+1 ! B;
note that these initial disks are not assumed to be fdisks. A map in K =B is s*
*aid
to be
(i)a qffibration if it has the RLP with respect to JfBand
(ii)a qfcofibration if it has the LLP with respect to all qacyclic qffibrat*
*ions,
that is, with respect to those qffibrations that are qequivalences.
Note that JfBconsists of relative IfBcell complexes and that a map is a qffib*
*ration
if and only if it has the "relative fdisk lifting property."
With these definitions in place, we have the following theorem. Recall the
definition of a wellgrounded model category from Definition 5.5.2 and recall f*
*rom
Propositions 5.3.7 and 5.4.9 that we have ground structures on K =B and KB with
respect to which the qequivalences are wellgrounded. Also recall the definiti*
*on of
a quasifibration from Definition 3.5.1.
Theorem 6.2.5. The category K =B of spaces over B is a wellgrounded model
category with respect to the qequivalences, qffibrations and qfcofibrations.*
* The
sets IfBand JfBare the generating qfcofibrations and the generating acyclic qf
cofibrations. All qfcofibrations are also f~cofibrations and all qffibratio*
*ns are
quasifibrations.
Using Proposition 5.1.3 and Proposition 5.1.4, we obtain the qfmodel struct*
*ure
on KB . We define a qffibration in KB to be a map which is a qffibration when
regarded as a map in K =B, and similarly for qfcofibrations.
Theorem 6.2.6. The category KB of exspaces over B is a wellgrounded model
category with respect to the qequivalences, qffibrations, and qfcofibrations*
*. The
sets IfBand JfBof generating qfcofibrations and generating acyclic qfcofibrat*
*ions
are obtained by adjoining disjoint sections to the corresponding sets of maps in
K =B. All qfcofibrations are f~cofibrations and all qffibrations are quasif*
*ibra
tions.
Since the qfmodel structures are wellgrounded, they are in particular prop*
*er
and topological. Furthermore, the qfcofibrant spaces over B are wellgrounded
82 6. THE qfMODEL STRUCTURE ON KB
and the qffibrant spaces over B are quasifibrant. Since qfcofibrations are q
cofibrations, we have an obvious comparison.
Theorem 6.2.7. The identity functor is a left Quillen equivalence from K =B
with the qfmodel structure to K =B with the qmodel structure, and similarly f*
*or
the identity functor on KB .
We state and prove two technical lemmas in x6.3, prove that K =B is a com
pactly generated model category in x6.4, and prove that the qffibrations are q*
*uasi
fibrations and the model structure is right proper in x6.5. The product condi*
*tion
of Theorem 5.5.1(iv) follows as usual by inspection of what happens on generati*
*ng
(acyclic) cofibrations and, as in the case A = * of Warning 6.1.7, the projecti*
*ons
cause no problems here.
6.3.Statements and proofs of the thickening lemmas
We need two technical "thickening lemmas". They encapsulate the idea that
no information about homotopy groups is lost if we restrict from the general di*
*sks
and cells used in the qmodel structure to the fdisks and fcells that we use *
*in the
qfmodel structure.
Lemma 6.3.1. Let (Sm , q) be a sphere over B. Then there is an hequivalence
~: (Sm , ~q) ! (Sm , q) in K =B such that (Sm , ~q) is an IfBcell complex wit*
*h two
cells in each dimension.
Lemma 6.3.2. Let (Dn, q) be a disk over B. Then there is an hequivalence
:(Dn, ~q) ! (Dn, q) relative to the upper hemisphere Sn1+such that (Dn, ~q)*
* is a
relative fdisk.
The rest of the section is devoted to the proofs of these lemmas. The reader
may prefer to skip ahead to x6.4 to see how they are used to prove Theorem 6.2.*
*5.
Proof of Lemma 6.3.1. To define the map ~: (Sm , ~q) ! (Sm , q), we begin
by defining some auxiliary maps for each natural number n m. They will in fact
be continuous families of maps, defined for each s 2 [1_2, 1]. The parameter s *
*will
show that ~ is an hequivalence.
First we define the map
OEn+:Dn \ Rn+! As [ s . Sn1+
from the upper half of the disk Dn to the union of the equatorial annulus
_______________
As = Dn1  s . Dn1= {(x, 0) 2 Rn :s x 1}
and the upper hemisphere
s . Sn1+= {(x, t) 2 Rn :t 0 and (x, t)}= s
to be the projection from the south pole en. Similarly, we define
OEn:Dn \ Rn! As [ s . Sn1
to be the projection from the north pole en. The map OEn+is drawn schematically
in the following picture. Each point in the upper half of the larger disk lies *
*on a
6.3. STATEMENTS AND PROOFS OF THE THICKENING LEMMAS 83
unique ray from en. The map OEn+sends it to the intersection of that ray with
As [ s . Sn1; two such points of intersection are marked with dots in the pict*
*ure.
________________________________________________*
*_____________________________________________________________________________*
*%%
_________________________________________________*
*_________________________________________________________________________%
__________________________________________________*
*________________________________________________________________________%
7___________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________%%o7
___________________________________________________*
*_____________________________________________________________________________*
*_______%7
___________________________________________________*
*_____________________________________________________________________________*
*________________________________%%77os.Dn
___________________________________________________*
*_____________________________________________________________________________*
*_______%77
___________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________%%77
__________________________________________________*
*________________________________________________________________________%77
_________________________________________________*
*_________________________________________________________________________%%7
______________________________________________*
*_____________________________________________________________________________*
*__7eDn
*
* *
* n
Next we use the maps OEn to define a continuous family of maps fns:Dn ! Dn
for s 2 [1_2, 1] by induction on n. We let f0s:D0 ! D0 be the unique map and we
define f1s:D1 ! D1 by
8
>1 if t ,s
:1 if t s;
it maps [s, s] homeomorphically to [1, 1]. We define fns:Dn ! Dn by
8
>>>s1 . (x, t) if (x, t), s
>>>1 n n
fn1s(OEn+(x, t))if (x, t) s, t 0 and OEn+(x,,t) s
>>> 1 n n
>>:s . OE (x,it)f (x, t) s, t 0 and OE,(x, t) = s
fn1s(OEn(x, t))if (x, t) s, t 0 and OEn(x,.t) s
The map fnsis drawn in the following picture. The smaller ball s . Dn is mapped
homeomorphically to Dn by radial expansion from the origin. Next comes the
region in the upper half of the larger ball that is inside the cone and outside*
* the
smaller ball. Each segment of a ray from the south pole en that lies in that r*
*egion
is mapped to a point which is determined by where we mapped the intersection of
that raysegment with the smaller ball (which was radially from the origin to t*
*he
boundary of Dn). Third is the region in the upper half of the larger ball that *
*is
outside the cone. Each segment of a ray from the south pole en that lies in th*
*at
region is first projected to the annulus in the equatorial plane of the two bal*
*ls; we
then apply the previously defined map fn1sto map the projected points to the
equator of Dn. The lower half of the ball is mapped similarly.
_____________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*__________________________________________________
______________________________________________________*
*_______________________________________________________
///___________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________fflffl__________*
*_____________________________________________________________________________*
*_______________________________@
/___________________________________________________*
*_________________________ffl__________________________________________________
____________________________________________________*
*__________//fflffl___________________________________________________________*
*__
_____________________________________________________*
*_____________________________________________________________________________*
*_________________________________________________//fflffl____________________*
*_____________________________________________________________________________*
*_______________________________@
______________________________________________________*
*_________________________________________________________/ffl
______________________________________________________*
*_____________________________________________________________________________*
*__________//fflffl
______________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*________________/fflsDn
______________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_______________________________@
_____________________________________________________*
*__________________________________________________________//fflffl
_____________________________________________________*
*_____________________________________________________________________________*
*_________________________________________________/ffl
____________________________________________________*
*__________//fflffl
___________________________________________________*
*_________________________/ffl
_________________________________________________*
*_____________________________________________________________________________*
*______________________________________________________________//fflfflffl
en Dn
It is clear that fnsgives a homotopy from fn1=2to the identity and, given any d*
*isk
(Dn, q) in K =B, the map fnsinduces an hequivalence from the fdisk (Dn, qOfn1*
*=2)
to the disk (Dn, q).
84 6. THE qfMODEL STRUCTURE ON KB
Finally we define the required cell structure on the domain of the desired m*
*ap
~: (Sm , ~q) ! (Sm , q). For each n m, the boundary sphere (Sn, q O fn+11=2*
*Sn) is
constructed from two copies of the fdisk (Dn, q O fn1=2) by gluing them along *
*their
boundary. The inclusions (Dn, q O fn1=2) ! (Sn, q O fn+11=2Sn) of the two cel*
*ls are
given by projecting Dn to the upper hemisphere from the south pole en+1 and,
similarly, by projecting Dn to the lower hemisphere from the north pole en+1. T*
*he
map
~ = fm+11=2Sm :(Sm , q O fm+11=2Sm ) ! (Sm , q).
is then the required fcell sphere approximation.
Proof of Lemma 6.3.2. Define s:Dn ! Dn for s 2 [1_2, 1] by
8
>>>s1 . (x, t) if (x, t), s
<(x, t)1 . (x, t) if (x, t) s, t 0 and,x s
s(x, t) = > 1 n+1 n+1
>>:s . OE (x, t) if (x, t) s, t 0 and OE (x,,t) =*
* s
OEn+1(x, t)1 . OEn+1(x,it)f (x, t) s, t 0 and OEn+1(x*
*,,t) s
where OEnis the projection as in the previous proof. Then s maps s . Dn home
omorphically to Dn, it is radially constant on the region in the upper half spa*
*ce
between the disks Dn and s . Dn with respect to projection from the origin, and
it is radially constant on the region in the lower half space between the two d*
*isks
with respect to projection from the north pole.
6.4.The compatibility condition for the qfmodel structure
This section is devoted to the proof that K =B is a compactly generated model
category. Since our generating sets IfBand JfBcertainly satisfy conditions (i) *
*and
(iii) of Theorem 5.5.1, it only remains to verify the compatibility condition (*
*ii).
That is, we must show that a map has the RLP with respect to IfBif and only if
it is a qequivalence and has the RLP with respect to JfB. Let p: E ! Y have
the RLP with respect to IfB. Since all maps in JfBare relative IfBcell complex*
*es, p
has the RLP with respect to JfB. To show that ssn(p) is injective, let ff: Sn *
*! E
represent an element in ssn(E) such that p O ff: Sn ! Y is nullhomotopic. Then
there is a nullhomotopy fi :CSn ! Y that gives rise to a lifting problem
Sn ___________ff__________//E
i p
fflffl fflffl
Dn+1 _____//Dn+1 ~=CSn__fi_//Y
where :Dn+1 ! Dn+1 is defined by
( 1
,_
(x) = 2x if x 21
x1 . xif x _2.
Then i is an fdisk and we are entitled to a lift, which can be viewed as a nul*
*lho
motopy of ff after we identify Dn+1 with CSn.
To show that ssn(p) is surjective, choose a representative ff: Sn ! Y of an
element in ssn(Y ). The projection of Y induces a projection q :Sn ! B and by
Lemma 6.3.1 there is an hequivalence ~: (Sn, ~q) ! (Sn, q) such that (Sn, ~q)*
* is
6.4. THE COMPATIBILITY CONDITION FOR THE qfMODEL STRUCTURE 85
an IfBcomplex with two cells in each dimension. We may therefore assume that
the source of ff is an IfBcell complex. Inductively, we can then solve the li*
*fting
problems for the diagrams
Sk1D_____________//E
 DDD 
 DDD p
fflfflD!! fflffl
Sk __i___//Sk____//Y,
ffSk
where Sk1 ! Sk is the inclusion of the equator and i :Sk  ! Sk are the
inclusions of the upper and lower hemispheres. We obtain a lift Sn ! E.
Conversely, assume that p: E ! Y is an acyclic qffibration. We must show
that p has the RLP with respect to any cell i in IfB. We are therefore faced wi*
*th a
lifting problem
Sn ___ff_//E
i p
fflffl fflffl
Dn+1 __fi_//Y.
Identifying Dn+1 with CSn we see that fi gives a nullhomotopy of p O ff. Since
ssn(p) is injective there is a nullhomotopy fl :CSn ! E such that ff = fl O i,*
* but
it may not cover fi. Gluing fi and p O fl along p O ff gives ffi :Sn+1 ! Y suc*
*h that
ffiSn+1+= fi and ffiSn+1= pOfl. Surjectivity of ssn+1(p) gives a map : Sn+1*
* ! E
and a homotopy h: Sn+1 ^ I+ ! Y from p O to ffi. We now construct a diagram
()=Sn n+1 [fl
Sn+1+_____//Sn+1+[ H____//Sn+1 x 0 [ S x_1__//E
w 
jwww    p
www    
ww fflffl fflffl fflffl fflffl
Dn+2 ____//_Dn+2__,___//Dn+2____OE____//Sn+1 ^ I+___h____//_Y
where the downward maps, except p, are inclusions. Part of the bottom row of
the diagram is drawn schematically below. Let H be the region on Sn+1be
tween the equator Sn and the circle through e1 and en+2 with center on the
line R . (e1  en+2). Let , be a homeomorphism whose restriction to Sn+1+maps it
homeomorphically to Sn+1+[ H. Define OE: Dn+2 ! Dn+2=Sn ~=Sn+1 ^ I+ as the
composite of the quotient map that identifies the equator Sn of Dn+2 to a point
and a homeomorphism that maps the upper hemisphere Sn+1+to Sn+1 x0, maps H
to Sn+1x 1, and is such that (h O OE O ,)Sn+1= fi. The map is defined as a*
*bove.
_____________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*____
_______________________________________________________________________*
*_______________________________pO____________________________________________*
*____________________________________________________________________pO
pOf_______________________________________________________________________*
*_______________l____________________________________________pO_______________*
*_____________________________________________________________________________*
*________________fi
________________________________________________________________________*
*____________________________________,________________________________________*
*__________________OE_________________________________________________________*
*_______________________________________________
________________________________________________________________________*
*_____________________________________________________________________________*
*___pOff!____________________________________________________________________*
*___________________________________________!________________________________*
*_______________________________@
________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_______________________________@
_______________________________________________________________________*
*___________________________________________________________pOflpOffH_________*
*_____________________________________________________________________________*
*______________________
______________________________________________________________________*
*_____________________________________________________________________________*
*__________________________________________________________________________pOfl
fi ____________________________________________________________________*
*_____________________________________________________________________________*
*___________________________________________________________________________fi*
*_____________________________________________________________________________*
*_____________________
Dn+2 Dn+2 Sn+1 ^ I+
Since the restriction Sn ! Sn+1~=Dn+1 of j agrees with the fcofibration i in
our original lifting problem, we see that j is a JfBcell. Since p is a qffib*
*ration
86 6. THE qfMODEL STRUCTURE ON KB
we get a lift in the outer trapezoid. Denote its restriction to Sn+1~= Dn+1 by
k :Dn+1 ! E. Then k solves our original lifting problem.
6.5.The quasifibration and right properness properties
We have now established the qfmodel structures on both K =B and KB . We
will derive the right properness of K =B, and therefore of KB , from the fact t*
*hat
every qffibration is a quasifibration.
Proposition 6.5.1. If p: E ! Y is a qffibration in K =B, then p is a quasi
fibration. Therefore, for any choice of e 2 E, there results a long exact seque*
*nce of
homotopy groups
. ..! ssn+1(Y, y) ! ssn(Ey, e) ! ssn(E, e) ! ssn(Y, y) ! . ..! ss0(Y, y),
where y = p(e) and Ey = p1(y).
Proof. We must prove that p induces an isomorphism
ssn(p): ssn(E, Ey, e) ! ssn(Y, y)
for all n 1 and verify exactness at ss0(E, e). We begin with the latter. Let *
*e02 E
and suppose that p(e0) is in the component of y0. Let fl :I ! Y be a path in
Y from p(e0) to y0 such that fl is constant at p(e0) for time t 1_2. Let q b*
*e the
projection of Y . Then (I, q O fl) is a relative fdisk, and we obtain a lift ~*
*fl:I ! E
such that fl = p O ~fl. But then e0 is in the same component as the endpoint of*
* ~fl,
which lies in Ey.
Now assume that n 1. Recall that an element of ssn(X, A, *) can be repre
sented by a map of triples (Dn, Sn1, Sn1+) ! (X, A, *). We begin by showing
surjectivity. Let ff: (Dn, Sn1) ! (Y, y) represent an element of ssn(Y, y). W*
*e can
view Dn as a disk over B, and Lemma 6.3.2 gives an approximation :Dn ! Dn
by a relative fdisk. Then we can solve the lifting problem
Sn1+__ce_//E==
  p
  f~f 
fflffl fflffl
Dn __ffO_//Y,
where the top map is the constant map at e 2 E. A lift is a map of triples
~ff:(Dn, Sn1, Sn1+) ! (E, Ey, e) such that p*([~ff]) = [ff].
For injectivity, let ff: (Dn, Sn1, Sn1+) ! (E, Ey, e) represent an elemen*
*t of
ssn(E, Ey, e) such that p*([ff]) = 0. Then there is a homotopy h: Dn x I ! Y r*
*el
Sn1 such that hDn x 0 = p O ff and h maps the rest of the boundary of Dn x I *
*to
y. Let A = Dn x {0, 1} [ Sn1+x I @(Dn x I) and define fi :A ! E by setting
fi(x) = ff(x) if x 2 Dnx0 and fi(x) = e otherwise. We then have a homeomorphism
of pairs OE: (Dn x I, A) ! (Dn+1, Sn+) and an approximation :Dn+1 ! Dn+1
by an fdisk by Lemma 6.3.2. We can now solve the lifting problem
1
Sn+fiO(OEA)//_E==
z
 zz 
 z ~ff 
fflfflz fflffl
Dn+1 hOOE1O//_Y,
6.5. THE QUASIFIBRATION AND RIGHT PROPERNESS PROPERTIES 87
and this shows that [ff] = 0 in ssn(E, Ey, e).
Corollary 6.5.2. The qfmodel structure on K =B is right proper.
Proof. Since qffibrations are preserved under pullbacks, this is a five le*
*mma
comparison of long exact sequences as in Proposition 6.5.1.
CHAPTER 7
Equivariant qf type model structures
Introduction
We return to the equivariant context in this chapter, letting G be a Lie gro*
*up
throughout. Actually, our definitions of the q and qfmodel structures work for
arbitrary topological groups G, but we must restrict to Lie groups to obtain st*
*ruc
tures that are Gtopological and behave well with respect to change of groups a*
*nd
smash products. A discussion of details special to the noncompact Lie case is
given in x7.1, but after that the generalization from compact to noncompact Lie
groups requires no extra work. However, we alert the reader that passage to sta*
*ble
equivariant homotopy theory raises new problems in the case of noncompact Lie
groups that will not be dealt with in this book; see x11.6.
The equivariant qmodel structure on GKB is just the evident over and under *
*q
model structure. However, the equivariant generalization of the qfmodel struct*
*ure
is subtle. In fact, the subtlety is already relevant nonequivariantly when we s*
*tudy
base change along the projection of a bundle. The problem is that there are so *
*few
generating qfcofibrations that many functors that take generating qcofibratio*
*ns
to qcofibrations do not take generating qfcofibrations to qfcofibrations. We*
* show
how to get around this in x7.2. For each such functor that we encounter, we find
an enlargement of the obvious sets of (acyclic) generating qfcofibrations on t*
*he
target of the functor so that it is still a model category, but now the functor*
* does
send generating (acyclic) qfcofibrations to (acyclic) gfcofibrations.
The point is that there are many different useful choices of Quillen equiva
lent qftype model structures, and they can be used in tandem. For all of our
choices, the weak equivalences are the G equivalences and all cofibrations are*
* both
qcofibrations and fcofibrations. Given a finite number of adjoint pairs with *
*com
posable left adjoints such that each is a Quillen adjunction with its own choice
of qftype model structure, we can successively expand generating sets in target
categories of the left adjoints to arrange that the composite be one of Quillen*
* left
adjoints with respect to well chosen qftype model structures.
In x7.2, we describe the qf(C )model structure associated to a "generating *
*set"
C of G complexes. Each such model structure is Gtopological. In x7.3, we show
that external smash products are Quillen adjunctions when C is a "closed" gener
ating set, as can always be arranged, and we show that all base change adjuncti*
*ons
(f!, f*) are Quillen adjunctions. We show further that there are generating set*
*s for
which (f*, f*) is a Quillen adjunction when f is a bundle with cellular fibers.*
* In
x7.4, we show similarly that various change of group functors are given by Quil*
*len
adjunctions when the generating sets are well chosen. In x7.5, we show that HoG*
*KB
has the properties required for application of the Brown representability theor*
*em.
88
7.1. FAMILIES AND NONCOMPACT LIE GROUPS 89
Those adjunctions between our basic functors that are not given by Quillen adjo*
*int
pairs in any choice of qfmodel structure are studied in Chapter 9.
7.1. Families and noncompact Lie groups
Two sources of problems in the equivariant homotopy theory of general topolo*
*gy
groups G are that we only know that orbit types G=K are HCW complexes for
H G when G is a Lie group and K is a compact subgroup and we only know
that a product of orbits G=H x G=K is a GCW complex when G is a Lie group
and K (or H) is a compact subgroup. This motivates us to restrict to Lie groups,
for which these conclusions are ensured by Theorem 3.3.2 and Lemma 3.3.3.
The compactness requirements force us to restrict orbit types when we prove
properties of our model structures, and the family G of all compact subgroups of
our Lie group G plays an important role. We recall the relevant definitions, wh*
*ich
apply to any topological group G and are familiar and important in a variety of
contexts. They provide a context that allows us to work with noncompact Lie
groups with no more technical work than is required for compact Lie groups.
A family F in G is a set of subgroups that is closed under passage to subgro*
*ups
and conjugates. An F space is a Gspace all of whose isotropy groups are in F .
An F equivalence is a Gmap f such that fH is a weak equivalence for all H 2 *
*F .
If X is an F space, then the only nonempty fixed point sets XH are those for
groups H 2 F . In particular, an F equivalence between F spaces is the same as
a qequivalence. For based Gspaces, the definition of an F space must be alte*
*red
to require that all isotropy groups except that of the Gfixed base point must *
*be in
F . The notion of an F equivalence remains unchanged.
A map in GK =B or GKB is an F equivalence if its map of total Gspaces
is an F equivalence. If B is an F space, then so is any Gspace X over B and
any fiber Xb. The only orbits that can then appear in our parametrized theory a*
*re
of the form G=H with H 2 F and the only nonempty fixed point sets XH are
those for groups H 2 F . In particular, H must be subconjugate to some Gb. An
F equivalence of Gspaces over an F space B is the same as a qequivalence.
It is wellknown that equivariant qtype model structures generalize natural*
*ly
to families. One takes the weak equivalences to be the F equivalences, and one
restricts the orbits G=H that appear as factors in the generating (acyclic) cof*
*ibra
tions to be those such that H 2 F . The resulting cell complexes are called F *
*cell
complexes. Restricting tensors from Gspaces to F spaces, we obtain a restrict*
*ion
of the notion of a Gtopological model category to an F topological model cate*
*gory
that applies here; see Remark 10.3.5.
Proper Gspaces are particularly wellbehaved G spaces, where G is the fami*
*ly
of compact subgroups of our Lie group G, and G cell complexes are proper G
spaces. Restricting base Gspaces to be proper, or more generally to be G spac*
*es,
has the effect of restricting all relevant orbit types G=H to ones where H is c*
*ompact.
However, this is too restrictive for some purposes. For example, we are interes*
*ted
in developing nonparametrized equivariant homotopy theory for noncompact Lie
groups G. Here B = * is a Gspace which, in the unbased sense, is not a G spac*
*e.
We therefore do not make the blanket assumption that B is a G space. We
give the qmodel structure in complete generality, in Theorem 7.2.3, but after *
*that
we restrict to G model structures throughout. That is, our weak equivalences w*
*ill
be the G equivalences. This ensures that, after cofibrant approximation, our t*
*otal
90 7. EQUIVARIANT qfTYPE MODEL STRUCTURES
Gspaces are G spaces. This convention enables us to arrange that all of our m*
*odel
categories are Gtopological. Everything in this chapter applies more generally*
* to
the study of parametrized F homotopy theory for any family F ; see Remark 7.2.*
*14.
The reader may prefer to think in terms of either the case when B = * or the
case when B is proper. Indeed, in order to resolve the problems intrinsic to t*
*he
parametrized context that are described in the Prologue, which we do in Chapter
9, it seems essential that we restrict to proper actions on base spaces. The re*
*ason
is that Stasheff's Theorem 3.4.2 relating the equivariant homotopy types of fib*
*ers
and total spaces plays a fundamental role in the solution. Alternatively, the r*
*eader
may prefer to focus just on compact Lie groups, reading qequivalence instead of
G equivalence and Gspace instead of G space.
7.2.The equivariant q and qfmodel structures
Recall from Definition 4.5.9 that the sets I and J of generating cofibrations
and generating acyclic cofibrations of Gspaces are defined as the sets of all *
*maps
of the form G=H x i, where i is in the corresponding set I or J of maps of spac*
*es.
Definition 7.2.1. Starting from the sets I and J of maps of Gspaces, define
sets IB and JB of maps of exGspaces over B in exactly the same way that their
nonequivariant counterparts were defined in terms of the sets I and J of maps of
spaces in Definition 6.1.1. Note that if B is a G space, then only orbits G=H *
*with
H compact appear in the sets IB and JB .
Taking Y = B in the usual composite adjunction
(7.2.2) GK (G=H x T, Y ) ~=HK (T, Y ) ~=K (T, Y H)
for nonequivariant spaces T and Gspaces Y , we can translate back and forth
between equivariant homotopy groups and cells for Gspaces over B on the one ha*
*nd
and nonequivariant homotopy groups and cells for spaces over BH on the other.
Maps in each of the equivariant sets specified in Definition 7.2.1 correspond by
adjunction to maps in the nonequivariant set with the same name. Systematically
using this translation, it is easy to use Theorem 4.5.5 to generalize the qmod*
*el
structures on K =B and KB to corresponding model structures on GK =B and
GKB . We obtain the following theorem.
Theorem 7.2.3 (qmodel structure). The categories GK =B and GKB are
compactly generated proper G topological model categories whose qequivalences*
*, q
fibrations, and qcofibrations are the maps whose underlying maps of total Gsp*
*aces
are qequivalences, qfibrations, and qcofibrations. The sets IB and JB are t*
*he
generating qcofibrations and generating acyclic qcofibrations, and all qcofi*
*brations
are ~hcofibrations. If B is a G space, then the model structure is Gtopologi*
*cal.
To show that the qmodel structures are G topological, and Gtopological if*
* B
is a G space, we must inspect the maps i j in GK =B, where i is a generating
qcofibration in GK =B and j is a generating cofibration in GK . They have the
form
i j :G=H x G=K x @(Dm x Dn) ! G=H x G=K x Dm x Dn
given by the product of G=H xG=K with the inclusion of the boundary of Dm xDn.
By Lemma 3.3.3, G=H x G=K is a proper Gspace if H or K is compact. Since we
7.2. THE EQUIVARIANT q AND qfMODEL STRUCTURES 91
are assuming that G is a Lie group, we can then triangulate G=H x G=K as a G 
CW complex and use the triangulation to write i j as a relative IB cell comple*
*x.
The case when either i or j is acyclic works in the same way. As explained in
Warning 6.1.7, there is no problem with projection maps in this external contex*
*t.
Moreover, if i is an fcofibration, then so is i j, as we see from the fiberwis*
*e NDR
characterization.
One might be tempted to generalize the qfmodel structure to the equivariant
context in exactly the same way as we just did for the qmodel structure. This
certainly works to give a model structure. However, there is no reason to think
that it is either G or G topological. The problem is that we need i j above to*
* be a
qfcofibration when i is a generating qfcofibration, and triangulations into f*
*cells
are hard to come by. Therefore the GCW structure on G=H x G=K will rarely
produce a relative IfBcell complex. This means that we must be careful when
selecting the generating (acyclic) qfcofibrations if we want the resulting mod*
*el
structure to be Gtopological. We will build the solution into our definition *
*of
qftype model structures, but we need a few preliminaries.
We shall make repeated use of the adjunction
(7.2.4) GK (C x T, Y ) ~=K (T, MapG (C, Y ))
for nonequivariant spaces T and Gspaces C and Y . This is a generalization of
(7.2.2). Taking Y = B, we note in particular that it gives a correspondence bet*
*ween
maps f :T ! T 0over Map G(C, B) and Gmaps idx f :C x T ! C x T 0over B.
Lemma 7.2.5. If C is a G cell complex, then the functor MapG (C, ): GK  !
K preserves all qequivalences.
Proof. The functor Map (C, ) is a Quillen right adjoint since the qmodel
structure on GK is G topological. The Gfixed point functor is also a Quillen
right adjoint, for example by Proposition 7.4.3 below. The composite Map G(C, )
therefore preserves qequivalences between qfibrant Gspaces. However, every G
space is qfibrant.
Observe that Lemma 3.3.3 gives that the collection of G cell complexes is c*
*losed
under products with arbitrary orbits G=H of G.
Definition 7.2.6. Let OG denote the set of all orbits G=H of G. Any set C
of G cell complexes in GK that contains all orbits G=K with K 2 G and is closed
under products with arbitrary orbits in OG is called a generating set. It is a *
*closed
generating set if it is closed under finite products. The closure of a generati*
*ng set
C is the generating set consisting of the finite products of the G cell comple*
*xes in
C . We define sets of generating qf(C )cofibrations and acyclic qf(C )cofibra*
*tions
in GK =B associated to any generating set C as follows.
(i)Let IfB(C ) consist of the maps
(idx i)(d): C x Sn1 ! C x Dn
such that C 2 C , d: C x Dn ! B is a Gmap, i is the boundary inclusion,
and the corresponding map i over Map G(C, B) is a generating qfcofibration
in K =Map G(C, B); that is, i must be an fcofibration.
(ii)Similarly let JfB(C ) consist of the maps
(idx i)(d): C x Sn+! C x Dn+1
92 7. EQUIVARIANT qfTYPE MODEL STRUCTURES
such that C 2 C , d: C x Dn+1 ! B is a Gmap, i is the inclusion, and the
corresponding map i over Map G(C, B) is a generating acyclic qfcofibration
in K =Map G(C, B).
Adjoining disjoint sections, we obtain the corresponding sets IfB(C ) and JfB(C*
* ) in
GKB .
Fix a generating set C . We define a qftype model structure based on C ,
called the qf(C )model structure. Its weak equivalences are the G equivalenc*
*es,
which are the same as the qequivalences when B is a G space. We define the
qf(C )fibrations.
Definition 7.2.7. A map f in GK =B is a qf(C )fibration if Map G(C, f) is
a qffibration in K =Map G(C, B) for all C 2 C . A map in GKB is a qf(C )
fibration if the underlying map in GK =B is one. In either category, a map f is*
* a
G quasifibration if fH is a quasifibration for H 2 G .
Theorem 7.2.8 (qfmodel structure). For any generating set C , the categories
GK =B and GKB are wellgrounded (hence Gtopological) model categories. The
weak equivalences and fibrations are the G equivalences and the qf(C )fibrati*
*ons.
The sets IfB(C ) and JfB(C ) are the generating qf(C )cofibrations and the gen*
*erating
acyclic qf(C )cofibrations. All qf(C )cofibrations are both qcofibrations a*
*nd f~
cofibrations, and all qf(C )fibrations are G quasifibrations.
Proof. Recall from Proposition 5.4.9 that the qequivalences in GK =B and
GKB are wellgrounded with respect to the ground structure given in Defini
tion 5.3.6 and Proposition 5.3.7. It follows that the G equivalences are also *
*well
grounded. It suffices to verify conditions (i)(iv) of Theorem 5.5.1. The acycl*
*icity
condition (i) is obvious.
Consider the compatibility condition (ii). By the adjunction (7.2.4), a map
f has the RLP with respect to IfB(C ) if and only if Map G(C, f) has the RLP
with respect to IfMapG(C,B)for all C 2 C . By the compatibility condition for
the nonequivariant qfmodel structure, that holds if and only if Map G(C, f) is*
* a
qequivalence and has the LLP with respect to JfMapG(C,B)for all C 2 C . By
Lemma 7.2.5, Map G(C, f) is a qequivalence if f is one. Conversely, if Map G(C*
*, f)
is a qequivalence for all C 2 C , then the case C = G=K shows that fK is a q
equivalence for every compact K and thus f is a G equivalence. By the adjuncti*
*on
again, we see that f has the RLP with respect to IfB(C ) if and only if f is a
G equivalence which has the RLP with respect to JfB(C ).
The fiberwise NDR characterization of f~cofibrations given in Lemma 5.2.4
shows that IfB(C ) and JfB(C ) consist of ~fcofibrations, as stipulated in (ii*
*i). More
precisely, if (u, h), u: Dn ! I and h: Dn x I ! Dn, represents (Dn, Sn1) as a
fiberwise NDRpair over MapG (C, B), then the map v = uOss :CxDn ! Dn ! I
and the homotopy given by the maps idxhtover B corresponding to the htrepresent
(C x Dn, C x Sn1) as a fiberwise NDR pair over B.
Since Map G(G=K, f) ~= fK is a nonequivariant qffibration for any qf(C )
fibration f, every qf(C )fibration is a G quasifibration by Proposition 6.5.1*
*. That
the model structure is right proper follows as in Corollary 6.5.2.
Finally, we must verify the product condition (iv). The relevant maps i j,
i: C x Sm1  ! C x Dm and j :G=H x Sn1 ! G=H x Dn,
7.2. THE EQUIVARIANT q AND qfMODEL STRUCTURES 93
are of the form
C x G=H x k :C x G=H x @(Dm x Dn) ! C x G=H x Dm x Dn,
where k is the boundary inclusion. Now C x G=H 2 C by the closure property
of the generating set, so we don't need to triangulate. The projection of the t*
*ar
get factors through the projection of the target C x Dm of i. To see that the
corresponding map k over Map G(C x G=H, B) is an ~fcofibration, let (u, h) rep*
*re
sent (Dm , Sm1 ) as a fiberwise NDRpair over Map G(C, B) and let (v, j) repre*
*sent
(Dn, Sn1) as an NDRpair; we can think of the latter as a fiberwise NDRpair o*
*ver
* = Map G(G=H, *). Then the usual product pair representation (for example, [71,
p. 43]) exhibits k as a fiberwise NDR over Map G(C, B) x MapG (G=H, *) and thus,
by the factorization of the projection of i j, also over Map G(C x G=H, B x *).
Theorem 7.2.9. If C C 0is an inclusion of generating sets, then the identi*
*ty
functor is a left Quillen equivalence from GK =B with the qf(C )model structure
to GK =B with the qf(C 0)model structure. The identity functor is also a left
Quillen equivalence from GK =B with the qf(C )model structure to GK =B with
the qmodel structure. Both statements also hold for the identity functor on GK*
*B .
Proof. The first statement is obvious. For the second, if idCxi is a genera*
*ting
qf(C )cofibration, then C is a G cell complex and we can use the triangulatio*
*n to
write idCx i as a relative IB cell complex.
Theorem 7.2.10. For any C , the identity functor is a left Quillen adjoint
from GK =B with the qf(C )model structure to GK =B with the fmodel structure.
Similarly, the identity functor is a left Quillen adjoint from GKB with the qf(*
*C )
model structure to GKB with the fpmodel structure.
The last result, which implements the philosophy of x4.1, is false for the q*
*model
structures.
Remark 7.2.11. The smallest generating set C is the set of all (nonempty)
finite products of orbits G=H of G such that at least one of the factors has H
compact. Clearly it is a closed generating set. Henceforward, by the qfmodel
structure, we mean the qf(C )model structure associated to this choice of C . *
*In
the nonequivariant case, this is the qfmodel structure of the previous chapter.
Remark 7.2.12. In the nonparametrized setting, the G model structure associ
ated to the qmodel structure and the qf(C )model structures on GK = GK =* co
incide, and similarly for GK*. This holds since the fcofibrations and hcofibr*
*ations
over a point coincide and since the C 2 C for any choice of C are G cell compl*
*exes.
Of course, the qf(C )model structures have more generating (acyclic) cofibrati*
*ons.
Remark 7.2.13. It might be useful to combine the various qf(C )model struc
tures by taking the union of the qf(C )cofibrations over some suitable collect*
*ion
of generating sets C and so obtain a "closure" of the qfmodel structure whose
cofibrations are as close as possible to being the intersection of the qcofibr*
*ations
with the ~fcofibrations. We do not know whether or not that can be done.
Remark 7.2.14. As noted in the introduction, we can generalize the q and
qf(C )model structures to the context of families F . We generalize the qmodel
structure to the F model structure by taking the F equivalences and F fibrat*
*ions
and by restricting the sets IB and JB to be constructed from orbits G=H with
94 7. EQUIVARIANT qfTYPE MODEL STRUCTURES
H 2 F . The resulting model structure will then be (F \ G )topological and F 
topological if the base space B is a G space.
To generalize the qf(C ) model structure, we take the weak equivalences to be
the F \ G equivalences and we require the generating set C to consist of F \ G*
* 
cell complexes, to contain the orbits G=K for K 2 F \ G , and to be closed under
products with orbits G=K where K 2 F . With that modification, everything else
above goes through unchanged.
7.3. External smash product and base change adjunctions
The following results relate the q and qf(C )model structures to smash prod
ucts and base change functors and show that various of our adjunctions are given
by Quillen adjoint pairs and therefore induce adjunctions on passage to homotopy
categories. For uniformity, we must understand the qmodel structure to mean the
associated G model structure, although many of the results do apply to the full
qmodel structure. Those results that refer to qequivalences by name work equa*
*lly
well for G equivalences. Most of the results in this section and the next appl*
*y both
to the G model structure and to the qf(C )model structure for any generating *
*set
C . We agree to omit the q or qf(C ) from the notations in those cases. In other
cases, we will have to restrict to well chosen generating sets C .
With these conventions, our first result is clear from the fact that our mod*
*el
structures are Gtopological.
Proposition 7.3.1. For a based GCW complex K, the functor () ^B K
preserves cofibrations and acyclic cofibrations, hence the functor FB (K, ) pr*
*eserves
fibrations and acyclic fibrations. Thus (() ^B K, FB (K, )) is a Quillen adjo*
*int
pair of endofunctors of GKB .
For the rest of our results, recall from Lemma 5.4.6 that a left adjoint that
takes generating acyclic cofibrations to acyclic cofibrations preserves acyclic*
* cofi
brations. The following two results apply to the qf(C )model structure for any
closed generating set C .
Proposition 7.3.2. If i: X ! Y and j :W ! Z are cofibrations over base
Gspaces A and B, then
i j :(Y Z W ) [XZW (X Z Z) ! Y Z Z
is a cofibration over A x B which is acyclic if either i or j is acyclic.
Proof. It suffices to inspect i j for generating (acyclic) cofibrations as *
*was
done for the case A = * in the proof of Theorem 7.2.8. For generating cofibrati*
*ons,
the argument there generalizes without change to this setting. The assumption t*
*hat
C is closed avoids the need for triangulations here. For the acyclicity, it su*
*ffices
to work in the qmodel structure, for which the conclusion is both more general
and easier to prove. There it is easily checked using triangulations of product*
*s of
G cell complexes that if i is a generating cofibration and j is a generating a*
*cyclic
cofibration, then i j is an acyclic cofibration.
Of course, by Warning 6.1.7, the analogue for internal smash products fails.
Taking W = *B and changing notations, we obtain the following special case.
Corollary 7.3.3. Let Y be a cofibrant exspace over B. Then the functor
() Z Y from exspaces over A to exspaces over A x B preserves cofibrations and
7.3. EXTERNAL SMASH PRODUCT AND BASE CHANGE ADJUNCTIONS 95
acyclic cofibrations, hence the functor F~(Y, ) from exspaces over A x B to e*
*x
spaces over A preserves fibrations and acyclic fibrations. Thus (() Z Y, ~F(Y,*
* ))
is a Quillen adjoint pair of functors between GKA and GKAxB .
The next two results apply to the qf(C )model structures for any C , provid*
*ed
that we use the same generating set C for both GKA and GKB .
Proposition 7.3.4. Let f :A ! B be a Gmap. Then the functor f!preserves
cofibrations and acyclic cofibrations, hence (f!, f*) is a Quillen adjoint pair*
*. The
functor f! also preserves qequivalences between wellsectioned exspaces. If f*
* is a
qfibration, then the functor f* preserves all qequivalences.
Proof. If (D, p) is a space over A, then f!((D, p) q A) = (D, f O p) q B.
Therefore f!takes generating (acyclic) qcofibrations over A to such maps over *
*B.
If (u, h) represents (Dn, Sn1) as a fiberwise NDRpair over Map G(C, A), then,
after composing the projection maps with Map G(C, A) ! Map G(C, B), it also
represents (Dn, Sn1) as a fiberwise NDRpair over Map G(C, B). It follows that
f! also preserves the generating (acyclic) qfcofibrations. Recall that the we*
*ll
sectioned exspaces are those that are ~fcofibrant and that fcofibrations are*
* h
cofibrations. Since f!X is defined by a pushout in GK , the gluing lemma in GK
implies that f!preserves qequivalences between wellsectioned exspaces.
If f is a qfibration and k :Y ! Z is a qequivalence of exspaces over B,
consider the diagram
f*k mmf*Z6_____________//Z677oo
mmmmm jj okoooo''
f*Y m______jj____//_Yoo '''
66 j 33 ''
666 jj 333 ''
66 jj 33 ''
aejjae6 s'sss3
A _______f______//B.
The relation (A xB Z) xZ Y ~= A xB Y shows that the top square is a pullback,
and the pullback f*Z ! Z of f is a qfibration. Since the qmodel structure on
the category of Gspaces is right proper, it follows that f*k is a qequivalenc*
*e.
Proposition 7.3.5. If f :A ! B is a qequivalence, then (f!, f*) is a Quill*
*en
equivalence.
Proof. The conclusion holds if and only if the induced adjunction on homo
topy categories is an adjoint equivalence [44, 1.3.3], so it suffices to verify*
* the usual
defining condition for a Quillen adjunction in either model structure. The cond*
*ition
for the other model structure follows formally. We choose the qmodel structure.
Let X be a qcofibrant exspace over A and Y be a qfibrant exspace over B, so
that A ! X is a qcofibration and Y ! B is a qfibration of Gspaces. Since t*
*he
model structure on the category of Gspaces is left and right proper, inspectio*
*n of
the defining diagrams in Definition 2.1.1 shows that the canonical maps X ! f!X
and f*Y  ! Y of total spaces are qequivalences. For an exmap k :f!X ! Y
96 7. EQUIVARIANT qfTYPE MODEL STRUCTURES
with adjoint "k:X ! f*Y , the commutative diagram
X ______//f!X
"k k
fflffl fflffl
f*Y ______//Y
of total spaces then implies that k is a qequivalence if and only if "kis a q
equivalence.
In view of Counterexample 0.0.1, we can at best expect only a partial and
restricted analogue of Proposition 7.3.4 for (f*, f*). We first give a result f*
*or the
qmodel structure and then show how to obtain the analogue for the qf(C )model
structures using well chosen generating sets C .
Proposition 7.3.6. Let f :A ! B be a Gbundle such that B is a G space
and each fiber Ab is a Gbcell complex. Then (f*, f*) is a Quillen adjoint pair*
* with
respect to the qmodel structures. Moreover, if the total space of an exGspac*
*e Y
over B is a G cell complex, then so is the total space of f*Y .
Proof. Since f is a qfibration, f* preserves qequivalences. It therefore *
*suf
fices to show that f* takes generating cofibrations in IB to relative IA cell *
*com
plexes. Observe first that if OE: G=H ! B is a Gmap with OE(eH) = b, then
H Gb and the pullback Gbundle OE*f :f*(G=H, OE) ! G=H of f along OE is
Ghomeomorphic to G xH Ab ! G=H. We can triangulate orbits in a Gbcell
decomposition of Ab as HCW complexes, by Theorem 3.3.2, and so give Ab the
structure of an Hcell complex. Then G xH Ab has an induced structure of a G c*
*ell
complex and thus so does f*(G=H, OE).
For a space d: E ! B over B with associated exspace E qB over B, we have
f*(E q B) = f*E q A. Let E = G=H x Dn and let i: G=H ! G=H x Dn be
the inclusion i(gH) = (gH, 0). The composite d O i is a map OE as above. Since *
*the
identity map on G=H x Dn is homotopic to the composite i O ss :G=H x Dn !
G=H x Dn, where ss is the projection, the pullback Gbundle d*f :f*(E, d) ! E
is equivalent to the pullback bundle (OE O ss)*f :f*(E, OE O ss) ! E. But the *
*latter
is the product of OE*f :f*(G=H, OE) ! G=H and the identity map of Dn as we see
from the following composite of pullbacks
f*(G=H x Dn, OE O ss)___//f*(G=H, OE)__//f*(G=H x Dn, d)____//A
(OEOss)*f OE*f d*f f
fflffl fflffl fflffl fflffl
G=H x Dn _____ss____//_G=H____i_____//_G=H x Dn__d___//_B.
The G cell structure on f*(G=H, OE) gives a canonical decomposition of the inc*
*lusion
f*(G=H, OE) x Sn1 ! f*(G=H, OE) x Dn as a relative G cell complex. The last
statement follows by applying this analysis inductively to the cells of Y .
The previous result fails for the qfmodel structure. In fact, it already f*
*ails
nonequivariantly for the unique map f :A ! *, where A is a CWcomplex. The
proof breaks down when we try to use a cell decomposition of A (the fiber over *
**) to
decompose cells AxSn1 ! AxDn over A as relative IfAcell complexes. Similarly,
the equivariant proof above breaks down when we try to use the Gcell structure
7.4. CHANGE OF GROUP ADJUNCTIONS 97
of f*(G=H, OE) to obtain a relative IfAcell complex. Note, however, that there
is no problem when the fibers are homogeneous spaces G=H; the nonequivariant
analogue is just the trivial case when f is a homeomorphism, but principal bund*
*les
and projections G=H x B ! B give interesting equivariant examples. For the
general equivariant case, we choose a closed generating set C (f) that depends
on the Gbundle f and a given closed generating set C . Using the qf(C )model
structures on GKA and GKB , we then recover the Quillen adjunction.
Construction 7.3.7. Let f :A ! B be a Gbundle such that B is a G space
and each fiber Ab is a Gbcell complex and let C be a closed generating set. We
construct the set C (f) inductively. Let C (f)0 = C and suppose that we have
constructed a set C (f)n of G cell complexes in GK that is closed under both f*
*inite
products and products with arbitrary orbits G=H of G. Let
An = {f*(C, OE)  C 2 C (f)n and OE 2 GK (C,}B).
Then let C (f)n+1 consist of all finite products of spaces in C (f)n [ An. Note*
* that
C (f)n+1 contains C (f)n and that the f*(C, OE) are GScell complexes by the la*
*st
statement of Proposition 7.3.6. Finally, let C (f) = C (f)n. Clearly C (f) C
is a closed generating set that contains f*(C, OE) for all C 2 C (f) and all G*
*maps
OE: C ! B.
Proposition 7.3.8. Let f :A ! B be a Gbundle such that B is a G space
and all fibers Ab are Gbcell complexes. Then (f*, f*) is a Quillen adjoint pai*
*r with
respect to the qf(C (f))model structures on GKA and GKB .
Proof. Reexamining the proof of Proposition 7.3.6, but starting with a map
d: E = C x Dn ! B where C 2 C (f), we see that
f*E ~=f*(C, OE) x Dn
where OE = d O i. Since f*(C, OE) is a G cell complex in C (f), it remains on*
*ly
to show that f*(C, OE) x Sn1 ! f*(C, OE) x Dn is an fcofibration. Let (u, h)
represent (Dn, Sn1) as a fiberwise NDRpair over Map G(C, B). Applying f*
to the corresponding maps ht:C x Dn  ! C x Dn over B, we obtain maps
f*ht:f*E ! f*E over A. Under the displayed isomorphism, these maps give
a homotopy f*h: Dn x I ! Dn that, together with u, represents (Dn, Sn1) as
a fiberwise NDRpair over Map G(f*(C, OE), A).
7.4. Change of group adjunctions
We consider change of groups in the q and the qfmodel structures, starting
with the former. The context of the following results is given in xx2.3 and 2.4.
Proposition 7.4.1. Let ` :G ! G0 be a homomorphism of Lie groups. The
restriction of action functor
`*: G0KB ! GK`*B
preserves qequivalences and qfibrations. If B is a G 0space, then it also pr*
*eserves
qcofibrations.
Proof. Since (`*A)H = A`(H) for any subgroup H of G and a map f :
X ! Y of Gspaces is a qequivalence or qfibration if and only if each fH is
a qequivalence or qfibration, it is clear that `* preserves qequivalences an*
*d q
fibrations. To study qcofibrations, recall that ` factors as the composite of*
* a
98 7. EQUIVARIANT qfTYPE MODEL STRUCTURES
quotient homomorphism, an isomorphism, and an inclusion. If ` is an inclusion
and H0 is a compact subgroup of G0, then we can triangulate G0=H0 as a GCW
complex by Theorem 3.3.2. If ` is a quotient homomorphism with kernel N and H0
is a subgroup of G0, then H0= H=N for a subgroup H of G and `*(G0=H0) = G=H
so that no triangulations are required. Thus in both of these cases, `* takes g*
*ener
ating qcofibrations to qcofibrations. Since `* is also a left adjoint in both*
* cases,
it preserves qcofibrations in general.
Remark 7.4.2. We did not require `*B to be a G space in Proposition 7.4.1.
However, if the kernel of ` is compact and B is a G 0space, then `*B is a G s*
*pace.
Indeed, ` is then a proper map and Gb = `1(G0b) is compact since G0bis compact.
The restriction to compact kernels is the price we must pay in order to stay in
the context of compact isotropy groups. We might instead consider G0spaces B
such that the isotropy groups of both B as a G0space and `*B as a Gspace are
compact, but the assumption on `*B would be unnatural. Note however that one
of the main reasons for restricting to compact isotropy groups is to obtain GCW
structures. If X is a G0CW complex where G0= G=N is a quotient group of G,
then `*X is a GCW complex with the same cells since the relevant orbits G0=H0
can be identified with G=H, where H0= H=N.
For the qfmodel structures, and to study adjunctions, it is convenient to c*
*on
sider quotient homomorphisms and inclusions separately. For the former, we con
sider the adjunctions of Proposition 2.4.1.
Proposition 7.4.3. Let ffl: G ! J be a quotient homomorphism of G by a
normal subgroup N. For a Gspace B, consider the functors
()=N :GKB ! JKB=N and ()N :GKB ! JKBN .
Let j :BN  ! B be the inclusion and p: B ! B=N be the quotient map. Then
(()=N, p*ffl*) and (j!ffl*, ()N ) are Quillen adjoint pairs with respect to t*
*he qmodel
structures on both GKB and JKB=N . Let CG and CJ be generating sets of Gcell
complexes and Jcell complexes. Consider GKB with the qf(CG )model structure
and JKB=N and JKBN with the qf(CJ)model structure. Then
(i)(()=N, p*ffl*) is a Quillen adjunction if C=N 2 CJ for C 2 CG .
(ii)(j!ffl*, ()N ) is a Quillen adjunction if "*C 2 CG for C 2 CJ.
Proof. Since (j!, j*) and (p!, p*) are Quillen adjoint pairs in both the q *
*and
the qf contexts, it suffices to consider the case when N acts trivially on B, so
that j and p are identity maps. Then "* is right adjoint to ()=N and left adjo*
*int
to ()N . The properties of "* in the previous result give the conclusion for *
*the
qmodel structures. The functors "* and ()N preserve qequivalences. Since
Map G(C, "*f0) ~=Map G(C=N, f0) and Map J(C0, fN ) ~=Map G(ffl*C0, f)
for a Jmap f0 and a Gmap f, the conditions on generating sets in (i) and (ii)
ensure that "* and ()N preserve the relevant qffibrations.
Remark 7.4.4. In (i), we can take CJ to consist of all finite products of qu*
*o
tients C=N with C 2 CG and orbits J=H to arrange that CJ be closed and contain
these C=N. In (ii), we can take CG to consist of all products of pullbacks "*C *
*for
C 2 CJ with finite products of orbits G=H. This set will be closed if CJ is clo*
*sed
since "* preserves products.
7.4. CHANGE OF GROUP ADJUNCTIONS 99
Using Proposition 7.4.3 in conjunction with the additional change of group
relations of Propositions 2.4.2 and 2.4.3, we obtain the following compendium of
equivalences in homotopy categories.
Proposition 7.4.5. Let A and B be Gspaces. Let j :BN ! B be the in
clusion and p: B ! B=N be the quotient map, and let f :A ! B be a Gmap.
Then, for exGspaces X over A and Y over B,
(p!Y )=N ' Y=N, (f!X)=N ' (f=N)!(X=N),
(j*Y )N ' Y N, (f*Y )N ' (fN )*(Y N),
(p*Y )N ' Y=N, (f!X)N ' (fN )!(XN ),
where, for the last equivalence on the left, B must be an Nfree Gspace.
Proof. The equivalences displayed in the first line come from isomorphisms
between Quillen left adjoints and are therefore clear. Similarly the equivalenc*
*es in
the second line come from isomorphisms between Quillen right adjoints. The first
equivalence in the third line (in which we have changed notations from Proposi
tion 2.4.3) comes from an isomorphism between a Quillen right adjoint on the le*
*ft
hand side, by Proposition 7.3.6, and a Quillen left adjoint on the right hand s*
*ide and
therefore also descends directly to an equivalence on homotopy categories. For *
*the
last equivalence, note that ()N preserves all qequivalences and also preserve*
*s well
grounded exspaces and that (fN )!preserves qequivalences between wellgrounded
exspaces. Letting Q and R denote cofibrant and fibrant replacement functors, as
usual, it follows that the maps
(R(f!X))N  (f!X)N ~=(fN )!(XN )  (fN )!(Q(XN ))
are qequivalences on exspaces X that are qffibrant and qfcofibrant. As note*
*d in
the proof of Proposition 2.4.2, the point set level isomorphism (f!X)N ~=(fN )!*
*(XN )
is only valid for an exspace X whose section is a closed inclusion. However, i*
*f X
is qfcofibrant, then it is compactly generated and this holds by Lemma 1.1.3(i*
*).
Thus the equivalence holds in general in the homotopy category.
The context for the next result is given in Definition 2.3.7 and Proposition*
* 2.3.9.
Proposition 7.4.6. Let ': H ! G be the inclusion of a subgroup and let A
be an Hspace. The adjoint equivalence ('!, *'*) relating HKA and GK'!Ais a
Quillen equivalence in the qmodel structures and also in the qf(CH ) and qf(CG*
* )
model structures for any generating sets CH and CG of Hcell complexes and Gce*
*ll
complexes such that '!C = GxH C 2 CG for C 2 CH . If A is proper and completely
regular, then the functor '!is also a Quillen right adjoint with respect to the*
* q and
qfmodel structures.
Proof. Recall that :A ! '*'!A = G xH A is the natural inclusion of H
spaces and that ( !, *) is a Quillen adjunction in both the q and qf contexts.*
* The
functor '* preserves qequivalences and qfibrations. It takes qf(CG )fibratio*
*ns to
qf(CH )fibrations when '!C 2 CG for C 2 CH since
Map H (C, '*f) ~=Map G('!C, f).
To show that ('!, *'*) is a Quillen equivalence, we may as well check the
defining condition in the qmodel structure. Let X be a qcofibrant exHspace *
*over
A and Y be a qfibrant exGspace over '!A. Consider a Gmap f :'!X ! Y . We
100 7. EQUIVARIANT qfTYPE MODEL STRUCTURES
must show that f is a qequivalence if and only if its adjoint Hmap "f:X ! **
*'*Y
is a qequivalence. Since '! preserves acyclic qcofibrations, we can extend f*
* to
f0:'!RX ! Y , where RX is a qfibrant approximation. Since f0is a qequivalence
if and only if f is one, and similarly for their adjoints, we may assume withou*
*t loss
of generality that X is qfibrant. Recall from Proposition 2.3.9 that '! and *
**'*
are inverse equivalences of categories and observe that *'*Y can be viewed as *
*the
restriction, Y A , of Y along the inclusion of Hspaces :A ! G xH A. From t*
*hat
point of view, "f:X ! *'*Y is just the map X ! Y A of exHspaces over A
obtained by restriction of '*f along .
Now f is a qequivalence if and only if f restricts to a qequivalence f[g,a*
*]on
each fiber, meaning that this restriction is a weak equivalence after passage to
fixed points under all subgroups of the isotropy group of [g, a]. For a 2 A, t*
*he
isotropy subgroup Ha H of a coincides with the isotropy subgroup G[e,a] G of
[e, a] 2 G xH A. For g 2 G, the isotropy subgroup of [g, a] is gHag1. Since the
action by g 2 G induces a homeomorphism between the fibers over [e, a] and over
[g, a], we see that f is a qequivalence if and only if each of the restriction*
*s f[e,a]is
a qequivalence. But that holds if and only if "fis a qequivalence.
For the last statement, recall the description of '!in Remark 2.4.4 as the c*
*om
posite (p*ss*"*())H , where ": GxH ! H and ss :GxA ! A are the projections
and p: G x A ! G xH A is the quotient map. Since G x A is completely regu
lar, p is a bundle with fiber G=Ha over [g, a], and Ha is compact since A is pr*
*oper.
Therefore, by Propositions 7.3.6 and 7.3.8, p* is a Quillen right adjoint with *
*respect
to the q and qfmodel structures. In view of Proposition 7.4.3, this displays '*
*!as a
composite of Quillen right adjoints.
Remark 7.4.7. We can take CG to consist of all finite products of the '!C wi*
*th
C 2 CH and orbits G=K to arrange that CG be closed and contain these '!C.
We shall prove that ('!, *'*) descends to a closed symmetric monoidal equiv
alence of homotopy categories in Proposition 9.4.8 below. The first statement of
Proposition 7.4.6 implies that the description of '* in terms of base change th*
*at is
given in Proposition 2.3.10 descends to homotopy categories.
Corollary 7.4.8. The functor '*: HoGKB ! HoHK'*B is the composite
*
Ho GKB __~__//HoGK'!'*B' HoHK'*B
7.5. Fiber adjunctions and Brown representability
For a point b in B, we combine the special case "b:G=Gb ! B of Proposi
tion 7.3.4 with the special case ': Gb ! G and A = *, hence :* ! G=Gb, of
Proposition 7.4.6 to obtain the following result concerning passage to fibers. *
*Re
call from Example 2.3.12 that the fiber functor ()b:GKB ! GbK* is given by
*'*"b*= b*'*. By conjugation, its left adjoint ()b therefore agrees with "b!'*
*!.
Proposition 7.5.1. For b 2 B, the pair of functors (()b, ()b) relating GbK*
and GKB is a Quillen adjoint pair.
We use this result to verify the formal hypotheses of Brown's representabili*
*ty
theorem [14] for the category HoGKB . Of course, this verification is independe*
*nt
of the choice of model structure. The category GKB has coproducts and homotopy
pushouts, hence homotopy colimits of directed sequences. The usual constructions
7.5. FIBER ADJUNCTIONS AND BROWN REPRESENTABILITY 101
of homotopy pushouts as double mapping cylinders and of directed homotopy col
imits as telescopes makes clear that if the total spaces of their inputs are co*
*mpactly
generated, as they are after qcofibrant approximation, then so are the total s*
*paces
of their outputs. We need a few preliminaries.
Definition 7.5.2. For n 0, b 2 B, and H Gb, let Sn,bHbe the exGspace
((Gb=H xSn)+ )b over B. Explicitly, by Construction 1.4.6, Sn,bH= (G=H xSn)qB,
with the obvious section and with the projection that maps G=H x Sn to the point
b and maps B by the identity map. Equivalently, taking d to be the constant map
at b, Sn,bHis the quotient exGspace associated to the generating cofibration *
*i(d),
i: G=H x Sn1 ! G=H x Dn. Therefore, Sn,bHis cofibrant in both the q and the
qfmodel structures. Let DB be the "detecting set" of all such exGspaces Sn,b*
*H.
Let [X, Y ]G,B denote the set of maps X ! Y in HoGKB .
Lemma 7.5.3. Each X in DB is compact, in the sense that
colim[X, Yn]G,B ~=[X, hocolimYn]G,B
for any sequence of maps Yn ! Yn+1 in GKB .
Proof. If X = Sn,bH, then [X, Y ]G,B ~=[Gb x Sn)+ , Yb]Gb. In GbK*, every
object is fibrant and the target is the set of homotopy classes of Gbmaps (Gb x
Sn)+ ! Yb, which is the set of unbased nonequivariant homotopy classes of maps
Sn ! Yb. Using cofibrant replacement, we can arrange that the (Yn)b have total
spaces in U . Then the conclusion follows from Lemma 1.1.5.
The following result says that the set DB detects qequivalences.
Proposition 7.5.4. A map , :Y ! Z in GKB is a qequivalence if and only
if the induced map ,*: [X, Y ]G,B ! [X, Z]G,B is a bijection for all X 2 DB .
Proof. We may assume that Y and Z are fibrant. By the evident long exact
sequences of homotopy groups and the five lemma, , is a qequivalence if and on*
*ly
if each Yb ! Zb is a qequivalence. This is detected by the based Gbspaces
(Gb=H x Sn)+ and the conclusion follows by adjunction.
Theorem 7.5.5 (Brown). A contravariant setvalued functor on the category
HoGKB is representable if and only if it satisfies the wedge and MayerVietoris
axioms.
CHAPTER 8
Exfibrations and exquasifibrations
To complete the foundations of parametrized homotopy theory, we are faced
with two problems that were discussed in the Prologue. In our preferred qfmodel
structure, the base change adjunction (f!, f*) is a Quillen pair for any map f *
*and is
a Quillen equivalence if f is an equivalence. As shown by Counterexample 0.0.1,*
* this
implies that the base change adjunction (f*, f*) cannot be a Quillen adjoint pa*
*ir.
Some such defect must hold for any model structure. Therefore, we cannot turn to
model theory to construct the functor f* on the level of homotopy categories. T*
*he
same counterexample illustrates that passage to derived functors is not functor*
*ial
in general, so that a relation between composites of functors that holds on the
pointset level need not imply a corresponding relation on homotopy categories.
In any attempt to solve those two problems, one runs into a third one that
concerns a basic foundational problem in exspace theory. Model theoretical con
siderations lead to the use of Serre fibrations as projections, or to the even *
*weaker
class of qffibrations. However, only Hurewicz fibrations are considered in mos*
*t of
the literature. There is good reason for that. Fiberwise smash products, suspen
sions, cofibers, function spaces, and other fundamental constructions in exspa*
*ce
theory do not preserve Serre fibrations.
The solutions to all three problems are obtained by the use of exfibrations.
Recall that these are the wellsectioned hfibrant exspaces. We study their pr*
*op
erties in x8.1. They seem to give the definitively right kind of "fibrant exsp*
*ace"
from the point of view of classical homotopy theory, and they behave much better
under the cited constructions than do Serre fibrations, as we show in x8.2. Many
variants of this notion appear in the literature. Precisely this variant, with*
* this
name, appears in Monica Clapp's paper [18], and we are indepted to her work
for an understanding of the centrality of the notion. Perversely, as we noted *
*in
Remark 5.2.7, it is unclear how it fits into the model categorical framework.
We construct an elementary exfibrant approximation functor in x8.3. It plays
a key role in bridging the gap between the model theoretic and classical worlds.
In a different context, the classification of sectioned fibrations, the first a*
*uthor
introduced this construction in [64, x5]. We record some its properties in x8.4.
We define quasifibrant exspaces and exquasifibrations and show that they
inherit some of the good properties of exfibrations in x8.5. They will play a *
*key
role in the stable theory.
Everything in this chapter works just as well equivariantly as nonequivarian*
*tly
for any topological group G of equivariance.
8.1. Exfibrations
Under various names, the following notions were in common use in the 1970's.
We shall see shortly that these definitions agree with those given in Definitio*
*n 5.2.5.
102
8.1. EXFIBRATIONS 103
Definition 8.1.1. Let (X, p, s) be an exspace over B.
(i)(X, p, s) is wellsectioned if s is a closed inclusion and there is a retra*
*ction
ae: X x I ! X [B (B x I) = Ms
over B.
(ii)(X, p, s) is wellfibered if there is a coretraction, or pathlifting func*
*tion,
': Np = X xB BI ! XI
under BI, where BI maps to Np via ff ! (sff(0), ff).
(iii)(X, p, s) is an exfibration if it is both wellsectioned and wellfibered.
The requirement in (i) that the retraction ae be a map over B ensures that it
restricts on fibers to a retraction that exhibits the nondegeneracy of the base*
*point
s(b) in Xb for each b 2 B. In view of Theorem 5.2.8(i), we have the following
characterization of wellsectioned exspaces, in agreement with Definition 5.2.*
*5.
Lemma 8.1.2. An exspace X is wellsectioned if and only if X is ~fcofibran*
*t.
We use the term "wellsectioned" since it goes well with "wellbased". The
category of wellsectioned exspaces is the appropriate parametrized generaliza*
*tion
of the category of wellbased spaces, and restricting to wellsectioned exspac*
*es is
analogous to restricting to wellbased spaces.
Note that the section of X provides a canonical way of lifting a path in B t*
*hat
starts at b to a path in X that starts at s(b). The requirement in Definition 8*
*.1.1(ii)
that the pathlifting function ' be a map under BI says that '(sff(0), ff)(t) =*
* s(ff(t))
for all ff 2 BI and t 2 I. That is, ' is required to restrict to the canonical
lifts provided by the section, so that paths in X that start in s(B) remain in
s(B). In contrast with Lemma 8.1.2, the wellfibered condition does not by itse*
*lf
fit naturally into the model theoretic context of Chapter 5. However, we have t*
*he
following characterization of exfibrations, which again is in agreement with t*
*he
original definition we gave in Definition 5.2.5.
Lemma 8.1.3. If X is wellfibered, then X is hfibrant. If X is wellsection*
*ed,
then X is an exfibration if and only if X is hfibrant.
Proof. The first statement is clear since the coretraction ' is a pathlift*
*ing
function. This gives the forward implication of the second statement, and the
converse is a special case of the following result of Eggar [38, 3.2].
Lemma 8.1.4. Let i: X ! Y be an ~fcofibration of exspaces over B, where
Y is hfibrant. Then any map ': X xB BI ! Y Isuch that the composite
X xB BI __'_//_Y_I__//Y xB BI
is the inclusion can be extended to a coretraction Y xB BI ! Y I.
Proof. The inclusion X xB BI ! Y xB BI is an ~hcofibration by Proposi
tion 4.4.5. Therefore there is a lift in the diagram
f
(Y xB BI) x {0} [ (X xB BI) x I____//Y55kk
 k k k 
 k k 
fflfflkk fflffl
(Y xB BI) x I_______g_____//_B,
104 8. EXFIBRATIONS AND EXQUASIFIBRATIONS
where f(y, !, 0) = y, f(x, !, t) = '(x, !)(t), and g(y, !, t) = !(t). The adjo*
*int
Y xB BI ! Y Iof is the required extension to a coretraction.
Remark 8.1.5. We comment on the terminology.
(1) We are following [29, 51] and others in saying that an ~fcofibrant exs*
*pace
is wellsectioned; the term "fiberwise wellpointed" is also used. For a based *
*space,
the terms "nondegenerately based" and "wellbased" or "wellpointed" are used
interchangeably to mean that the inclusion of the basepoint is an hcofibration*
*. In
contrast, for an exspace, the term "fiberwise nondegenerately pointed" is used*
* in
[29, 51] to indicate a somewhat weaker condition than wellsectioned.
(2) The term "wellfibered" is new but goes naturally with wellsectioned. T*
*he
concept itself is old. We believe that it is due to Eggar [38, 3.3], who calls*
* a
coretraction under BI a special lifting function.
(3) Becker and Gottlieb [2] may have been the first to use the term "ex
fibration", but for a slightly different notion with sensible CW restrictions. *
* As
noted in the introduction, precisely our notion is used by Clapp [18]. Earlier,
in [64, x5] and [65], the first author called exfibrations "T fibrations", an*
*d he
studied their classification and their fiberwise localizations and completions.*
* The
equivariant generalization appears in Waner [97]. A more recent treatment of the
classification of exfibrations has been given by Booth [9].
8.2. Preservation properties of exfibrations
We have a series of results that show that exfibrations behave well with re*
*spect
to standard constructions. In some of them, one must use the equivariant versio*
*n of
Lemma 5.2.4 to verify that the given construction preserves wellsectioned obje*
*cts.
In all of them, if we only assume that the input exspaces are wellsectioned, *
*then
we can conclude that the output exspaces are wellsectioned. It is the fact th*
*at
the given constructions preserve wellfibered objects that is crucial. Few if a*
*ny of
these results hold with Serre rather than Hurewicz fibrations as projections.
Proposition 8.2.1. Exfibrations satisfy the following properties.
(i)A wedge over B of exfibrations is an exfibration.
(ii)If X, Y and Z are exfibrations and i is an f~cofibration in the followi*
*ng
pushout diagram of exspaces over B, then Y [X Z is an exfibration.
X ___i___//_Y
 
 
fflffl fflffl
Z ____//_Y [X Z
(iii)The colimit of a sequence of ~fcofibrations Xi ! Xi+1 between exfibrati*
*ons
is an exfibration.
If the input exspaces are only assumed to be wellsectioned, then the output e*
*x
spaces are wellsectioned.
Proof. The last statement is clear. Using it, we see that the colimits in (*
*i),
(ii), and (iii) are wellsectioned, hence it suffices to prove that they are h*
*fibrant.
This is done by constructing path lifting functions for the colimits from path *
*lifting
functions for their inputs. In (i), we start with path lifting functions under*
* BI
for the wedge summands and see that they glue together to define a path lifting
8.2. PRESERVATION PROPERTIES OF EXFIBRATIONS 105
function under BI for the wedge. Part (ii) is due to Clapp [18, 1.3], and we om*
*it
full details. She starts with a path lifting function for X and uses Lemma 8.1.*
*4 to
extend it to a path lifting function for Y . She also starts with a path liftin*
*g function
for Z. She then uses a representation (h, u) of (Y, X) as a fiberwise NDR pair *
*to
build a path lifting function for the pushout from the given path lifting funct*
*ion
for Z and a suitably deformed version of the path lifting function for Y . In (*
*iii),
Lemma 8.1.4 shows that we can extend a path lifting function for Xi to a path
lifting function for Xi+1. Inductively, this allows the construction of compat*
*ible
path lifting functions for the Xi that glue together to give a path lifting fun*
*ction
for their colimit.
Although of little use to us, since the fhomotopy category is not the right*
* one
for our purposes, many of our adjunctions give Quillen adjoint pairs with respe*
*ct to
the fmodel structure. For example, the following result, which should be compa*
*red
with Proposition 7.3.4, implies that (f!, f*) is a Quillen adjoint pair in the *
*fmodel
structures and that it is a Quillen equivalence if f is an hequivalence.
Proposition 8.2.2. Let f :A ! B be a map, let X be a wellsectioned ex
space over A, and let Y be a wellsectioned exspace over B. Then f!X and f*Y a*
*re
wellsectioned. If Y is an exfibration, then so is f*Y , and the functor f* pr*
*eserves
fequivalences. If f is an hequivalence, then (f!, f*) induces an equivalence*
* of
fhomotopy categories.
Proof. It is easy to check that representations of (X, A) and (Y, B) as fib*
*er
wise NDRpairs induce representations of (f!X, B) and (f*Y, A) as fiberwise NDR
pairs. As a pullback, the functor f* preserves both ffibrant and hfibrant ex
spaces, and f* preserve fequivalences since it preserves fhomotopies. For the
last statement, if f is a homotopy equivalence with homotopy inverse g, then st*
*an
dard arguments with the CHP imply that f* induces an equivalence of fhomotopy
categories with inverse g*; see, for example, [64, 2.5]. It follows that g* is *
*equivalent
to f!and that (f!, f*) is a Quillen equivalence.
The following result appears in [38] and [64, 3.6]. It also leads to a Quil*
*len
adjoint pair with respect to the fmodel structure; compare Corollary 7.3.3.
Proposition 8.2.3. Let X and Y be wellsectioned exspaces over A and B.
Then X Z Y is a wellsectioned exspace over A x B. If X and Y are exfibration*
*s,
then X Z Y is an exfibration.
Proof. Representations of (X, A) and (Y, B) as fiberwise NDRpairs deter
mine a representation of (X Z Y, A x B) as a fiberwise NDRpair, by standard
formulas [71, p. 43]. Similarly, path lifting functions for X and Y can be used*
* to
write down a path lifting function for X Z Y .
Corollary 8.2.4. If X and Y are exfibrations over B, then so is X ^B Y .
Corollary 8.2.5. If X is an exfibration over B and K is a wellbased space,
then X ^B K is an exfibration over B.
Proposition 8.2.6. Let X and Y be wellsectioned and let f :X ! Y be
an exmap that is an hequivalence. Then f ^B id:X ^B Z ! Y ^B Z is an
hequivalence for any exfibration Z. In particular, f ^B id:X ^B K ! Y ^B K
is an hequivalence for any wellbased space K.
106 8. EXFIBRATIONS AND EXQUASIFIBRATIONS
Proof. As observed by Clapp [18, 2.7], this follows from the gluing lemma by
comparing the defining pushouts.
As in ordinary topology, function objects work less well, but we do have the
following analogue of Corollary 8.2.5.
Proposition 8.2.7. If X is an exfibration over B and K is a compact well
based space, then FB (K, X) is an exfibration over B.
Proof. Let (h, u) represent (X, B) as a fiberwise NDRpair. Then (j, v) rep
resents (FB (K, X), B) as a fiberwise NDRpair, where
v(f) = supk2Ku(f(k)) and jt(f)(k) = ht(f(k))
for f 2 FB (K, X). Note for this that FB (K, B) = B and that, by Proposi
tion 1.3.20, FB (K, X) is hfibrant.
8.3. The exfibrant approximation functor
We describe an elementary exfibrant replacement functor P . It is just the
composite of a whiskering functor W with a version of the mapping path fibration
functor N. The functor P replaces exspaces by naturally hequivalent exfibrat*
*ions.
From the point of view of model theory, P can be thought of as a kind of qfibr*
*ant
replacement functor that gives Hurewicz fibrations rather than just Serre fibra*
*tions
as projections. The nonequivariant version of P appears in [64, 5.3, 5.6], and
the equivariant version appears in [97, x3]. With motivation from the theory of
transports in fibrations, those sources work with Moore paths of varying length.
Surprisingly, that choice turns out to be essential for the construction to wor*
*k.
We therefore begin by recalling that the space of Moore paths in B is given *
*by
B = {(~, l) 2 B[0,1]x [0, 1)  ~(r) = ~(l) for r} l
with the subspace topology. We write ~ for (~, l) and l~ for l, which is the le*
*ngth
of ~. Let e: B ! B be the endpoint projection e(~) = ~(l~). The composite of
Moore paths ~ and ~ such that ~(l~) = ~(0) is defined by l~~ = l~ + l~ and
(
(~~)(r) = ~(r) if r l~,
~(r  l~) if r l~.
Embed B and BI in B as the paths of length 0 and 1. For a Moore path ~ in
B and real numbers u and v such that 0 u v, let ~vudenote the Moore path
r 7! ~(u + r) of length v  u.
Definition 8.3.1. Consider an exspace X = (X, p, s) over B.
(i)Define the whiskering functor W by letting
W X = (X [B (B x I), q, t),
where the pushout is defined with respect to i0: B ! B x I. The projection
q is given by the projection p of X and the projection B x I ! B, and the
section t is given by t(b) = (b, 1).
(ii)Define the Moore mapping path fibration functor L by letting
LX = (X xB B, q, t),
8.3. THE EXFIBRANT APPROXIMATION FUNCTOR 107
where the pullback is defined with respect to the map B ! B given by
evaluation at 0. The projection q is given by q(x, ~) = e(~) and the sectio*
*n t
is given by t(b) = (s(b), b), where b is viewed as a path of length 0.
Thus W X is obtained by growing a whisker on each point in the section of X,
and the endpoints of the whiskers are used to give W X a section. Similarly, LX*
* is
obtained by attaching to x 2 X all Moore paths in B starting at p(x). The end
points of the paths give the projection. In the language of x4.3, W X is the st*
*andard
mapping cylinder construction of the section of X, thought of as a map in GK =B.
The section t of W X is just the fcofibration in the standard factorization ae*
* O t
of s through its mapping cylinder. In particular, W X is wellsectioned. Simila*
*rly,
LX is a modification of the mapping path fibration Np in GK . The projection p
of X factors through the projection q of LX, which is an hfibration; a path li*
*fting
function , :LX xB BI ! (LX)I is given by ,((x, ~), fl)(t) = (x, flt0~). Thus *
*LX
is hfibrant, but it need not be wellfibered.
We can display all of this conveniently in the following diagram. The third
square on the top is a pushout and the second square on the bottom is a pullbac*
*k.
That defines the maps OE and ss, and the maps ae and ' are induced by the unive*
*rsal
properties from the identity map of X.
B FFF
FFFFF
i1 FFFFFFF
fflfflFFF
B _______B_______B__i0//_B x Ipr_//_B
    
  s  
fflffl'fflfflsfflfflsfflfflOEfflfflae
X ` ` `//LX____//_X____//W X` ` `//X
    
  p  
fflffl fflfflpfflffl0fflffl fflffl
B CCCC_// B____//_B_______B________B
CCCCCCe
CCCCCCfflffl
B
Thus ae projects whiskers on fibers to the original basepoints and ' is the inc*
*lusion
x 7! (x, p(x)), where p(x) is the path of length zero. Note that OE is not a m*
*ap
under B and ss is not a map over B. They give an inverse fequivalence to ae and
an inverse hequivalence to '.
Proposition 8.3.2. The map ae: W X  ! X is a natural fequivalence of
exspaces and W X is wellsectioned. The map ': X ! LX is a natural h
equivalence of exspaces and LX is hfibrant. Therefore W takes fequivalences
to fpequivalences and L takes hequivalences to fequivalences.
The last statement follows from Proposition 5.2.2. We think of ae and ' as g*
*iving
a wellsectioned approximation and an hfibrant approximation in the category of
exspaces. We will combine them to obtain the promised exfibrant approximation,
but we first insert a technical lemma.
Lemma 8.3.3. If X is an exspace with a closed section, then W LX is an
exfibration. If X is wellfibered, then W X is an exfibration.
108 8. EXFIBRATIONS AND EXQUASIFIBRATIONS
Proof. A path lifting function , :NW LX = W LX xB BI ! (W LX)I for
W LX is obtained by letting
8
><(x, flt0~) 2 LX if z = (x, ~) 2,LX
,(z, fl)(t) = >(fl(t), u  t) 2 B xiIf z = (b, u) and,t u
:(s(fl(u)), flt
u) 2 LXif z = (b, u) and t u.
It is easy to verify that, as a map of sets, , gives a welldefined section of *
*the canon
ical retraction ss :(W LX)I ! W LX xB BI. Continuity is a bit more delicate,
but if the section of X is closed, then one verifies that
= {(z, fl)  z is the equivalence class of (s(b), b)}~ (b, 0)
is a closed subset of WLX and hence N is a closed subset of NW LX. To see the
implication, note that () x BI preserves closed inclusions and Z xB BI Z x BI
is a closed inclusion because B is in U (see Remark 1.1.4). Continuity follows *
*since
we are then piecing together continuous functions on closed subsets.
If X is wellfibered and , :X xB BI ! XI is a pathlifting function under
BI, we can define a path lifting function ~,:W X xB BI ! (W X)I for W X by
(
~,(x, fl) = ,(x, fl)if x 2 X,
(fl, u) if x = (b,.u)
To check that ~,is continuous, we use the fact that the functor N() = BI xB ()
commutes with pushouts to write NW X as a pushout. We then see that ~,is the
map obtained by passage to pushouts from a pair of continuous maps.
Recall that the sections of exspaces in GUB are closed, by Lemma 1.1.3. Sin*
*ce
we shall only need to apply the constructions of this section to exspaces in G*
*UB ,
the closed section hypothesis need not concern us.
Definition 8.3.4. Define the exfibrant approximation functor P by the nat
ural zigzag of hequivalences OE = (ae, W ') displayed in the diagram
ae W'
X oo___W X _____//W LX = P X.
By Proposition 8.3.2, P takes hequivalences between arbitrary exspaces to fp
equivalences. If X has a closed section, then P X is an exfibration. If X is*
* an
exfibration, then it has a closed section, and the above display is a natural *
*zigzag
of fpequivalences between exfibrations.
8.4. Preservation properties of exfibrant approximation
One advantage of exfibrant approximation over q or qffibrant approximation
is that there are explicit commutation natural transformations relating it to m*
*any
constructions of interest. The following result is an elementary illustrative e*
*xample.
Lemma 8.4.1. Let D be a small category, X :D ! GKB be a functor, and
! :colimW Xd ! W colimXd and :colimLXd ! LcolimXd
be the evident natural maps. Then ! is a map over colimXd and is a map under
colimXd, so that the following diagrams commute. All maps in these diagrams are
8.4. PRESERVATION PROPERTIES OF EXFIBRANT APPROXIMATION 109
hequivalences.
colimW XdN________!_________//NWpcolimXd
NNNN pppppp
colimaeN''NNNNwwppaepp
colimXd
colimXdNN
colim'ppppppp NNN'NNN
xxpppp NNN&&
colimLXd _________________//_LcolimXd
Let ~ = W O ! :colimP Xd ! P colimXd. Then the following diagram of h
equivalences commutes.
colimae colimW'
colimXd oo___colimW Xd _____//colimP Xd
  
 ! ~
 fflffl fflffl
colimXd ooae_W colimXd _W'__//P colimXd
The analogous statements for limits also hold.
Proof. This is clear from the construction of limits and colimits in Propos*
*i
tion 1.2.9. The relevant hequivalences of total spaces are natural and piece t*
*ogether
to pass to limits and colimits.
Warning 8.4.2. We would like an analogue of the previous result for tensors.
In particular, we would like a natural map (LX) ^ K ! L(X ^ K) under X ^ K
for exspaces X over B and based spaces K. Inspection of definitions makes clear
that there is no such map. The obvious map that one might write down, as in the
erroneous [64, 5.6], is not welldefined. In Part III, this complicates the ext*
*ension
of P to a functor on spectra over B.
Lemma 8.4.3. Let f :A ! B be a map.
(i)Let X be an exspace over A. Then there are natural maps
! :f!W X ! W f!X and :f!LX ! Lf!X
of exspaces over B such that ! is a map over f!X and is a map under f!X.
Let ~ = W O ! :f!P X ! P f!X. Then the following diagram commutes.
f!X of!aef!WoX__f!W'//_f!P X
  
 ! ~
 fflffl fflffl
f!X ooae_W f!X _W'__//P f!X
(ii)Let Y be an exspace over B. Then there are natural maps
! :W f*Y ! f*W Y and :Lf*Y ! f*LY
of exspaces over A, the first an isomorphism, such that ! is a map over f*Y
and is a map under f*Y . Let ~ = ! O W :P f*Y  ! f*P Y . Then the
110 8. EXFIBRATIONS AND EXQUASIFIBRATIONS
following diagram commutes.
f*Y ooae_W f*Y _W'__//P f*Y
  
 ! ~
 fflffl fflffl
f*Y of*aef*WoY__f*W'//_f*P Y
If Y is an exfibration, then ~ is an fpequivalence.
(iii)Let X be an exspace over A. Then there are natural maps
! :W f*X ! f*W X and :Lf*X ! f*LX
of exspaces over B such that ! is a map over f*X and is a map under f*X.
Let ~ = ! O W :P f*X ! f*P X. Then the following diagram commutes.
f*X ooae_W f*X _W'__//P f*X
  
 ! ~
 fflffl fflffl
f*X of*aef*WoX__f*W'//_f*P X
Proof. Again, the proof is by inspection of definitions. Since f! does not
preserve exfibrations, we do not have an analogue for f! of the last statement
about f* in (ii).
Warning 8.4.4. We offer another example of the technical dangers lurking in
this subject. The maps ~ in the previous proposition are not hequivalences in
general, the problem in (ii), say, being that f* does not preserve hequivalenc*
*es
in general. If ~: P f*Y  ! f*P Y were always an hequivalence, then one could
prove by the methods in x9.3 below that the relations (2.2.10) descend to homot*
*opy
categories for all pullbacks of the form displayed in Proposition 2.2.9. In vie*
*w of
Counterexample 0.0.1, that conclusion is false. This is another pitfall we fell*
* into,
and it invalidated much work in an earlier draft.
8.5. Quasifibrant exspaces and exquasifibrations
By analogy with the fact that an exfibration is a wellsectioned hfibrant *
*ex
space, we adopt the following terminology.
Definition 8.5.1. An exspace X is quasifibrant if its projection p is a qua*
*si
fibration. An exquasifibration is a wellsectioned quasifibrant exspace.
If X is quasifibrant, there is a long exact sequence of homotopy groups
. ..! ssHq+1(B, b) ! ssHq(Xb, x) ! ssHq(X, x) ! ssHq(B, b) ! . ..! ssH0(B*
*, b)
for any b 2 B, x 2 Xb and H Gb. Using this and the long exact sequences of the
pairs (X, Xb), five lemma comparisons give the following observations.
Lemma 8.5.2. Let f :X ! Y be a qequivalence of exspaces over B. Then
each map of fibers f :Xb ! Yb is a qequivalence if and only if each map of
pairs f :(X, Xb) ! (Y, Yb) is a qequivalence. If X and Y are quasifibrant, th*
*en
these maps of pairs are qequivalences. Conversely, if these maps of pairs are*
* q
equivalences and either X or Y is quasifibrant, then so is the other.
8.5. QUASIFIBRANT EXSPACES AND EXQUASIFIBRATIONS 111
Working in GUB , we obtain the following result. The same pattern of proof
gives many other results of the same nature that we leave to the reader.
Proposition 8.5.3. The following statements hold.
(i)A wedge over B of exquasifibrations is an exquasifibration.
(ii)If f : X ! Y is a map such that X is an exquasifibration and Y is quasif*
*i
brant, then the cofiber CB f is quasifibrant.
(iii)If X is an exquasifibration and K is a wellbased space, then X ^B K is an
exquasifibration.
Proof. This follows from Lemma 8.5.2, the natural zigzag
X oo___W X _____//P X
of hequivalences, the corresponding preservation properties for exfibrations,*
* and
the properties of qequivalences given by the statement that they are wellgrou*
*nded;
see Definition 5.4.1 and Proposition 5.4.9. It is also relevant that in each c*
*ase
passage to fibers gives the nonparametrized analogue of the construction under
consideration. Since this result plays a vital role in our work, we give more c*
*omplete
details of (ii) and (iii); (i) works the same way.
The cofiber CB f is the pushout of the diagram
f
CB X oo___X _____//Y.
If X is wellsectioned, then the left arrow is an hcofibration and W X and P X*
* are
wellsectioned. Replacing f by W f and P f we obtain three such cofiber diagram*
*s.
Together with our original zigzag this gives a 3 x 3diagram. Applying the glu*
*ing
lemma, Definition 5.4.1(iii), we obtain a zigzag of qequivalences
CB foo___ CB W f_____//CB P f.
Similarly, on fibers we obtain zigzags of qequivalences
Cfb oo___C(W f)b ____//_CW (Lf)b.
There results a zigzag of qequivalences of pairs
(CB f, Cfb)oo__(CB W f, CW fb)____//(CB P f, CW (Lf)b).
Since CB P f is exfibrant and in particular quasifibrant, CB f is quasifibrant.
Similarly, by Definition 5.4.1(v), we have natural zigzags of qequivalences
X ^B K oo___W X ^B K _____//P X ^B K
and
Xb^ K oo___ W Xb^ K ____//_W (LX)b^ K.
We therefore have a zigzag of qequivalences of pairs
(X ^B K, Xb^ K) oo__(W X ^B K, W Xb^ K) ____//(P X ^B K, W (LX)b^ K).
Since P X ^B K is exfibrant and in particular quasifibrant, X ^B K is quasifib*
*rant.
CHAPTER 9
The equivalence between Ho GKB and hGWB
Introduction
We developed the pointset level properties of the category GKB of exGspac*
*es
over B in Chapters 1 and 2, and we developed those homotopical properties that
are accessible to model theoretic techniques in Chapter 4  7. In this chapter,*
* we
use exfibrations to prove that certain structure on the pointset level that s*
*eems
inaccessible from the point of view of model category theory nevertheless desce*
*nds
to homotopy categories. In particular, we prove that Ho GKB is closed symmetric
monoidal and that the right derived functor f* of the Quillen adjunction (f!, f*
**) in
the qfmodel structure is closed symmetric monoidal and has a right adjoint f*.
In x9.1 we use the exfibrant approximation functor to prove that our model
theoretic homotopy category of exGspaces over B is equivalent to the classical
homotopy category of exGfibrations over B. In x9.2, we discuss how to pass to
derived functors on either side of that equivalence in certain general cases. R*
*eplac
ing the modeltheoretic method of constructing derived functors by a more class*
*ical
method given in terms of exfibrant approximation, we construct the functors f*
and FB on homotopy categories in x9.3. By a combination of methods, we prove
that Ho GKB is a symmetric monoidal category and that the base change func
tor f* descends to a closed symmetric monoidal functor on homotopy categories
in x9.4. We also obtain such descent to homotopy categories results for change
of group adjunctions and for passage to fibers in that section. These results *
*are
central to the theory, and there seem to be no shortcuts to their proofs.
Everything is understood to be equivariant in this chapter, and we abbreviate
exGfibration and exGspace to exfibration and exspace throughout. We shall
retreat just a bit from allembracing generality. We assume that G is a Lie gro*
*up
and that all given base Gspaces B are proper and are of the homotopy types of
GCW complexes. The reader may prefer to assume that G is compact, but there
is no gain in simplicity. In view of the properties of the base change adjunct*
*ion
(f!, f*) given in Proposition 7.3.4, there would be no real loss of generality *
*if we
restricted further to base spaces that are actual GCW complexes, but that would
be inconveniently restrictive.
9.1.The equivalence of Ho GKB and hGWB
Recall that X ^B I+ is a cylinder object in the sense of the qfmodel struct*
*ure.
When we restrict to qffibrant and qfcofibrant objects, homotopies in the qf
model sense are the same as fphomotopies, by Lemma 5.6.1. The morphism set
[X, Y ]G,B in Ho GKB is naturally isomorphic to [RQX, RQY ]G,B, and this is the
set of fphomotopy classes of maps RQX ! RQY . Here R and Q denote the
functorial qffibrant and qfcofibrant approximation functors obtained from the
112
9.1. THE EQUIVALENCE OF HoGKB AND hGWB 113
small object argument. The total space of RQX has the homotopy type of a G
CW complex since B does. This leads us to introduce the following categories.
Definition 9.1.1. Define GVB to be the full subcategory of GKB whose ob
jects are wellgrounded and qffibrant with total spaces of the homotopy types *
*of
GCW complexes. Define GWB to be the full subcategory of GVB whose objects
are the exfibrations over B. Let hGWB denote the category obtained from GWB
by passage to fphomotopy classes of maps.
Note that the definition of GWB makes no reference to model category theory.
Recall that wellgrounded means wellsectioned and compactly generated. When
B = *, GW* is just the category of wellbased compactly generated Gspaces of t*
*he
homotopy types of GCW complexes, and it is standard that its classical homotopy
category is equivalent to the homotopy category of based Gspaces with respect *
*to
the qmodel structure. We shall prove a parametrized generalization.
We think of GVB as a convenient half way house between GKB and GWB .
It turns out to be close enough to the category of qfcofibrant and qffibrant
objects in GKB to serve as such for our purposes, while already having some of *
*the
properties of GWB . The following crucial theorem fails for the qmodel structu*
*re.
It is essential for this result that we allow the objects of VB to be wellsect*
*ioned
rather than requiring them to be qfcofibrant. This will force an assymmetry wh*
*en
we deal with left and right derived functors in Proposition 9.2.2 below.
Theorem 9.1.2. The qfcofibrant and qffibrant approximation functor RQ
and the exfibrant approximation functor P , together with the forgetful functo*
*rs I
and J, induce the following equivalences of homotopy categories.
_RQ_//_ __P__//
Ho GKB oo___Ho GVB oo___ hGWB
I J
Proof. For X in GKB , we have a natural zigzag of qequivalences in GKB
X oo___QX _____//RQX.
Therefore X and IRQX are naturally qequivalent in GKB . If X is in GVB , then
it is qffibrant and therefore so is QX. Then the above zigzag is in GVB and t*
*hus
X and RQIX are naturally qequivalent in GVB .
Since qequivalences in GVB are hequivalences, and P takes hequivalences to
fpequivalences, it is clear that P induces a functor on homotopy categories. C*
*on
versely, since fpequivalences are in particular qequivalences, the forgetful *
*functor
J induces a functor in the other direction. For X in GVB we have the natural
zigzag of hequivalences
ae
X oo___W X _W'__//P X
of Definition 8.3.4. However W X may not be in GVB since it may not be qffibra*
*nt.
Applying qffibrant approximation, we get a natural zigzag of qequivalences in
GVB connecting X and P X. It follows that X and JP X are naturally qequivalent
in GVB . Starting with X in GWB , the above display is a zigzag of fpequivale*
*nces
in GWB , by Proposition 8.3.2. It follows that X and JP X are naturally fp
equivalent in GWB .
114 9. THE EQUIVALENCE BETWEEN HoGKB AND hGWB
9.2. Derived functors on homotopy categories
Model category theory tells us how Quillen functors V :GKA ! GKB induce
derived functors on the homotopy categories on the left hand side of the equiva*
*lence
displayed in Theorem 9.1.2. We now seek an equivalent way of passing to derived
functors on the right hand side. We begin with an informal discussion. We focus
on functors of one variable, but functors of several variables work the same wa*
*y.
Following the custom in algebraic topology, we have been abusing notation
by using the same notation for pointset level functors and for derived homotopy
category level functors. We will continue to do so. However, the more accurate *
*no
tations of algebraic geometry, LV and RV for left and right derived functors, m*
*ight
clarify the discussion. As we have already seen in Counterexample 0.0.1, passag*
*e to
derived functors is not functorial in general, so that a relation between compo*
*sites
of functors that holds on the pointset level need not imply a corresponding re*
*lation
on passage to homotopy categories.
Recall that, model theoretically, if V is a Quillen right adjoint, then the *
*right
derived functor of V is obtained by first applying fibrant approximation R and
then applying V on homotopy categories, which makes sense since V preserves
weak equivalences between fibrant objects. The left derived functor of a Quill*
*en
left adjoint V is defined dually, via V Q. Problems arise when one tries to com*
*pose
left and right derived functors, which is what we must do to prove some of our
compatibility relations.
The equivalence of categories proven in Theorem 9.1.2 gives us a way of putt*
*ing
the relevant left and right adjoints on the same footing, giving a "straight" p*
*assage
to derived functors that is neither "left" nor "right". We need mild good behav*
*ior
for this to work.
Definition 9.2.1. A functor V :GKA ! GKB is good if it is continu
ous, takes wellgrounded exspaces to wellgrounded exspaces, and takes exspa*
*ces
whose total spaces are of the homotopy types of GCW complexes to exspaces with
that property. Since V is continuous, it preserves fphomotopies.
Proposition 9.2.2. Let V :GKA ! GKB be a good functor that is a left or
right Quillen adjoint. If V is a Quillen left adjoint, assume further that it p*
*reserves
qequivalences between wellgrounded exspaces. Then, under the equivalence of
categories in Theorem 9.1.2, the derived functor Ho GKA  ! Ho GKB induced
by V Q or V R is equivalent to the functor P V J :hGWA  ! hGWB obtained by
passage to homotopy classes of maps.
Proof. If V is a Quillen right adjoint, then it preserves qequivalences be*
*tween
qffibrant objects. If V is a Quillen left adjoint, then we are assuming that*
* it
preserves qequivalences between wellgrounded objects. Since GVA consists of
wellsectioned qffibrant objects, it follows in both cases that V :GVA ! GVB
passes straight to homotopy categories to give V :Ho GVA ! HoGVB . Since V
preserves GCW homotopy types on total spaces, V takes qequivalences to h
equivalences. Therefore P V takes qequivalences to fpequivalences and induc*
*es
a functor Ho GVA  ! hGWB . To show that P V J and either V Q or V R agree
under the equivalence of categories, it suffices to verify that the following d*
*iagram
9.3. THE FUNCTORS f* AND FB ON HOMOTOPY CATEGORIES 115
commutes.
V Q orV R
Ho GKA __________//_HoGKB
RQ  PRQ
fflffl fflffl
Ho GVA ____PV_____//_hGWB
We have a natural acyclic qffibration QX ! X and a natural acyclic qf
cofibration X  ! RX. If V is a Quillen left adjoint, then we have a zigzag
of natural qequivalences
RQV Q _____//RV Qoo___V Q ____//_V RQ
because V preserves acyclic qfcofibrations. If V is a Quillen right adjoint, t*
*hen
we have a zigzag of natural qequivalences
RQV R oo___ RQV RQ ____//_RV RQoo___V RQ
because V preserves qequivalences between qffibrant objects. In both cases, a*
*ll
objects have total spaces of the homotopy types of GCW complexes, so in fact
we have zigzags of hequivalences. Therefore, applying P gives us zigzags of *
*fp
equivalences in GWB , by Proposition 8.3.2.
Remark 9.2.3. When V preserves exfibrations, P V is naturally fpequivalent
to V on exfibrations, by Proposition 8.3.2. The derived functor of V can then *
*be
obtained directly by applying V and passing to equivalence classes of maps under
fphomotopy.
9.3. The functors f* and FB on homotopy categories
We use the equivalence between Ho GKB and hGWB to prove that, for any
map f :A ! B between spaces of the homotopy types of GCW complexes, the
(f*, f*) adjunction descends to homotopy categories. We begin by verifying that
f* satisfies the hypotheses of 9.2.2.
Proposition 9.3.1. Let f :A ! B be a map of base spaces. Then the base
change functor f* restricts to a functor f* :GWB ! GWA .
Proof. Consider Y in GWB . Since the total space of Y is of the homotopy ty*
*pe
of a GCW complex, the fibers Yb are of the homotopy types of GbCW complexes
by Theorem 3.4.2. The fiber (f*Y )a is a copy of Yf(a), and Ga acts through the
evident inclusion Ga Gf(a). Therefore (f*Y )a is of the homotopy type of a
GaCW complex. The total space of f*Y is therefore of the homotopy type of
a GCW complex, again by Theorem 3.4.2. Moreover, f*Y is an exfibration by
Proposition 8.2.2. Thus f* restricts to a functor f* :GWB ! GWA .
Theorem 9.3.2. For any map f :A ! B of base spaces, the right derived
functor f* :HoGKB ! Ho GKA has a right adjoint f*, so that
[f*Y, X]G,A ~=[Y, f*X]G,B
for X in GKA and Y in GKB .
116 9. THE EQUIVALENCE BETWEEN HoGKB AND hGWB
Proof. In view of the equivalence of categories in Theorem 9.1.2 and the
fact that f* descends directly to a functor f* :hGWB  ! hGWB on homotopy
categories, by Propositions 9.2.2 and 9.3.1, it suffices to construct a right a*
*djoint
f*: hGWA ! hGWB . We do that using the Brown representability theorem. By
Theorem 7.5.5, Ho GKB satisfies the formal hypotheses for Brown representabilit*
*y,
and therefore so does hGWB . In fact GWB has all of the relevant wedges and hom*
*o
topy colimits since these constructions preserve exfibrations by Proposition 8*
*.2.1
and Corollary 8.2.5 and since they clearly preserve GCW homotopy types on the
total space level and stay within GUB . The objects in the detecting set DB of
Definition 7.5.2 are not in GWB , but we can apply the exfibrant approximation
functor P to them to obtain a detecting set of objects in hGWB . Therefore a co*
*n
travariant setvalued functor on hGWB is representable if and only if it satisf*
*ies the
wedge and MayerVietoris axoms.
For a fixed exfibrant space X over A, consider the functor ss(f*Y, X)G,A on
Y in GWB , where ss denotes fphomotopy classes of maps. Since the functor
ss(W, X)G,A on W in GWA is represented and is computed using homotopy classes
of maps, it clearly satisfies the wedge and MayerVietoris axioms. It therefore*
* suf
fices to show that the functor f* preserves wedges and homotopy pushouts, since
that will imply that the functor ss(f*Y, X)G,A of Y also satisfies the wedge a*
*nd
MayerVietoris axioms. We can then conclude that there is an object f*X 2 GWB
that represents this functor. It follows formally that f* is a functor of X and*
* that
the required adjunction holds.
Because f* :GKB ! GKA is a left adjoint, it preserves colimits, and this im
plies that f* :GWB ! GWA preserves the relevant homotopy colimits. Moreover,
f* preserves fphomotopies and so induces a functor on homotopy categories that
still preserves these homotopy colimits.
We agree to write ' for natural equivalences on homotopy categories.
Remark 9.3.3. For composable maps f and g, g* O f* ' (g O f)* on homotopy
categories since f* O g* ' (g O f)* on homotopy categories. The latter equivale*
*nce
is clear since f* and g* are derived from Quillen right adjoints. More sophisti*
*cated
commutation laws are proven in the next section.
Applying Theorem 9.3.2 to diagonal maps and composing with the homotopy
category level adjunction between the external smash product and function exsp*
*ace
functors, we obtain the following basic result; compare Lemma 2.5.4.
Theorem 9.3.4. Define ^B and FB on Ho GKB as the composite (derived)
functors
X ^B Y = *(X Z Y ) and FB (X, Y ) = ~F(X, *Y ).
Then
[X ^B Y, Z]G,B ~=[X, FB (Y, Z)]G,B
for X, Y , and Z in Ho GKB .
Proof. The displayed adjunction is the composite of adjunctions for the (de
rived) external smash and function exspace functors and for the (derived) adjo*
*int
pair ( *, *).
Remark 9.3.5. The referee points out that the exspace analogue of [11, 7.2]
shows that we can work directly with the pointset topology to show that the
9.4. COMPATIBILITY RELATIONS FOR SMASH PRODUCTS AND BASE CHANGE 117
(^B , FB ) adjunction on the original category GKB is continuous and so descends
to (classical) fphomotopy categories to give the adjunction
hGKB (X ^B Y, Z) ~=hGKB (X, FB (Y, Z)).
Presumably similar pointset topological arguments work to show that, for a map
f :A ! B, we have an adjunction
hGKA (f*X, Y ) ~=hGKB (X, f*Y ).
These adjunctions do not imply our Theorems 9.3.2 and 9.3.4. By definition, our
category hGWB is a full subcategory of hGKB , but it is not an equivalent full
subcategory. The objects of GWB are very restricted, and general function ex
spaces FB (Y, Z) are not fphomotopy equivalent to such objects. The force of o*
*ur
theorems is that, after restricting to our subcategories hGWB , we still have r*
*ight
adjoints in these categories. It is this fact that we need to obtain right adjo*
*ints in
our preferred homotopy categories Ho GKB .
9.4. Compatibility relations for smash products and base change
We first prove that HoGKB satisfies the associativity, commutativity and uni*
*ty
conditions required of a symmetric monoidal category. We then show that all of
the isomorphisms of functors in Proposition 2.2.1 and some of the isomorphisms
of functors in Proposition 2.2.9 still hold after passage to homotopy categorie*
*s.
Finally, we relate change of groups and passage to fibers to the symmetric mono*
*idal
structure on homotopy categories. In some of our arguments, it is natural to wo*
*rk
in Ho GKB . In others, it is natural to work in the equivalent category hGWB .
Proposition 9.4.1. For maps f :A ! B and g :A0 ! B0 of base spaces
and for exspaces X over B and Y over B0,
(9.4.2) (f*Y Z g*Z) ' (f x g)*(Y Z Z)
in Ho GKA . For exspaces W over A and X over A0,
(9.4.3) (f!W Z g!X) ' (f x g)!(W Z X)
in Ho GKB .
Proof. For (9.4.2), we work with exfibrations, starting in hGWBxB0 . By
Propositions 8.2.2 and 8.2.3, the functors we are dealing with preserve exfibr*
*ations
and therefore descend straight to homotopy categories. The conclusion is thus i*
*m
mediate from its pointset level analogue. For (9.4.3), we work with model theo*
*retic
homotopy categories, starting in Ho GKAxA0. Since (f x g)!' (f x id)!O (idx g)!,
we can proceed in two steps and so assume that g = id. By Corollary 7.3.3 and
Proposition 7.3.4, we are then composing Quillen left adjoints. Starting with q*
*f
cofibrant objects, we do not need to apply qfcofibrant approximation, and the
conclusion follows directly from its pointset level analogue.
We use this to complete the proof that Ho GKB is symmetric monoidal.
Theorem 9.4.4. The category Ho GKB is closed symmetric monoidal under
the functors ^B and FB .
118 9. THE EQUIVALENCE BETWEEN HoGKB AND hGWB
Proof. In view of Theorem 9.3.4, we need only prove the associativity, com
mutativity, and unity of ^B up to coherent natural isomorphism. The external
smash product has evident associativity, commutativity, and unity isomorphisms,
and these descend directly to homotopy categories since the external smash prod*
*uct
of qfcofibrant exspaces over A and B is qfcofibrant over AxB. To see that th*
*ese
isomorphisms are inherited after internalization along *, we use (9.4.2). For *
*the
associativity of ^B , we have
*( *(X Z Y ) Z Z) ' *( x id)*((X Z Y ) Z Z) ' (( x id) )*((X Z Y ) Z Z)
' ((idx ) )*(XZ(Y ZZ)) ' *(idx )*(XZ(Y ZZ)) ' *(XZ *(Y ZZ)).
The commutativity of ^B is similar but simpler. For the unit, we observe that
S0B' r*S0, r :B ! *. Therefore, since (idx r) = id,
X ^B S0B' *(X Z r*S0) ' *(idx r)*(X Z S0) ' ((idx r) )*(X) = X.
We turn next to the derived versions of the base change compatibilities of
Propositions 2.2.1 and 2.2.9. Observe that the functor f!is good since the sect*
*ion
of a wellsectioned exspace is an hcofibration and since GCW homotopy types
are preserved under pushouts, one leg of which is an hcofibration. Moreover,
f! preserves qequivalences between wellsectioned exspaces by Proposition 7.3*
*.4.
Therefore Proposition 9.2.2 applies to f!.
Theorem 9.4.5. For a Gmap f :A ! B, f* :Ho GKB  ! Ho GKA is a
closed symmetric monoidal functor.
Proof. Since f*S0B ~=S0Ain GKA and S0B is qffibrant, f*S0B ' S0Ain
Ho GKA . We must prove that the isomorphisms (2.2.2) through (2.2.6) descend to
equivalences on homotopy categories. Categorical arguments in [40, xx2, 3] show
that it suffices to show that the two isomorphisms (2.2.2) and (2.2.5) descend *
*to
equivalences on homotopy categories. These two isomorphisms do not involve the
right adjoints f* or * and are therefore more tractable than the other three. *
*First
consider (2.2.2):
f*(Y ^B Z) ~=f*Y ^A f*Z.
If Y and Z are in GWB , then the two sides of this isomorphism are both in GWA ,
by Proposition 8.2.2 and Proposition 8.2.3. Therefore the pointset level isomo*
*r
phism descends directly to the desired homotopy category level equivalence. Nex*
*t,
consider (2.2.5):
f!(f*Y ^A X) ~=Y ^B f!X.
Assume that X is in GWA and Y is in GWB . The functor f!does not preserve ex
fibrations so, to pass to derived categories, we must replace it by P f!on both*
* sides.
By Proposition 8.2.6, the functor Y ^B () preserves hequivalences between wel*
*l
sectioned exspaces. Since P sends hequivalences to fpequivalences, we theref*
*ore
have fpequivalences, natural up to fphomotopy,
P(id^BOE) OE
P f!(f*Y ^A X) ~=P (Y ^B f!X)____//P (Y ^B P f!X)oo__Y ^B P f!X,
where OE = (ae, W ') is the zigzag of hequivalences of Definition 8.3.4. This *
*implies
the desired equivalence in the homotopy category.
9.4. COMPATIBILITY RELATIONS FOR SMASH PRODUCTS AND BASE CHANGE 119
The reader is invited to try to prove directly that the projection formula h*
*olds
in the homotopy category. Even the simple case of f :* ! B, the inclusion of a
point, should demonstrate the usefulness of Proposition 9.2.2.
Theorem 9.4.6. Suppose given a pullback diagram of Gspaces
g
C _____//D
i j
fflfflfflffl
A __f__//B
in which f (or j) is a qfibration. Then there are natural equivalences of func*
*tors
on homotopy categories
(9.4.7) j*f!' g!i*, f*j* ' i*g*, f*j!' i!g*, j*f* ' g*i*.
Proof. As in Proposition 2.2.9 the second and fourth equivalences are conju
gate to the first and third. However, since the situation is no longer symmetri*
*c, we
must prove both the first and third equivalences, assuming f is a qfibration.
First consider the desired equivalence f*j!' i!g*. We work with exfibration*
*s,
starting with X 2 hGWD . We must replace j!and i!by P j!and P i!before passing
to homotopy categories. By Proposition 7.3.4, f* preserves qequivalences since*
* f is
a qfibration. Moreover, our qequivalences are hequivalences since we are dea*
*ling
with total spaces of the homotopy types of GCW complexes. By the diagram in
Lemma 8.4.3(ii), we see that ~: P f* ! f*P is a natural hequivalence here. Th*
*is
would be false for arbitrary maps f, as observed in Warning 8.4.4. Since ~ is an
hequivalence between exfibrations, it is an fpequivalence. Therefore
f*P j!X ' P f*j!X ~=P i!g*X.
Now consider the desired equivalence j*f!X ' g!i*X in Ho GKD . Our assump
tion that f is a qfibration gives us no direct help with this. However, we may
factor j as the composite of a homotopy equivalence and an hfibration. Expand
ing our pullback diagram as a composite of pullbacks, we see that it suffices to
prove our commutation relation when j is an hfibration and when j is a homotopy
equivalence. The first case is immediate by symmetry from the first part. Thus
assume that j is a homotopy equivalence. Then i is also a homotopy equivalence.
By Proposition 7.3.4, (i!, i*) and (j!, j*) are adjoint equivalences of homotop*
*y cat
egories. Therefore
j*f!' j*f!i!i* ' j*j!g!i* ' g!i*.
Finally, we turn to a promised compatibility relationship between products a*
*nd
change of groups. We observed in Proposition 7.4.6 that the pointset level clo*
*sed
symmetric monoidal equivalence of Proposition 2.3.9 is given by a Quillen equiv*
*a
lence. The following addendem shows that the resulting equivalence on homotopy
categories is again closed symmetric monoidal.
Proposition 9.4.8. Let ': H  ! G be the inclusion of a subgroup and A
be an Hspace. The Quillen equivalence ('!, *'*) descends to a closed symmetric
monoidal equivalence between HoHKA and HoGK'!A.
Proof. Let : A ! A x A be the diagonal map. The isomorphisms
'* *(X Z Y ) ~= *'*(X Z Y ) ~= *('*X Z '*Y )
120 9. THE EQUIVALENCE BETWEEN HoGKB AND hGWB
descend to equivalences on homotopy categories, the first since it is between Q*
*uillen
right adjoints, the second since '* preserves all qequivalences. It follows t*
*hat
*'* is a symmetric monoidal functor on homotopy categories. Since it is also
an equivalence, it follows formally that it is closed symmetric monoidal.
Combined with Theorem 9.4.5 applied to the inclusion "b:G=Gb ! B, this
last observation gives us the following conclusion.
Theorem 9.4.9. The derived fiber functor ()b: HoGKB ! Ho GbKb is
closed symmetric monoidal, and it has a left adjoint ()b and a right adjoint b*
*().
We emphasize that this innocent looking result packages highly nontrivial a*
*nd
important information. It gives in particular that, for exGspaces X and Y , t*
*he
(derived) fiber FB (X, Y )b of the (derived) function space FB (X, Y ) is equiv*
*alent in
Ho GbKb to the (derived) function space F (Xb, Yb) of the (derived) fibers Xb a*
*nd
Yb. On the point set level, that is what motivated the definition of the inter*
*nal
function exspace. That it still holds on the level of homotopy categories is a
reassuring consistency result.
Part III
Parametrized equivariant stable
homotopy theory
Introduction
We develop rigorous foundations for parametrized equivariant stable homotopy
theory. The idea is to start with a fixed base Gspace B and to build a good
category, here denoted GSB , of Gspectra over B. We assume once and for all
that our base spaces B must be compactly generated and must have the homotopy
types of GCW complexes. By "good" we mean that GSB is a closed symmet
ric monoidal topological model category whose associated homotopy category has
properties analogous to those of the ordinary equivariant stable homotopy categ*
*ory.
Informally, the homotopy theory of GSB is specified by the homotopy theory
seen on the fibers of Gspectra over B. One compelling reason for taking the
parametrized stable homotopy category seriously, even nonequivariantly, is to b*
*uild
a natural home in which one can do stable homotopy theory while still keeping
track of fundamental groups and groupoids. Stable homotopy theory has tended to
ignore such intrinsically unstable data. This has the effect of losing contact *
*with
more geometric branches of mathematics in which the fundamental group cannot
be ignored.
For example, one basic motivation for the equivariant theory is that it gives
a context in which to better understand equivariant orientations, Thom isomor
phisms, and Poincar'e duality. There is no problem for Gsimply connected mani
folds M [59, IIIx6], but restriction to such M is clearly inadequate for applic*
*ations
to transformation group theory. Despite a great deal of work on the subject by
Costenoble and Waner, and some by May, [24, 25, 26, 27, 69], this circle of ide*
*as
is not yet fully understood. Costenoble and Waner [28] use our work to study th*
*is
problem for ordinary equivariant theories, and for general theories this is wor*
*k in
progress by the second author.
There are many problems that make the development far less than an obvi
ous generalization of the nonparametrized theory. Problems on the space level
were dealt with in Parts I and II, and we deal with the analogous spectrum level
problems here. We give some categorical preliminaries on enriched equivariant c*
*at
egories in Chapter 10. We define and develop the basic properties of our prefer*
*red
category of parametrized Gspectra in Chapter 11, study its model structures in
Chapter 12, and study adjunctions and compatibility relations in Chapter 13. All
of the problems that we faced on the space level are still there, but their sol*
*utions
are considerably more difficult. In Chapter 14, we go on to study further such
compatibilities that more fundamentally involve equivariance.
The theory of highly structured spectra is highly cumulative. We build on the
theory of equivariant orthogonal spectra of Mandell and May [61]. In turn, that
theory builds on the theory of nonequivariant orthogonal spectra. A selfcontai*
*ned
treatment of nonequivariant diagram spectra, including orthogonal spectra, is g*
*iven
by Mandell, May, Schwede, Shipley in [62]. The treatments of [61] and [62], like
122
INTRODUCTION 123
this one, are topological as opposed to simplicial. That seems to be essential *
*when
dealing with infinite groups of equivariance. It also allows use of orthogonal *
*spectra
rather than symmetric spectra. These are much more natural equivariantly and,
even nonequivariantly, they have the major convenience that their weak equiva
lences are exactly the maps that induce isomorphisms of homotopy groups.
The theory of equivariant parametrized spectra can be thought of as the push*
*out
over the theory of spectra of the theories of equivariant spectra and of nonequ*
*iv
ariant parametrized spectra. However, there is no nonequivariant precursor of t*
*he
present treatment of parametrized spectra in the literature. There are prelimin*
*ary
forms of such a theory [2, 3, 18, 19, 29], but these either do not go beyond su*
*s
pension spectra or are based on obsolescent technology. None of them go nearly
far enough into the theory for the purposes we have in mind, although the early
first approximation of Monica Clapp [18], written up in more detail with Dieter
Puppe [19], deserves considerable credit. Clapp gave the strongest previous ver*
*sion
of our fiberwise duality theorem, and her emphasis on exfibrations, together w*
*ith
some key technical results about them, have been very helpful. The reader prima*
*r
ily interested in classical homotopy theory should ignore all details of equiva*
*riance
in reading Chapters 1113. In fact, given [61], the equivariance adds few serio*
*us
difficulties to the passage from spectra to parametrized spectra, although it d*
*oes
add many interesting new features.
There are at least two possible alternative cumulative approaches. Rather th*
*an
building on the theory of orthogonal Gspectra of [61, 62], one can build on the
theory of Gspectra of [59], the theory of Smodules of [39], and the pushout of
these, the theory of SG modules of [61]. Po Hu [47] began work on the first st*
*age
of a treatment along these lines, using parametrized Gspectra, but she did not
address the foundational issues concerning smash products, function spectra, ba*
*se
change functors, and compatibility relations considered here. Moreover, followi*
*ng
the first author's misleading unpublished notes [72], she took the qmodel stru*
*cture
on exGspaces as her starting point, and the stable model structure cannot be
made rigorous from there. It appears to us that resolving all of these issues i*
*n that
framework is likely to be more difficult than in the framework that we have ado*
*pted.
In particular, homotopical control of the parametrized spectrification functor *
*and
of cofiber sequences seems problematic.
Alternatively, for finite groups G, one can build on the theory of symmetric
spectra of Hovey, Smith, and Shipley [46] and its equivariant generalization du*
*e to
Mandell [60]. Such an approach would avoid the pointset topological technicali*
*ties
of the present approach and would presumably lead to rather different looking
problems with fibrations and cofibrations. The problems with the stable homotopy
category level adjunctions that involve base change functors, smash products, a*
*nd
function spectra are intrinsic and would remain. Our solutions to these problems
do not seem to carry over to the simplicial context in an obvious way, and an
alternative simplicial treatment could prove to be quite illuminating.
In view of the understanding of unstable equivariant homotopy theory for
proper actions of noncompact Lie groups that was obtained in Part II, it might
seem that there should be no real difficulty in obtaining a good stable theory *
*along
the same lines as the theory for compact Lie groups. However, in contrast with *
*the
rest of this book, equivariant stable homotopy theory for noncompact Lie groups
is in preliminary and incomplete form, with still unresolved technical problems*
*. We
124 INTRODUCTION
leave its study to future work, explaining in x11.6 where some of the problems *
*lie.
Except in that section, G is asssumed to be a compact Lie group from Chapter 11
onwards.
A few other notes on terminology may be helpful. We shall not use the term
"exspectrum over B" since, stably, there is no meaningful unsectioned theory.
Instead, we shall use the term "spectrum over B". This is especially convenient
when considering base change. We write out "orthogonal Gspectrum over B"
until x11.4. However, since we consider no other kinds of Gspectra and work
equivariantly throughout, we later abbreviate this to "spectrum over B" when
there is no danger of confusion. That is, we work equivariantly throughout, but*
* we
only draw attention to this fact when it plays a significant mathematical role.
CHAPTER 10
Enriched categories and Gcategories
Introduction
To give context for the structure enjoyed by the categories of parametrized
orthogonal Gspectra that we shall define, we first describe the kind of equiva*
*riant
parametrized enrichments that we shall encounter. In fact, our categories have *
*sev
eral layers of enrichment, and it is helpful to have a consistent language, som*
*ewhat
nonstandard from a categorical point of view, to keep track of them. In xx10.1*
* and
10.2, we give some preliminaries on enriched categories, working nonequivariant*
*ly
in x10.1 and adding considerations of equivariance in x10.2. We discuss the rol*
*e of
the several enrichments in sight in our Gtopological model Gcategories in x10*
*.3.
In this chapter, G can be any topological group.
10.1.Parametrized enriched categories
As discussed in x1.2, all of our categories C are topological, meaning that
they are enriched over the category K* of based spaces (= kspaces). In contrast
with general enriched category theory and our further enrichments, the topologi*
*cal
enrichment is given just by a topology on the underlying set of morphisms, and *
*we
denote the space of morphisms X ! Y by C (X, Y ). We say that a topological
category C is topologically bicomplete if it is bicomplete and bitensored over *
*K*.
In fact, we shall have enrichments and bitensorings over the category KB of ex
spaces over B that imply the topological enrichment and bitensoring by restrict*
*ion
to exspaces B x T for T 2 K*.
Recall from x1.3 that KB is topologically bicomplete, with tensors and coten
sors denoted by K ^B T and FB (T, K) for T 2 K* and K 2 KB . (Since we shall
use letters like X, Y , and Z for spectra, we have changed the letters that we *
*use
generically for spaces and exspaces from those that we used earlier). It is a*
*lso
closed symmetric monoidal under its fiberwise smash product and function space
functors, which are also denoted by ^B and FB ; its unit object is S0B= B x S0.
It is therefore enriched and bitensored over itself. The two enrichments are re*
*lated
by natural based homeomorphisms
(10.1.1) KB (K, L) ~=KB (S0B, FB (K, L)).
This is the case T = S0 of the more general based homeomorphism
(10.1.2) K*(T, KB (K, L)) ~=KB (S0B^B T, FB (K, L))
for T 2 K* and K, L 2 KB . The Yoneda lemma, (10.1.1), and the bitensoring
adjunctions imply that the two bitensorings are related by the equivalent natur*
*al
isomorphisms of exspaces
(10.1.3) K ^B T ~=K ^B (S0B^B T ) and FB (T, K) ~=FB (S0B^B T, K).
125
126 10. ENRICHED CATEGORIES AND GCATEGORIES
These in turn imply the equivalent generalizations
(10.1.4)
K ^B (L ^B T ) ~=(K ^B L) ^B T and FB (T, FB (K, L)) ~=FB (K ^B T, L).
Formally, rather than defining the enrichments and bitensorings over K* indepen
dently, we can take (10.1.2) and (10.1.3) as definitions of these structures in*
* terms
of the enrichment and bitensoring over KB . Then (10.1.4) and the bitensoring
adjunction homeomorphisms
(10.1.5) KB (K ^B T, L) ~=K*(T, KB (K, L)) ~=KB (K, FB (T, L))
follow directly.
Remark 10.1.6. We shall be making much use of the functor S0B^B (), and
we henceforward abbreviate notation by setting
TB = B x T = S0B^B T
for a based space T , and similarly for maps. Observe that K ^B T and K ^B TB
are two names for the same exspace over B. When working on a formal conceptual
level, it is often best to think in terms of tensors over K* and use the first *
*name.
However, on a pragmatic level, to avoid confusion, it is best to view based spa*
*ces
as embedded in exspaces via S0B^B () and to use the second notation, working
only with tensors over KB .
We generalize and formalize several aspects of the discussion above.
Definition 10.1.7. A topological category C is topological over B if it is e*
*n
riched and bitensored over KB . It is topologically bicomplete over B if it is*
* also
bicomplete. We write PB (X, Y ) for the hom exspace over B, and we write X ^B K
and FB (K, X) for the tensor and cotensor in C , where X, Y 2 C and K 2 KB .
Explicitly, we require bitensoring adjunction homeomorphisms of based spaces
(10.1.8) C (X ^B K, Y ) ~=KB (K, PB (X, Y )) ~=C (X, FB (K, Y )).
By Yoneda lemma arguments, these imply unit and transitivity isomorphisms in C
(10.1.9) X ~=X ^B S0B and X ^B (K ^B L) ~=(X ^B K) ^B L.
and also bitensoring adjunction isomorphisms of exspaces
(10.1.10) PB (X ^B K, Y ) ~=FB (K, PB (X, Y )) ~=PB (X, FB (K, Y )).
Conversely, there is a natural homeomorphism
(10.1.11) C (X, Y ) ~=KB (S0B, PB (X, Y )),
and the isomorphisms (10.1.8) follow from (10.1.10) by applying KB (S0B, ).
If we do not require C to be topological to begin with, we can take (10.1.11)
as the definition of the space C (X, Y ) and so recover the topological enrichm*
*ent.
With the notation of Remark 10.1.6, we obtain tensors and cotensors with based
spaces T by setting
(10.1.12) X ^B T = X ^B TB and FB (T, X) = FB (TB , X).
The adjunction homeomorphisms
(10.1.13) C (X ^B T, Y ) ~=K*(T, C (X, Y )) ~=C (X, FB (T, Y ))
are obtained by replacing K by TB in (10.1.8) and using (10.1.2) and (10.1.11).
10.2. EQUIVARIANT PARAMETRIZED ENRICHED CATEGORIES 127
In the cases of interest, C is closed symmetric monoidal, and then the hom
exspaces PB (X, Y ) can be understood in terms of the internal hom in C by the
following definition and result.
Definition 10.1.14. Let C be a topological category over B with a closed
symmetric monoidal structure given by a product ^B and function object functor
FB , with unit object SB . We say that C is a topological closed symmetric mono*
*idal
category over B if the tensors and products are related by a natural isomorphism
X ^B K ~=X ^B (SB ^B K)
in C for K 2 KB and X 2 C .
Proposition 10.1.15. Let C be a topological closed symmetric monoidal cate
gory over B. Then, for K 2 KB and X, Y , Z 2 C , there are natural isomorphisms
FB (K, Y ) ~=FB (SB ^B K, Y ),
PB (X, Y ) ~=PB (SB , FB (X, Y )),
PB (X ^B Y, Z) ~=PB (X, FB (Y, Z))
in C and a natural homeomorphism of based spaces
KB (K, PB (X, Y )) ~=C (SB ^B K, FB (X, Y )).
10.2.Equivariant parametrized enriched categories
Turning to the equivariant generalization, we give details of the context of
topological Gcategories, continuous Gfunctors, and natural Gmaps that we fir*
*st
alluded to in x1.4. The discussion elaborates on that given in [61, IIx1]. Gene*
*rically,
we use notations of the form CG and GC to denote a category CG enriched over the
category GK* of based Gspaces and its associated "Gfixed category" GC with
the same objects and the Gmaps between them; GC is enriched over K*. We shall
write (CG , GC ) for such a pair, and we shall refer to the pair as a "Gcatego*
*ry".
In the terminology of enriched category theory, GC is the underlying topolog*
*ical
category of CG . The hom objects of CG are Gspaces CG (X, Y ); Gfunctors and
natural Gmaps just mean functors and natural transformations enriched over GK*.
Consistently with enriched category theory, the space GC (X, Y ) = CG (X, Y )G *
*can
be identified with the space of Gmaps S0 ! CG (X, Y ). We call the points of
CG (X, Y ) "arrows" to distinguish them from the points of GC (X, Y ), which we
call "Gmaps", or often just "maps", with the equivariance understood.
We cannot expect CG to have limits and colimits, but GC is usually bicomplet*
*e.
In many of our examples, both CG and GC are closed symmetric monoidal under
functors ^B and FB . For example, we have the closed symmetric monoidal G
category (KG,B, GKB ) of exGspaces over a Gspace B described in x1.4.
Definition 10.2.1. A Gcategory (CG , GC ) is Gtopological over B if CG is
enriched over GKB and bitensored over KG,B. It follows that GC is enriched over
KB and bitensored over GKB . We say that (CG , GC ) is Gtopologically bicomple*
*te
over B if, in addition, GC is bicomplete. We write PB (X, Y ) for the hom exG
space over B, and we write X ^B K and FB (K, X) for the tensor and cotensor in
CG , where X, Y 2 CG and K 2 KG,B. Explicitly, we require bitensoring adjunction
homeomorphisms of based Gspaces
(10.2.2) CG (X ^B K, Y ) ~=KG,B(K, PB (X, Y )) ~=CG (X, FB (K, Y )).
128 10. ENRICHED CATEGORIES AND GCATEGORIES
There result coherent unit and transitivity isomorphisms in GC
(10.2.3) X ~=X ^B S0B and X ^B (K ^B L) ~=(X ^B K) ^B L
and also bitensoring adjunction isomorphisms of exGspaces
(10.2.4) PB (X ^B K, Y ) ~=FB (K, PB (X, Y )) ~=PB (X, FB (K, Y )).
Conversely, there is a natural homeomorphism of based Gspaces
(10.2.5) CG (X, Y ) ~=KG,B(S0B, PB (X, Y )),
and the isomorphisms (10.2.2) follow from (10.2.4) by applying KG,B(S0B, ). Pa*
*s
sage to Gfixed points from (10.2.2) gives the bitensoring adjunction homeomor
phisms of based spaces
(10.2.6) GC (X ^B K, Y ) ~=GKB (K, PB (X, Y )) ~=GC (X, FB (K, Y )).
We warn the reader that we shall not always adhere strictly to the notational
pattern of Definition 10.2.1 for our several layers of enrichment. In particula*
*r, in
the domain categories for our equivariant diagram spaces and diagram spectra, o*
*nly
CG is of interest, not GC , and our notations will reflect that. On the other h*
*and,
when studying model categories, it is always the bicomplete category GC that is
of fundamental interest.
If (CG , GC ) is Gtopological over B, then it is automatically Gtopological
(over *). This enrichment is recovered by taking (10.1.11), read equivariantly,*
* as
the definition of the based Gspace CG (X, Y ). Just as in the nonequivariant c*
*ase,
for based Gspaces T and objects X of CG , the tensors and cotensors in CG and
GC are given on objects by
(10.2.7) X ^B T = X ^B TB and FB (T, X) = FB (TB , X),
using the notation of Remark 10.1.6 equivariantly. The required Ghomeomorphisms
(10.2.8) CG (X ^B T, Y ) ~=KG,*(T, CG (X, Y )) ~=CG (X, FB (T, Y ))
follow directly.
We have equivariant analogues of Definition 10.1.14 and Proposition 10.1.15.
Definition 10.2.9. Let (CG , GC ) be a Gtopological Gcategory over B with
a closed symmetric monoidal structure given by a product Gfunctor ^B and a
function object Gfunctor FB , with unit object SB . We say that (CG , GC ) is
a Gtopological closed symmetric monoidal Gcategory over B if the tensors and
products are related by a natural isomorphism
X ^B K ~=X ^B (SB ^B K)
in GC for K 2 GKB and X 2 GC .
Proposition 10.2.10. Let (CG , GC ) be a Gtopological closed symmetric mon
oidal Gcategory over B. Then, for K 2 KB and X, Y , Z 2 C , there are natural
isomorphisms
FB (K, Y ) ~=FB (SB ^B K, Y ),
PB (X, Y ) ~=PB (SB , FB (X, Y )),
PB (X ^B Y, Z) ~=PB (X, FB (Y, Z))
in GC and there is a natural homeomorphism of based Gspaces
KG,B(K, PB (X, Y )) ~=CG (SB ^B K, FB (X, Y )).
10.3. GTOPOLOGICAL MODEL GCATEGORIES 129
10.3.Gtopological model Gcategories
We explain what it means for a Gtopological Gcategory (CG , GC ) over B to
have a Gtopological model structure. This structure implies in particular that*
* the
homotopy category HoGC is bitensored over the homotopy category HoGK . We
need some notation. Throughout this section, we consider maps
i: W ! X, j :V ! Z, and p: E ! Y
in GC and a map k :K ! L in either GKB or GK*; in the latter case we apply
the functor ()B = B x () to k and so regard it as a map in GKB , as suggested
in Remark 10.1.6. We shall define the notion of a Gtopological model category *
*in
terms of the induced map
(10.3.1) CG (i, p): CG (X, E) ! CG (W, E) xCG(W,Y )CG (X, Y )
of based Gspaces. Passing to Gfixed points, this gives rise to a map
(10.3.2) GC (i, p): GC (X, E) ! GC (W, E) xGC(W,Y )GC (X, Y )
of based spaces, and we have the following motivating observation.
Lemma 10.3.3. The pair (i, p) has the lifting property if and only if the fu*
*nction
GC (i, p) is surjective.
Definition 10.3.4. Let (CG , GC ) be a Gtopological Gcategory over B such
that GC is a model category. We say that the model structure is Gtopological if
CG (i, p) is a fibration in GK* when i is a cofibration and p is a fibration an*
*d is
acyclic when, further, either i or p is acyclic.
Remark 10.3.5. The definition must refer consistently to either htype or q
type model structures. The resulting notions are quite different. We usually ha*
*ve in
mind a qtype model structure. In that case, the weak equivalences and fibratio*
*ns
are often characterized by conditions on the Hfixed point maps fH of a map f.
If F is a family of subgroups of G, such as the family G of compact subgroups,
we can restrict attention to those H 2 F . The resulting F equivalences and F 
fibrations usually specify another model structure on GC . In particular, we ha*
*ve
the F model structure on GK*. For the qftype model structures of x7.2, we must
start with a generating set C that contains the orbits G=H with H 2 F \ G and
consists of F \ G cell complexes. We say that an F model structure on GC is
F topological if the condition of the previous definition holds when we use th*
*e F 
notions of fibration, cofibration and weak equivalence throughout. The observat*
*ions
of this section generalize to F topological model categories for any family F .
In addition to the map of Gspaces displayed in (10.3.1), we have a map
(10.3.6) PB (i, p): PB (X, E) ! PB (W, E) xPB(W,Y )PB (X, Y )
of exGspaces over B.
Warning 10.3.7. We can define what it means for (CG , GC ) to be Gtopologic*
*al
over B, using the map PB (i, p) of exspaces rather than the map CG (i, p) of s*
*paces.
However, we know of no examples where this condition is satisfied. For example,
(KG,B, GKB ) is Gtopological, by Theorems 7.2.3 and 7.2.8, but, as Warning 6.1*
*.7
makes clear by adjunction, we cannot expect it to be Gtopological over B.
130 10. ENRICHED CATEGORIES AND GCATEGORIES
Just as in the classical theory of simplicial or topological model categorie*
*s, there
are various equivalent reformulations of what it means for GC to be Gtopologic*
*al.
To explain them, observe that the tensors and cotensors with exGspaces over B
give rise to induced maps
(10.3.8) i B k :(X ^B K) [W^BK (W ^B L) ! X ^B L
and
(10.3.9) FB (k, p): FB (L, E) ! FB (K, E) xFB(K,Y )FB (L, Y )
of exGspaces over B. If (CG , GC ) is closed symmetric monoidal, then we also
have the induced maps
(10.3.10) i B j :(X ^B V ) [W^BV (W ^B Z) ! X ^B Z
and
(10.3.11) FB (j, p): FB (Z, E) ! FB (V, E) xFB(V,Y )FB (Z, Y )
in GC . We have various adjunction isomorphisms relating these various product
maps and function object maps.
Proposition 10.3.12. If k is a map of exGspaces over B, then there are
adjunction isomorphisms
(10.3.13) PB (i B k, p) ~=FB (k, PB (i, p)) ~=PB (i, FB (k, p))
of maps of exGspaces over B and
(10.3.14) CG (i B k, p) ~=KG,B(k, PB (i, p)) ~=CG (i, FB (k, p))
of maps of based Gspaces. If k is a map of based Gspaces, then the last pair *
*of
isomorphisms can be rewritten as
(10.3.15) CG (i B k, p) ~=KG,*(k, CG (i, p)) ~=CG (i, FB (k, p)).
When (CG , GC ) is closed symmetric monoidal there are adjunction isomorphisms
(10.3.16) PB (i B k, p) ~=PB (i, FB (k, p))
of maps of exGspaces over B and
(10.3.17) CG (i B k, p) ~=CG (i, FB (k, p))
of maps of based Gspaces.
Together with Lemma 10.3.3, this implies the promised alternative equivalent
conditions that describe when a model category is Gtopological.
Proposition 10.3.18. Let (CG , GC ) be a Gtopological Gcategory over B such
that GC has a model structure. Then the following conditions are equivalent.
(i)The map i B k of (10.3.8) is a cofibration in GC if i is a cofibration in GC
and k is a cofibration in GK*. It is acyclic if either i or k is acyclic.
(ii)The map FB (k, p) of (10.3.9) is a fibration in GC if p is a fibration in *
*GC
and k is a cofibration in GK*. It is acyclic if either p or k is acyclic.
(iii)The map CG (i, p) of (10.3.1) is a fibration in GK* if i is a cofibration *
*in GC
and p is a fibration in GC . It is acyclic if either i or p is acyclic.
10.3. GTOPOLOGICAL MODEL GCATEGORIES 131
Proof. The third condition is our definition of the model structure being
Gtopological. We prove that the first condition is equivalent to the third. A
similar argument shows that the second condition is also equivalent to the thir*
*d.
The map CG (i, p) is a fibration if and only if (k, CG (i, p)) has the lifting *
*property
with respect to all acyclic cofibrations k in GK*. By Lemma 10.3.3 and the first
adjunction isomorphism in (10.3.15), that holds if and only if (i B k, p) has t*
*he
lifting property, that is, if and only if i B k is an acyclic cofibration. If e*
*ither i or
p is acyclic, then we take k to be a cofibration in GK* and argue similarly.
CHAPTER 11
The category of orthogonal Gspectra over B
Introduction
Intuitively, an orthogonal spectrum X over B consists of exspaces X(V ) ove*
*r B
and exmaps oe :X(V )^B SW ! X(V W ) for suitable inner product spaces V and
W . The orthogonal group O(V ) must act on X(V ), and oe must be (O(V )xO(W ))
equivariant. The orthogonal group actions enable the definition of a good exter*
*nal
smash product. Moreover, they will later allow us to define stable weak equival*
*ences
in terms of homotopy groups, as would not be possible if we only had actions by
symmetric groups.
Similarly, use of general inner product spaces allows us to build in actions
by a compact Lie group G without difficulty. For noncompact Lie groups, we
should ignore inner products and use linear isomorphisms, replacing the compact
orthogonal group O(V ) by the general linear group GL(V ). However, as we expla*
*in
in x11.6, there are more serious problems in generalizing to noncompact Lie gr*
*oups;
except in that section, we require G to be a compact Lie group.
Working equivariantly, we first describe X as a suitable diagram of exGspa*
*ces
in x11.1. The domain category for our diagrams is denoted IG and is independent
of B. We then build in the structure maps oe in x11.2, where we define the cate*
*gory
of orthogonal Gspectra over B. In x11.3, we show that it too can be described *
*as a
category of diagrams of exGspaces. The domain category here is denoted JG,B.
It does depend on B, as indicated by the notation. The formal properties of the
category of exGspaces over B carry over to the category of orthogonal Gspect*
*ra
over B, but there are some new twists. For example, our category of Gspectra
over B is enriched not just over based Gspaces, but more generally over exG
spaces over B. We discussed the relevant formalities in the previous chapter. T*
*his
enhanced enrichment is essential to the definition of function Gspectra over B.
We show in x11.4 that the base change functors and their properties also car*
*ry
over to these categories of parametrized Gspectra, and we discuss change of gr*
*oup
functors and restriction to fibers in x11.5.
11.1. The category of IG spaces over B
We recall the Gcategory (IG , GI ) from [61, II.2.1]. The objects and arrows
of IG are finite dimensional Ginner product spaces and linear isometric isomor
phisms. The maps of GI are Glinear isometries. More precisely, as dictated by
the general theory of [61, 62], we take IG (V, W ) as based with basepoint disj*
*oint
from the space of linear isometric isomorphisms V  ! W . As in [61, II.1.1], t*
*he
objects V run over the collection V of all representations that embed up to is*
*o
morphism in a given "Guniverse" U, where a Guniverse is a sum of countably
many copies of representations in a set of representations that includes the tr*
*ivial
132
11.1. THE CATEGORY OF IGSPACES OVER B 133
representation. We think in terms of a "complete Guniverse", one that contains
all representations of G, but the choice is irrelevant until otherwise stated. *
*As in
[61, II.2.2], we can restrict from V to any cofinal subcollection W that is c*
*losed
under direct sums.
Based Gspaces are exGspaces over *, and IG spaces are defined in [61,
II.2.3] as Gfunctors IG ! TG , where TG is the Gcategory of compactly gener
ated based Gspaces. One can just as well drop the weak Hausdorff condition,
which plays no necessary mathematical role in [61, 62 ], and allow general k
spaces. With the notations of Part II, we can thus change the target Gcategory
to KG,*. Then we generalize the definition to the parametrized context simply
by changing the target Gcategory to the category KG,B of exGspaces over a G
space B. Thus we define an IG space X over (and under) B to be a Gfunctor
X :IG ! KG,B. Using nonequivariant arrows and equivariant maps, we obtain
the Gcategory (IG KB , GI KB ) of IG spaces.
To unravel definitions, for each representation V 2 V we are given an exG
space X(V ) over B, for each arrow (linear isometric isomorphism) f :V ! W we
are given an arrow (nonequivariant map)
X(f): X(V ) ! X(W )
of exGspaces over B, and the continuous function
X :IG (V, W ) ! KG,B(X(V ), X(W ))
is a based Gmap. An arrow ff: X ! Y is just a natural transformation, and
a Gmap is a Gnatural transformation, for which each ffV :X(V ) ! Y (V ) is a
Gmap. For both arrows and Gmaps, the naturality diagrams
X(V )__ffV//_Y (V )
X(f) Y(f)
fflffl fflffl
X(W ) _ffW_//Y (W )
must commute for all arrows f :V ! W . The group G acts on the space
IG KB (X, Y ) of arrows by levelwise conjugation. The Gfixed category is denot*
*ed
by GI KB . It has objects the IG spaces X and maps the Gmaps.
To study the parametrized enrichment of the Gcategory of orthogonal G
spectra over B, it is convenient to extend the domain category IG , which is en*
*riched
over KG,*, to a new domain category IG,B that is enriched over KG,B. Departing
from the notational pattern of Definition 10.2.1 and using Remark 10.1.6, we de*
*fine
the hom exGspaces over B of IG,B by
(11.1.1) IG,B(V, W ) = IG (V, W )B B x IG (V, W ).
If X :IG ! KG,B is an IG space, then the given based Gmaps
X :IG (V, W ) ! KG,B(X(V ), X(W ))
correspond by adjunction (see (10.2.7) and (10.2.8)) to exGmaps
X(V ) ^B IG,B(V, W ) ! X(W ).
In turn, these correspond by the internal hom adjunction to exGmaps
X :IG,B(V, W ) ! FB (X(V ), X(W )).
134 11. THE CATEGORY OF ORTHOGONAL GSPECTRA OVER B
These give an equivalent version of the original Gfunctor X, but now in terms *
*of
categories enriched over the category GKB .
Lemma 11.1.2. The Gcategory (IG KB , GI KB ) of IG spaces is equivalent to
the Gcategory of IG,Bspaces, where an IG,Bspace is a Gfunctor X :IG,B !
KG,B enriched over GKB .
Proposition 11.1.3. The Gcategory (IG KB , GI KB ) is Gtopological over
B and thus also Gtopological. Therefore the category GI KB is topologically bi
complete over B.
Proof. We define tensor and cotensor IG spaces over B
X ^B K and FB (K, X)
levelwise, where K is an exGspace and X is an IG space. For IG spaces X
and Y , we must define a parametrized morphism exGspace PB (X, Y ) over B.
Parallelling a standard formal description of the Gspace IG KB (X, Y ), we def*
*ine
PB (X, Y ) to be the end
Z
(11.1.4) PB (X, Y ) = FB (X(V ), Y (V )).
IG,B
Explicitly, it is the equalizer displayed in the following diagram of exGspac*
*es.
PB (X, Y )


Q fflffl
V FB (X(V ), Y (V ))
"~"
Q ffflfflflffl
V,W FB (IG,B(V, W ), FB (X(V ), Y (W ))).
The products run over the objects and pairs of objects of a skeleton skIG of
IG . The (V, W )th coordinate of "~is given by the composite of the projection *
*to
FB (X(W ), Y (W )) and the Gmap
FB (X(W ), Y (W )) ! FB (IG,B(V, W ), FB (X(V ), Y (W )))
adjoint to the composite exGmap
FB (X(W ), Y (W )) ^B IG,B(V, W )
id^BX
fflffl
FB (X(W ), Y (W )) ^B FB (X(V ), X(W ))
O
fflffl
FB (X(V ), Y (W )).
The (V, W )th coordinate of "is the composite of the projection to FB (X(V ), Y*
* (V ))
and the Gmap
" V,W:FB (X(V ), Y (V )) ! FB (IG,B(V, W ), FB (X(V ), Y (W ))
11.1. THE CATEGORY OF IGSPACES OVER B 135
adjoint to the composite exGmap
IG,B(V, W ) ^B FB (X(V ), Y (V ))
Y^Bid
fflffl
FB (Y (V ), Y (W )) ^B FB (X(V ), Y (V ))
O
fflffl
FB (X(V ), Y (W )).
Passage to ends from the isomorphisms of exGspaces
FB (X(V ) ^B K, Y (V )) ~=FB (K, FB (X(V ), Y (V ))) ~=FB (X(V ), FB (K, Y (V *
*)))
gives natural isomorphisms of exGspaces
(11.1.5) PB (X ^B K, Y ) ~=FB (K, PB (X, Y )) ~=PB (X, FB (K, Y )).
With these constructions, we see that (IG KB , GI KB ) is Gtopological over B;
compare Definition 10.2.1 and the discussion following it. The last statement f*
*ollows
since GI KB is complete and cocomplete, with limits and colimits constructed
levelwise from the limits and colimits in GKB .
We have several kinds of smash products and function objects in this context.
For IG spaces X and Y over B, define the "external" smash product X ZB Y by
X ZB Y = ^B O (X x Y ): IG x IG ! KG,B.
Thus (X ZB Y )(V, W ) = X(V ) ^B Y (W ). Here we have used the word "external"
to refer to the use of pairs of representations, as is usual in the theory of d*
*iagram
spectra. It is standard category theory [30, 62] to use left Kan extension to i*
*nter
nalize this external smash product over B. This gives the internal smash product
X ^B Y of IG spaces over B, which is again an IG space over B. For an IG spa*
*ce
Y over B and an (IG xIG )space Z over B, define the external function IG space
over B, denoted ~FB(Y, Z), by
F~B(Y, Z)(V ) = PB (Y, Z),
where Z(W ) = Z(V, W ). It is mainly to allow this definition that we need *
*the
morphism exGspaces PB (, ). It is also formal to obtain an internal function
IG space functor FB on IG spaces over B by use of right Kan extension [30, 62*
*].
Using these internal smash product and function IG space functors, we obtain t*
*he
following result. Recall Definition 10.2.9 and Proposition 10.2.10.
Theorem 11.1.6. (IG KB , GI KB ) is a Gtopological closed symmetric mon
oidal Gcategory over B.
Remark 11.1.7. In the theory of exspaces, we also have the "external smash
product" of exspaces over different base spaces defined in x2.5. Using the two
different notions of "external" together, we obtain the definition of the "exte*
*rnal
external smash product" of an IG space X over A and an IG space Y over B; it
is an (IG x IG )space over A x B. We write X Z Y for the left Kan extension
internalization of this smash product. Thus X Z Y is an IG space over A x B.
Similarly, using the external function exspace construction ~Fof x2.5, for an *
*IG 
space Y over B and an IG space Z over AxB, we obtain the "internalized external
136 11. THE CATEGORY OF ORTHOGONAL GSPECTRA OVER B
function IG space" F~(Y, Z) over A. Notationally, use of Z and F~ without an
ensuing subscript always denotes these internalized external operations with re*
*spect
to varying base spaces. We shall return to these functors in Proposition 11.4.1*
*0.
Similarly, but more simply, we have the "external tensor" K Z Y of an exG
space K over A and an IG space Y over B, which again is an IG space over AxB.
When A = *, this is just the tensor of based Gspaces with IG spaces over B. T*
*he
case B = * shows how to construct an IG space over A from an exGspace over
A and an IG space. Since these external tensors can be view as special cases of
external smash products, via variants of Definition 10.2.9 and (11.2.6) below, *
*we
shall not treat them formally and shall not repeat the definitions on the Gspe*
*ctrum
level. However, we shall find several uses for them.
11.2. The category of orthogonal Gspectra over B
For a representation V of G and an IG space X, we define
(11.2.1) VBX = X ^B SVB and VBX = FB (SVB, X),
where SV is the onepoint compactification of V .
Definition 11.2.2. Define the Gsphere SB , written SG,B when necessary for
clarity, to be the IG space over B that sends V to SVB.
Clearly SVB^B SWB ~=SVB W , and the functor SB is strong symmetric monoidal,
where the monoidal structure on IG is given by direct sums. It follows that SB *
*is
a commutative monoid in the symmetric monoidal category GI KB , and we can
define SB modules X in terms of (right) actions X^B SB ! X. These SB modules
are our orthogonal Gspectra over B, but it is more convenient to give the defi*
*nition
using the equivalent reformulation in terms of the external smash product.
Definition 11.2.3. An IG spectrum, or orthogonal Gspectrum, over B is an
IG space X over B together with a structure Gmap
oe :X ZB SB ! X O
such that the evident unit and associativity diagrams commute. Thus we have
compatible equivariant structure maps
oe : WBX(V ) = X(V ) ^B SWB ! X(V W ).
Let SG,B denote the topological Gcategory of IG spectra over B and arrows
f :X ! Y that commute with the structure maps, with G acting by conjugation
on arrows. Let GSB denote the topological category of IG spectra over B and
Gmaps (equivariant arrows) between them.
Definition 11.2.4. Define the suspension orthogonal Gspectrum functor and
the 0th exGspace functor
1B:KG,B ! SG,B and 1B:SG,B ! KG,B
by ( 1BK)(V ) = VBK, with the evident isomorphisms as structure maps, and
1BX = X(0). Then 1B and 1B give left and right adjoints between KG,B and
SG,B and, on passage to Gfixed points, between GKB and GSB .
The category GSB is our candidate for a good category of parametrized G
spectra over B. It inherits all of the properties of the category GI KB of IG *
*spaces
that were discussed in the previous section and, in the case B = *, it is exact*
*ly
11.2. THE CATEGORY OF ORTHOGONAL GSPECTRA OVER B 137
the category GS of orthogonal Gspectra that is studied in [61]. We summarize
its formal properties in the following omnibus theorem. In the language of x10.*
*2,
much of it can be summarized by the assertion that the Gcategory (SG,B, GSB )
is a Gtopological closed symmetric monoidal Gcategory over B, but we prefer to
be more explicit than that.
Theorem 11.2.5. The Gcategory SG,B is enriched over GKB and is ten
sored and cotensored over KG,B. The category GSB is enriched over KB and is
tensored and cotensored over GKB . The Gcategory SG,B and the category GSB
admit smash product and function spectrum functors ^B and FB under which they
are closed symmetric monoidal with unit object SB . Let X and Y be orthogonal
Gspectra over B and K be an exGspace over B. The morphism exGspaces
PB (X, Y ) can be specified by
PB (X, Y ) = 1BFB (X, Y ),
and there are natural isomorphisms
1BK ~=SB ^B K and 1BX ~=PB (SB , X).
The tensors and cotensors are related to smash products and function Gspectra *
*by
natural isomorphisms
(11.2.6) X ^B K ~=X ^B 1BK and FB (K, X) ~=FB ( 1BK, X)
of orthogonal Gspectra. There are natural isomorphisms
(11.2.7) PB ( 1BK, X) ~=FB (K, 1BX)
and
(11.2.8) PB (X ^B K, Y ) ~=FB (K, PB (X, Y )) ~=PB (X, FB (K, Y ))
of exGspaces,
(11.2.9) SG,B(X ^B K, Y ) ~=KG,B(K, PB (X, Y )) ~=SG,B(X, FB (K, Y ))
of based Gspaces, and
(11.2.10) GSB (X ^B K, Y ) ~=GKB (K, PB (X, Y )) ~=GSB (X, FB (K, Y ))
of based spaces. Moreover, GSB is Gtopologically bicomplete over B.
Proof. For the enrichment, the Gspace SG,B(X, Y ) is the evident sub G
space of IG KB (X, Y ), and the space GSB (X, Y ) is the evident sub space of
GI KB (X, Y ). The tensors and cotensors in SG,B are constructed in IG KB and
given induced structure maps. The limits and colimits in GSB are constructed
in the same way. As in [61, IIx3], we think of orthogonal Gspectra over B as
SB modules, and we construct the smash product and function spectra functors
by passage to coequalizers and equalizers from the smash product and function
IG space functors, exactly as in the definition of tensor products and hom fun*
*ctors
in algebra. We have defined PB (X, Y ) in the statement, but we shall give a mo*
*re
intrinsic alternative description later. The first isomorphism of (11.2.6) is g*
*iven by
unit and associativity relations
X ^B K ~=(X ^B SB ) ^B K ~=X ^B 1BK.
138 11. THE CATEGORY OF ORTHOGONAL GSPECTRA OVER B
The second follows from the Yoneda lemma since
GSB (X, FB (K, Y ))~=GSB (X ^B K, Y )
~=GSB (X ^B 1BK, Y )
~=GSB (X, FB ( 1BK, Y )).
Now (11.2.7) and (11.2.8) follow from already established adjunctions. For part*
* of
the latter, we apply 1B to the composite isomorphism
FB (X ^B K, Y )~=FB (X ^B 1BK, Y )
~=FB (X, FB ( 1BK, Y ))
~=FB (X, FB (K, Y )).
Comparisons of definitions, seen more easily from (11.3.2) below, give
(11.2.11) SG,B(X, Y ) = KG,B(S0B, PB (X, Y ))
and
(11.2.12) GSB (X, Y ) ~=GKB (S0B, PB (X, Y )).
Therefore the isomorphisms (11.2.9) and (11.2.10) follow from (11.2.8).
As noted in x10.1, we obtain the following corollary by replacing K with TB
for a based Gspace T in the tensors and cotensors of the theorem. Of course, t*
*hese
tensors and cotensors with Gspaces could just as well be defined directly. It *
*will
be important in our discussion of model category structures to keep separately *
*in
mind the tensors and cotensors over exGspaces over B and over based Gspaces.
Corollary 11.2.13. The Gcategory SG,B is enriched over GK* and is ten
sored and cotensored over KG,*. The category GSB is enriched over KG,* and is
tensored and cotensored over GK*. Thus, for orthogonal Gspectra X and Y and
based Gspaces T ,
(11.2.14) SG,B(X ^B T, Y ) ~=KG,*(T, SG,B(X, Y )) ~=SG,B(X, FB (T, Y ))
and
(11.2.15) GSB (X ^B T, Y ) ~=GK*(T, SG,B(X, Y )) ~=GSB (X, FB (T, Y )).
We have the parallel definition of Gprespectra over B.
Definition 11.2.16. A Gprespectrum X over B consists of exGspaces X(V )
over B for V 2 V together with structure Gmaps oe : WBX(V ) ! X(V W )
such that oe is the identity if W = 0 and the following diagrams commute.
~= W Z
ZB WBX(V )_______// B X(V )
ZBoe oe
fflffl fflffl
ZBX(V W ) __oe//_X(V W Z)
Let PG,B denote the Gcategory of Gprespectra and nonequivariant arrows, and
let GPB denote its Gfixed category of Gprespectra and Gmaps. There result
forgetful functors
U: SG,B ! PG and U: GSB ! GPB .
11.3. ORTHOGONAL GSPECTRA AS DIAGRAM EXGSPACES 139
The categories PG,B and GPB enjoy the same properties that were speci
fied for SG,B and GSB in Theorem 11.2.5 and Corollary 11.2.13, except for the
statements about smash product and function spectra. Here, since we do not have
the internal hom functor FB , we must give an alternative direct description of
PB (X, Y ), as in (11.3.2) below.
11.3.Orthogonal Gspectra as diagram exGspaces
Arguing as in [62, x2] and [61, IIx4], we construct a new domain category JG*
*,B
which has the same object set V as IG and, like IG,B, is enriched over GKB . It
builds in spheres in such a way that the category of IG spectra over B is equi*
*valent
to the category of JG,Bspaces over B. Here, just as for IG,B in Lemma 11.1.2,
we understand a JG,Bspace to be an enriched Gfunctor X :JG,B ! KG,B.
Thus it is specified by exGspaces X(V ) and exGmaps
X :JG,B(V, W ) ! FB (X(V ), X(W )).
To construct JG,B, recall from [61, IIx4] that we have a topological Gcategory
JG with object set V such that the category of IG spectra is equivalent to the
category of JG spaces. We define
(11.3.1) JG,B(V, W ) = JG (V, W )B ,
just as we defined IG,B in (11.1.1), and the desired equivalence of categories *
*follows.
Rather than repeat either of the different constructions of JG given in [62] and
[61], we shall shortly give a direct description of JG,B. The intuition is that*
* an
extension of an IG,Bspace to a JG,Bspace builds in an action by SB .
The alternative description of GSB as the category of enriched Gfunctors
JG,B ! KG,B and enriched Gnatural transformations leads to a more concep
tual proof of Theorem 11.2.5: it is a specialization of general results about d*
*iagram
categories of enriched functors. In analogy with (11.1.4) we could have defined
PB (X, Y ) to be the end
Z
(11.3.2) PB (X, Y ) = FB (X(V ), Y (V ))
JG,B
and derived the isomorphism (11.2.8) just as we derived (11.1.5) in the previous
section. By the Yoneda lemma, the two definitions of PB (X, Y ) agree. With
this description of PB , some of the adjunctions in Theorem 11.2.5 become more
transparent.
This leads to an alternative description of JG,B in terms of IG,B, following*
* [62,
2.1]. We have the represented functors V *:IG ! KG,B specified by V *(W ) =
IG,B(V, W ). If X is an IG space, such as V *, then the smash product X ^B SB
in the category of IG spaces is a "free" orthogonal Gspectrum over B. Let
(11.3.3) JG,B(V, W ) = PB (W *^B SB , V *^B SB ),
with the evident composition. Then we can mimic the arguments of [62, xx2, 23] *
*to
check that the category of JG,Bspaces is equivalent to the category of IG spe*
*ctra
over B. An enriched Yoneda lemma argument [53, 2.4] shows that this description
of JG,B coincides up to isomorphism with our original one.
Although we will not have occasion to quote it formally, we record the follo*
*wing
consequence of the identification of IG spectra over B with JG,Bspaces.
140 11. THE CATEGORY OF ORTHOGONAL GSPECTRA OVER B
Lemma 11.3.4. For any enriched Gfunctor T :KG,B ! KG,B and orthogonal
Gspectrum X over B, the composite functor T OX is an orthogonal Gspectrum over
B. Similarly, an enriched natural transformation , :T ! T 0induces a natural G
map , :T O X ! T 0O X.
Proof. The enriched functor T is given by maps
T :FB (K, L) ! FB (T (K), T (L)).
Composing levelwise with X gives maps
JG,B(V, W ) ! FB (T (X(V )), T (X(W )))
that specify T O X. It is a direct categorical implication of the fact that T i*
*s an
enriched functor that there are natural maps of exGspaces
T (K) ^B L ! T (K ^B L) and T FB (K, L) ! FB (K, T (L))
for exGspaces K and L. This explains more concretely why the structure maps
of X induce structure maps for T O X. Similarly, since , is enriched, it is giv*
*en by
maps from the unit exGspace S0Bto FB (T (K), T 0(K)) such that the appropriate
diagrams commute. We specialize to K = X(V ) to obtain , :T O X ! T 0O X.
The following functors relating exGspaces to orthogonal Gspectra over B
play a central role in our theory. In particular, they give "negative dimension*
*al"
spheres 1VS0B= SVB.
Definition 11.3.5. Let V *= VB*denote the represented JG,Bspace specified
by V *(W ) = JG,B(V, W ). Define the shift desuspension functor
FV :KG,B ! SG,B
by letting FV K = V *^B K for an exGspace K. Let EvV :SG,B ! KG,B be
the functor given by evaluation at V . The alternative notations
1VK = FV K and 1VK = EvV
are often used. In particular, F0 = 10= 1B and Ev0 = 10= 1B.
Lemma 11.3.6. The functors FV and EvV are left and right adjoint, and there
is a natural isomorphism
FV K ^B FW L ~=FV W(K ^B L).
Proof. The first statement is clear, and the verification of the second sta*
*te
ment is formal, as in [62, x1].
11.4.The base change functors f*, f!, and f*
From now on, we drop the adjective "orthogonal" (or prefix IG ), and we gen
erally take the equivariance for granted, referring to orthogonal Gspectra ove*
*r B
just as spectra over B. We return G to the notations when considering change of
groups, or for emphasis, but otherwise Gactions are tacitly assumed throughout.
We first show that the results on base change functors proven for exspaces
in x2.2 extend to parametrized spectra. We then show that the results in x2.5
relating external and internal smash product and function exspaces also extend*
* to
parametrized spectra. Let A and B be base Gspaces.
11.4. THE BASE CHANGE FUNCTORS f*, f!, AND f* 141
Theorem 11.4.1. Let f :A ! B be a Gmap. Let X be in SG,A and let Y
and Z be in SG,B. There are Gfunctors
f!:SG,A ! SG,B, f* :SG,B ! SG,A, f*: SG,A ! SG,B
and Gadjunctions
SG,B(f!X, Y ) ~=SG,A(X, f*Y ) and SG,A(f*Y, X) ~=SG,B(Y, f*X).
On passage to Gfixed points levelwise, there result functors
f!:GSA ! GSB , f* :GSB ! GSA , f*: GSA ! GSB
and adjunctions
GSB (f!X, Y ) ~=GSA (X, f*Y ) and GSA (f*Y, X) ~=GSB (Y, f*X).
The functor f* is closed symmetric monoidal. Therefore, by definition and impli
cation, f*SB ~=SA and there are natural isomorphisms
(11.4.2) f*(Y ^B Z) ~=f*Y ^A f*Z,
(11.4.3) FB (Y, f*X) ~=f*FA (f*Y, X),
(11.4.4) f*FB (Y, Z) ~=FA (f*Y, f*Z),
(11.4.5) f!(f*Y ^A X) ~=Y ^B f!X,
(11.4.6) FB (f!X, Y ) ~=f*FA (X, f*Y ).
Proof. We define the functors f*, f!, and f* levelwise. This certainly gives
welldefined functors on IG spaces that satisfy the appropriate adjunctions th*
*ere.
We shall show shortly that these functors preserve IG spectra. For a based G
space T , f*(TB ) ~=TA , and this implies f*SB ~= SA . If we replace IG spect*
*ra
by IG spaces and replace the internal smash product and function object functo*
*rs
(^ and F ) by their external precursors (Z and ~F), then everything is immediate
by levelwise application of the corresponding results for exspaces. Still wor*
*king
with IG spaces, we first show how to internalize the isomorphisms (11.4.2) and
(11.4.5) by use of the universal property of left Kan extension. Indeed, noting*
* that
(f*X) O ~=f*(X O ), and similarly for f* and f!, we have
IG KA (f*(Y ^B Z), X)~=IG KB (Y ^B Z, f*X)
~=(IG x IG )KB (Y ZB Z, f*X O )
~=(IG x IG )KA (f*(Y ZB Z), X O )
~=(IG x IG )KA (f*Y ZA f*Z, X O )
~=IG KA (f*Y ^A f*Z, X)
and
IG KB (f!X ^B Y, Z)~=(IG x IG )KB (f!X ZB Y, Z O )
~=(IG x IG )KB (f!(X ZA f*Y ), Z O )
~=(IG x IG )KA (X ZA f*Y, f*Z O )
~=IG KA (X ^A f*Y, f*Z)
~=IG KA (f!(X ^A f*Y ), Z).
142 11. THE CATEGORY OF ORTHOGONAL GSPECTRA OVER B
As explained in [40, xx23], the remaining isomorphisms on the IG space level
follow formally.
We must show that our functors on IG spaces preserve IG spectra. The given
structure map oe :Y ZB SB ! Y O gives rise via the external version of (11.4*
*.2)
to the required structure map
f*Y ZA SA ~=f*(Y ZB SB ) ! f*Y O .
Similarly, the given structure map oe :X Z SA ! X O gives rise to the requir*
*ed
structure map
f!X ZB SB ~=f!(X ZA SA ) ! f!X O .
As in [40, (3.6)], there is a canonical natural map, not usually an isomorphism,
ss :f*X ZB Y ! f*(X ZA f*Y ).
Taking Y = SB , we see that oe also induces the required structure map
f*X ZB SB ! f*(X ZA SA ) ! f*X O .
Now the spectrum level adjunctions follow directly from their IG space analogu*
*es.
The spectrum level isomorphisms (11.4.2) and (11.4.5) follow from their IG spa*
*ce
analogues by comparisons of coequalizer diagrams, and the remaining isomorphisms
again follow formally.
Remark 11.4.7. Since the base change functors are defined levelwise, they
commute with the evaluation functors EvV . These commutation relations for the
right adjoints f* and f* imply conjugate commutation isomorphisms
f*FV ~=FV f* and f!FV ~=FV f!
of left adjoints. In particular,
f* 1B~= 1Af* and f! 1A~= 1Bf!.
Via (11.2.6), these isomorphisms and the isomorphisms of the theorem imply iso
morphisms relating base change functors to tensors and cotensors. For example
(11.4.5) implies isomorphisms
f!(f*Y ^A K) ~=Y ^B f!K and f!(f*L ^A X) ~=L ^B f!X.
Here K and L are exspaces over A and B and X and Y are spectra over A and B.
The following result is immediate from its precursor Proposition 2.2.9 for e*
*x
spaces.
Proposition 11.4.8. Suppose given a pullback diagram of Gspaces
g
C _____//D
i j
fflfflfflffl
A __f__//B.
Then there are natural isomorphisms of functors
(11.4.9) j*f!~=g!i*, f*j* ~=i*g*, f*j!~=i!g*, j*f* ~=g*i*.
Returning to Remark 11.1.7, we have the following important results on exter
nal smash product and function spectra and their internalization by means of ba*
*se
change along diagonal maps.
11.5. CHANGE OF GROUPS AND RESTRICTION TO FIBERS 143
Proposition 11.4.10. Let X be a spectrum over A, Y be a spectrum over B,
and Z be a spectrum over A x B. There is an external smash product functor that
assigns a spectrum X ZY over AxB to X and Y and an external function spectrum
functor that assigns a spectrum ~F(Y, Z) over A to Y and Z, and there is a natu*
*ral
isomorphism
GSAxB (X Z Y, Z) ~=GSA (X, ~F(Y, Z)).
The internal smash products are determined from the external ones via
X ^B Y ~= *(X Z Y ) and FB (X, Y ) ~=~F(X, *Y ),
where X and Y are spectra over B and : B ! B x B is the diagonal map.
Proof. It is not hard to start from Remark 11.1.7 and construct these func
tors directly. We instead follow Lemma 2.5.3 and observe that the spectrum level
external functors can and, up to isomorphism, must be defined in terms of the
internal functors as
X Z Y ~=ss*AX ^AxB ss*BY and F~(Y, Z) ~=ssA *FAxB (ss*BY, Z),
where ssA :A x B ! A and ssB :A x B ! B are the projections. The dis
played adjunction is immediate from the adjunctions (ss*A, ssA *), (ss*B, ssB **
*), and
(^AxB , FAxB ). The second statement follows formally, as in Lemma 2.5.4.
Proposition 11.4.11. For exspaces K over A and L over B, there is a natural
isomorphism
1AxB(K Z L) ~= 1AK Z 1BL.
Proof. This is most easily seen using adjunction and the Yoneda lemma. Us
ing external function objects, we see that ~F( 1BL, Z) ~=~F(L, Z) for Z 2 GSAxB*
* .
This has zeroth exspace ~F(L, Z(0)) over A.
11.5. Change of groups and restriction to fibers
We give the analogues for parametrized spectra of the results concerning cha*
*nge
of groups and restriction to fibers that were given for parametrized exspaces *
*in
x2.3. We shall say more about change of groups in Chapter 14. Fix an inclusion
': H ! G of a (closed) subgroup H of G and let A be an Hspace and B be a
Gspace. We index Hspectra over A on the collection '*V of Hrepresentations
'*V with V 2 V . As we discuss in xx14.2 and 14.3, when V is the collection of *
*all
representations of G, we can change indexing to the collection of all represent*
*ations
of H since our assumption that G is compact ensures that every representation
of H is a direct summand of a representation '*V . We have an evident forgetful
functor
(11.5.1) '*: GSB ! HS'*B.
On the space level, we write '!ambiguously for both the based and unbased induc
tion functors G+ ^H () and G xH (), and similarly for coinduction '*. Context
should make clear which is intended. Applying the unbased versions to retracts,
we defined induction and coinduction functors '! and '* on exspaces in Defini
tion 2.3.7. These functors extend to the spectrum level. Recall that SG,B denot*
*es
the Gsphere spectrum over B.
Proposition 11.5.2. Levelwise application of '!and '* gives functors
'!:HSA ! GS'!A and '*: HSA ! GS'*A.
144 11. THE CATEGORY OF ORTHOGONAL GSPECTRA OVER B
Proof. We must show that the structure Hmaps oe :X ZSH,A ! X O of an
Hspectrum X over A induce structure Gmaps for the IG spaces '!X and '*X. It
is clear that '!(X O ) ~='!X O and '*(X O ) ~='*X O . Using (2.3.4), we see th*
*at
SG,'!A~='!SH,A. Since the functor '!on the exspace level is symmetric monoidal
by Proposition 2.3.9, its levelwise IG space analogue commutes up to isomorphi*
*sm
with the external smash product Z. Thus oe induces a structure Gmap
'!X Z'!ASG,'!A~='!(X ZA SH,A) ! '!(X O ) ~='!X O .
For '*, let ~ : '*'* ! Id be the counit of the space level adjunction ('*, '*)
(see (2.3.2)). For an Hspace A, ~ is the Hmap Map H(G, A) ! A given by
evaluation at the identity element of G. Applied to an exspace K over A, thoug*
*ht
of as a retract, ~ gives a map '*'*K ! K of total spaces over and under the map
~ : '*'*A ! A of base spaces in the category of retracts of x2.5. We can apply
this to X levelwise. We also have the projection pr : ~*SH,A ! SH,A over ~.
Together, these maps give
'*('*X Z'*ASG,'*A) ~='*'*X Z'*'*A~*SH,A~Zpr//_X ZA SH,A.
For the isomorphism, we have used the facts that '* is strong monoidal and that
'*SG,'*A~= SH,'*'*A~=~*SH,A. The adjoint of the composite of this map with
the structure map oe : X ZA SH,A ! X O gives the required structure map
'*X Z'*ASG,'*A! '*X O .
As on the exspace level, the categories HSA and GSGxHA can be used inter
changeably. The following result is immediate from Proposition 2.3.9.
Proposition 11.5.3. Let :A ! '*'!A be the natural inclusion of Hspaces.
Then '!:HSA ! GS'!Ais a closed symmetric monoidal equivalence of categories
with inverse the composite * O '*: GS'!A! HS'*'!A! HSA .
In particular, if A = * then maps * to the identity coset eH 2 G=H and we *
*see
that HS and GSG=H can be used interchangeably. Arguing as in Proposition 2.3.1,
we could more easily prove this directly.
Corollary 11.5.4. The category HS is equivalent as a closed symmetric
monoidal category to GSG=H . Under this equivalence,
'* ~=r*, '!~=r!, and '* ~=r*,
where r :G=H ! *.
Looking at the fiber Xb(V ) = X(V )b over b of a Gspectrum X over B, we
see a Gbspectrum Xb of the sort that has been studied in [61], where Gb is the
isotropy group of b. Our homotopical analysis of parametrized Gspectra will be
based on the idea of applying the results of [61] fiberwise. By the previous re*
*sult,
we can think of this fiber as a Gspectrum over G=Gb. The following spectrum le*
*vel
analogues of Example 2.3.12 and Example 2.3.13 analyze the relationships among
passage to fibers, base change, and change of groups.
Example 11.5.5. For b 2 B, we write b: * ! B for the Gbmap that sends *
to b and "b:G=Gb ! B for the induced inclusion of orbits. Under the equivalence
GSG=Gb ~=GbS , "b*may be interpreted as the fiber functor GSB ! GbS that
sends Y to Yb. Its left and right adjoints "b!and "b*may be interpreted as the *
*functors
11.6. SOME PROBLEMS CONCERNING NONCOMPACT LIE GROUPS 145
that send a Gbspectrum X to the Gspectra Xb and bX over B obtained by level
wise application of the corresponding exspace level adjoints of Construction 1*
*.4.6
and Example 2.3.12. With these notations, the isomorphisms of Theorem 11.4.1
specialize to the following natural isomorphisms, where Y and Z are in GSB and
X is in GbS .
(Y ^B Z)b ~=Yb^ Zb,
FB (Y, bX) ~=bF (Yb, X),
FB (Y, Z)b ~=F (Yb, Zb),
(Yb^ X)b ~=Y ^B Xb,
FB (Xb, Y ) ~=bF (X, Yb).
Example 11.5.6. Let f :A ! B be a Gmap and let ib:Ab ! B be the
inclusion of the fiber over b, which is a Gbmap. As in Example 2.3.13, we have*
* the
compatible pullback squares
fb GxGbfb
Ab ____//_{b} G xGb Ab _____//G=Gb
ib b "b "b
fflffl fflffl fflffl fflffl
A ___f__//B A ____f____//_B.
Applying Proposition 11.4.8 to the righthand square and interpreting the concl*
*u
sion in terms of fibers, we obtain canonical isomorphisms of Gbspectra
(f!X)b ~=fb!i*bX and (f*X)b ~=fb*i*bX,
where X is a Gspectrum over A, regarded on the righthand sides as a Gbspectr*
*um
over A by pullback along ': Gb ! G.
11.6.Some problems concerning noncompact Lie groups
In equivariant stable homotopy theory, the key idea is that the onepoint co*
*m
pactification of a representation V of dimension n is a Gsphere and that smash*
*ing
with that sphere should be a selfequivalence of the equivariant stable homotopy
category. That is, the idea is to invert Gspheres in just the way that we in
vert spheres when constructing the nonequivariant stable homotopy category. For
compact Lie groups of equivariance, the philosophy and its implementation and
applications are well understood. When we invert representation spheres, we inv*
*ert
other homotopy spheres as well, and the relevant Picard group is analyzed in [4*
*1].
For noncompact Lie groups, the present work seems to be the first attempt to
consider foundations for equivariant stable homotopy theory. The philosophy is *
*less
clear, and its technical implementation is problematic. The need for such a the*
*ory is
evident, however. The focus on finite dimensional representations is intrinsic *
*to the
philosophy but fails to come to grips with basic features of the representation*
* theory
of noncompact Lie groups. A theory based on finite dimensional representations
should still have its uses, but there are real difficulties to obtaining even t*
*hat much.
In particular, a focus on spheres associated to linear representations, rather *
*than
on less highly structured homotopy spheres, may be misplaced.
146 11. THE CATEGORY OF ORTHOGONAL GSPECTRA OVER B
A noncompact semisimple Lie group will generally have no nontrivial finite
dimensional unitary or orthogonal representations, hence our theory of "orthogo
nal" Gspectra is clearly too restrictive. This is easily remedied. The use of *
*linear
isometries in the definition of orthogonal spectra is a choice dictated more by*
* the
history than by the mathematics. In the alternative approach to equivariant sta*
*ble
homotopy theory based on LewisMay spectra and EKMM [39, 59, 61], use of
orthogonal complements is certainly convenient and perhaps essential. However,
the diagram orthogonal spectra of [61, 62] could just as well have been develop*
*ed
in terms of diagram "general linear spectra". In the few places where complemen*
*ts
are used, they can by avoided. For consistency with the previous literature, we*
* have
chosen to give our exposition in the compact case using the word "orthogonal" a*
*nd
the language from the cited references, but for general Lie groups of equivaria*
*nce,
we should eliminate all considerations of isometries.
More precisely, for the complete case, we redefine I by taking V to be the
collection of all finite dimensional representations V of G. More generally, we*
* can
index on any subcollection that contains the trivial representation and is clos*
*ed
under finite direct sums. Since we are only interested in a skeleton of I , we *
*may
as well restrict to orthogonal representations in V when G is compact. We repla*
*ce
linear isometries by linear isomorphims when defining the Gspaces I (V, W ). T*
*hus
we replace orthogonal groups by general linear groups. Otherwise, the formal de*
*fi
nitional framework developed in this chapter (or, in the nonparametrized case, *
*[61,
II]) goes through verbatim for general topological groups G.
However, we emphasize the formality. When considering change of groups, for
example, the significance changes drastically. As noted at the start of the pre*
*vious
section, for an inclusion ': H  ! G of a (closed) subgroup H of G, we index
Hspectra on the collection '*V of Hrepresentations '*V with V 2 V . We also
pointed out the relevance of the compact case of the following result.
Proposition 11.6.1. If G is either a compact Lie group or a matrix group and
W is a representation of a subgroup H, then there is a representation V of G and
an embedding of W as a subrepresentation of '*V .
This is clear in the compact case and is given by [81, 3.1] for matrix group*
*s.
However, the following striking counterexample, which we learned from Victor
Ginzburg, shows just how badly this basic result fails in general.
Counterexample 11.6.2 (Ginzburg). Let H be the Heisenberg group of 3 x 3
matrices 0 1
1 a c
@ 0 1 b A
0 0 1
where a, b, and c are real numbers. Embed R in H as the subgroup of matrices
with a = b = 0. Embed Z in R as usual. Then R is a central subgroup of H.
Define G = H=Z. Then T = R=Z is a circle subgroup of G. Moreover, T is the
center of G and coincides with the commutator subgroup [G, G]. Let V be any
finite dimensional (complex linear) representation of G. Since T is compact, t*
*he
action of T on V is semisimple, and since T is central, any weight space of T i*
*s a
Gsubmodule. Therefore V is a direct sum of Gsubmodules Visuch that T acts on
each Viby scalar matrices. Since T = [G, G], this scalar action of T on Viis tr*
*ivial:
the determinant of g is 1 for any g 2 [G, G]. Therefore no nontrivial 1dimensi*
*onal
11.6. SOME PROBLEMS CONCERNING NONCOMPACT LIE GROUPS 147
character of T can embed in V . Reinterpreting in terms of real representations*
*, as
we may, we conclude that, for ': T ! G, '*V is the trivial T universe.
For a compact Lie group G and inclusion ': H G, '*X is a dualizable H
spectrum if X is a dualizable Gspectrum, and an Hspectrum indexed on the triv*
*ial
Huniverse is dualizable if and only if it is a retract of a finite HCW spectr*
*um built
up from trivial orbits. We conclude that duality theory (in the nonparametrized
context) cannot work as one would wish in the context of the previous example.
Looking ahead, much of the theory of the following three chapters also works
formally in the context of noncompact Lie groups. However, there is at least o*
*ne
serious technical difficulty. Our theory is based on the use of onepoint compa*
*ctifi
cations SV . If V is a linear representation of a noncompact Lie group G, ther*
*e is
no reason to think that G acts smoothly and properly on SV , even if the isotro*
*py
groups of V are compact. In fact, if Illman's Theorem 3.3.2 were to apply, then
SV would be a Gcell complex, hence it would be built up from noncompact orbits
G=H given by compact subgroups H. However, as closed subsets of SV , the closed
cells would have to be compact. That is, the putative GCW structure would con
tradict the compactness of SV . Said another way, we see no reason to believe t*
*hat
the SV are qcofibrant Gspaces. Therefore, the functors () ^ SV need not be
Quillen left adjoints and the functors V and VBneed not preserve fibrant obje*
*cts
in the relevant model structures. Compare, for example, Proposition 12.2.2 and *
*the
derivation of the long exact sequences (12.3.2) and (12.3.3) below. What seems *
*to
be needed, for a start, is something like a model structure on Gspaces such th*
*at X
is cofibrant if '*X is Hcofibrant for all inclusions ': H ! G of compact subgr*
*oups.
CHAPTER 12
Model structures for parametrized Gspectra
Introduction
We define and study two model structures on the category GSB of (orthogonal)
Gspectra over B. We emphasize that, except for the theory of smash products,
everything in this chapter applies equally well to the category GPB of Gprespe*
*ctra
over B. That fact will become important in the next chapter.
We start in x12.1 by defining a "level model structure" on GSB , based on the
qfmodel structure on GKB . In x12.2, we record analogues for this model struct*
*ure
of the results on external smash product and base change functors that were giv*
*en
for GKB in x7.2. The level model structure serves as a stepping stone to the st*
*able
model structure, which we define in x12.3. It has the same cofibrations as the *
*level
model structure, and we therefore call these "scofibrations". An essential po*
*int
in our approach is a fiberwise definition of the homotopy groups of a parametri*
*zed
Gspectrum that throws much of our work onto the theory of nonparametrized
orthogonal Gspectra developed by Mandell and the first author in [61]. We defi*
*ne
homotopy groups using the level qffibrant replacement functor provided by the
level model structure, and we define stable equivalences to be the ss*isomorph*
*isms.
It is essential to think in terms of fibers and not total spaces since the tota*
*l spaces
of a parametrized spectrum do not assemble into a spectrum. We show in x12.4
that the ss*isomorphisms give a wellgrounded subcategory of weak equivalences,
and we complete the proofs of the model axioms in x12.5. We return to the conte*
*xt
of x12.2 in x12.6, where we prove that various Quillen adjoint pairs in the lev*
*el
model structures are also Quillen adjoint pairs in the stable model structures.
The basic conclusion is that GSB is a wellgrounded model category under the
stable structure. Although not very noticeable on the surface, essential use is*
* made
of the qfmodel structure on GKB throughout this chapter. It is possible to obt*
*ain
a level model structure on GSB from the qmodel structure on GKB , as we explain
in Remark 12.1.8. However this model structure is not wellgrounded and therefo*
*re
does not provide the necessary tools to work out the technical details of x12.4*
*. The
results there are crucial to prove that the relative cell complexes over B defi*
*ned in
terms of the appropriate generating acyclic scofibrations are acyclic.1 It was*
* our
fruitless attempt to obtain a stable model structure starting from the level q*
*model
structure that led us to the construction of the qfmodel structure on GKB and *
*to
the notion of a wellgrounded model category.
When there are no issues of equivariance, we generally abbreviate Gspectrum
over B, exGspace, and Gspace to spectrum over B, exspace, and space; G is a
compact Lie group throughout.
____________
1In [47, 3.4], such acyclicity of relative cell complexes is assumed withou*
*t proof.
148
12.1. THE LEVEL MODEL STRUCTURE ON GSB 149
12.1. The level model structure on GSB
After changing the base space from * to B, the level model structure works in
much the same way as in the nonparametrized case of [61].
Definition 12.1.1. Let f :X ! Y be a map of spectra over B. With one
exception, for any type of exspace and any type of map of exspaces, we say th*
*at
X or f is a level type of spectrum over B or a level type of map of spectra ove*
*r B if
each X(V ) or f(V ): X(V ) ! Y (V ) is that type of exspace or that type of m*
*ap.
Thus, for example, we have level h, level f and level fpfibrations, cofibratio*
*ns
and equivalences from x5.1 together with the corresponding fibrant and cofibrant
objects. We have level qequivalences and level q and qffibrations from x7.1 a*
*nd
we have level exfibrations and level exquasifibrations from x8.1 and x8.5. T*
*he
exceptions concern cofibrations and cofibrant objects. We shall never be intere*
*sted
in "level qcofibrations" or "level qfcofibrations", nor in "level qcofibrant*
*" or "level
qfcofibrant" objects, since these do not correspond to cofibrations and cofibr*
*ant
objects in the model structures that we consider. Instead we have the following
definitions.
(i)f is an scofibration if it satisfies the LLP with respect to the level acy*
*clic
qffibrations.
(ii)f is a level acyclic scofibration if it is both a level qequivalence and*
* an
scofibration.
To reiterate, in the phrase "level acyclic qffibration", the adjective "level"*
* applies
to "acyclic qffibration", but in the phrase "level acyclic scofibration" it a*
*pplies
only to "acyclic"; the cofibrations are not defined levelwise.
Definition 12.1.2. A spectrum X over B is wellsectioned if it is level well
sectioned, so that each exspace X(V ) is ~fcofibrant. It is wellgrounded if*
* it is
level wellgrounded, so that each X(V ) is wellsectioned and compactly generat*
*ed.
The discussion of x4.3 applies to the category GSB of Gspectra over B with
homotopies defined in terms of the cylinders X ^B I+ . In particular, we have t*
*he
notion of a Hurewicz cofibration in GSB , abbreviated cylcofibration, defined *
*in
terms of these_cylinders, and we also have the notion of strong Hurewicz cofibr*
*ation,
abbreviated cylcofibration.
Lemma 12.1.3. A cylcofibration of spectra over B is a level fpcofibration *
*and
a cylfibration of spectra over B is a level fpfibration. A cylcofibration be*
*tween
wellsectioned spectra over B is a level fcofibration and therefore both a lev*
*el h
cofibration and a level fpcofibration.
Proof. By the mapping cylinder retraction criterion of Hurewicz cofibration*
*s,
a cylcofibration of spectra over B is a level fpcofibration. The statement ab*
*out fi
brations follows similarly from the path lifting function characterization of H*
*urewicz
fibrations. An fpcofibration between wellsectioned exspaces is an fcofibrat*
*ion
by Proposition 5.2.3, and all fcofibrations are hcofibrations.
Recall the notions of a ground structure and of a wellgrounded subcategory *
*of
weak equivalences from Definitions 5.3.2, 5.3.3, and 5.4.1.
Proposition 12.1.4. The wellgrounded spectra over B give GSB a ground
structure whose ground cofibrations, or gcofibrations, are the level hcofibra*
*tions.
150 12. MODEL STRUCTURES FOR PARAMETRIZED GSPECTRA
The level qequivalences specify a wellgrounded subcategory of weak equivalenc*
*es
with respect to this ground structure. In the gluing and colimit lemmas, one ne*
*ed
only assume that the relevant maps are level hcofibrations, not necessarily al*
*so
cylcofibrations.
Proof. That we have a ground structure follows levelwise from the ground
structure on exspaces in Proposition 5.3.7. That the level qequivalences are *
*well
grounded follows levelwise from Proposition 5.4.9.
We construct the level model structure on GSB from the qfmodel structure
on GKB specified in Remark 7.2.11, but all results apply verbatim starting from
the qf(C )model structure for any closed generating set C (as defined in Defin*
*i
tion 7.2.6). We shall need the extra generality for the reasons discussed in Ch*
*apter
7. Recall that IfBand JfBdenote the sets of generating qfcofibrations and gene*
*r
ating acyclic qfcofibrations in GKB . We use the shift desuspension functors FV
of Definition 11.3.5 to obtain corresponding sets on the spectrum level. We need
the following observations.
Lemma 12.1.5. The functor FV enjoys the following properties.
(i)If K is a wellgrounded exspace over B, then FV K is wellgrounded. If K is
an exfibration, then FV K is a level exfibration.
(ii)If i: K ! L is an hequivalence between wellgrounded exspaces over B,
then FV i is a level hequivalence.
(iii)If i: K ! L is an fpcofibration, then FV i is a cylcofibration and ther*
*efore
a level fpcofibration. If, further, K and L are wellsectioned, then FV i *
*is a
level fcofibration_and_therefore a level hcofibration._
(iv)If i: K ! L is an fpcofibration,_then FV i is a cylcofibration.
(v)If i: K ! L is_an_fcofibration between wellgrounded_exspaces over B,
then FV i_is_a cylcofibration which_is a level fcofibration and therefore*
* both
a level fpcofibration and a level hcofibration.
Proof. By Definition 11.3.5, (FV K)(W ) = JG (V, W )B ^B K, and the G
space JG (V, W ) is wellbased. Now (i) holds by Corollary 8.2.5 and (ii) holds
by Proposition 8.2.6. Since FV is left adjoint to the evaluation functor Ev V
and since cylfibrations are level fpfibrations, (iv) and the first statement *
*of (iii)
follow from the definitions by adjunction. The second statement of (iii)_follo*
*ws
from_Proposition_5.2.3. The first half of (v) follows from (iv) since fcofibra*
*tions
are fpcofibrations, and the second half follows from (iii) since_FV i is a lev*
*el f
cofibration between wellgrounded spectra and therefore a level fcofibration by
Theorem 5.2.8(ii).
Definition 12.1.6. Define F IfBto be the set of maps FV i with V in a skelet*
*on
skIG of IG and i in IfB. Define F JfBto be the set of maps FV j with V in skIG
and j in JfB.
Recall the notion of a wellgrounded model structure from Definition 5.5.2.
Among other properties, such model structures are compactly generated, proper,
and Gtopological.
Theorem 12.1.7. The category GSB is a wellgrounded model category with
respect to the level qequivalences, the level qffibrations and the scofibrat*
*ions. The
12.1. THE LEVEL MODEL STRUCTURE ON GSB 151
sets F IfBand F JfBgive the generating scofibrations_and the generating level_*
*acyclic
scofibrations._ All scofibrations are level fcofibrations, hence level fp an*
*d level
hcofibrations, and all scofibrant spectra over B are wellgrounded.
___
Proof. By Lemma 12.1.5, the maps_in F IfB and F JfB are cylcofibrations
between wellgrounded objects and fcofibrations. Moreover, the maps in F JfB
are level acyclic. Therefore, to prove the model axioms, we need only verify t*
*he
compatibility condition (ii) in Theorem 5.5.1. Adjunction arguments show that a
map is a level qffibration if and only if it has the RLP with respect to F JfB*
*and
that it is a level acyclic qfibration if and only if it has the RLP with respe*
*ct to
F IfB. This implies that the classes of scofibrations and of F IfBcofibratio*
*ns (in
the sense of Definition 4.5.1(iii)) coincide. Therefore, if a map has the RLP w*
*ith
respect to F IfB, then it is a level acyclic qffibration. The required compati*
*bility
condition now follows from its analogue for GKB . Condition (iv) in Theorem 5.5*
*.1
holds by its exspace level analogue and the fact that (FV K) ^B T ~=FV (K ^B T*
* )
for an exspace K over B and a based space T . Right properness follows directly
from the space level analogue.
Remark 12.1.8. Just as in Definition 12.1.6, we can also define sets F IB and
F JB based on the generating sets IB and JB for the qmodel structure on GKB . *
*We
can then use Theorem 4.5.5 to prove the analogue of Theorem 12.1.7 stating that
GSB is a cofibrantly generated model category under the level qmodel structure.
Since the compatibility condition holds by the same proof as for the level qfm*
*odel
structure, we need only verify the acyclicity condition to show this.
For a generating acyclic qcofibration j 2 JB , we have FV j = V *^B j, where
V *(W ) = JG,B(V, W ). This map is a level hequivalence by Lemma 12.1.5(ii).
Although j is an hcofibration, it is not immediate that FV j is a level hcofi*
*bration.
(This holds for j 2 JfBby Lemma 12.1.5(iii), since j is then an fpcofibration).
Indeed, for general exspaces K and hcofibrations f, K ^B f need not be an h
cofibration. However, since JG,B(V, W ) = JG (V, W )B , we see directly that FV*
* j
is indeed a level hcofibration. By inspection of the definition of wedges over*
* B in
terms of pushouts, the gluing lemma in K then applies to show that wedges over B
of maps in F JB are level acyclic hcofibrations. Since pushouts and colimits i*
*n SB
are constructed levelwise on total spaces, it follows that relative F JB comple*
*xes
are acyclic hcofibrations since the qmodel structure on K is wellgrounded.
Remark 12.1.9. As in the nonparametrized case [61], "positive" model struc
tures would be needed to obtain a comparison with the as yet undeveloped alter
native approach to parametrized stable homotopy theory based on [39, 59]. Such
model structures can be defined as in [61, p. 44], starting from the subsets (F*
* IfB)+
and (F JfB)+ that are obtained by restricting to those V such that V G 6= 0. One
then defines the positive level versions of all of the types of maps specified *
*in Def
inition 12.1.1 by restricting to those levels V such that V G 6= 0. The posit*
*ive
level analogue of Theorem 12.1.7 holds, where the positive scofibrations are t*
*he
scofibrations that are isomorphisms at all levels V such that V G = 0; compare
[61, III.2.10]. However, we shall make no use of the positive model structure i*
*n this
paper, and we will make little further reference to it.
The same proof as in [61, I.2.10, II.4.10, III.2.12] gives the following res*
*ult.
152 12. MODEL STRUCTURES FOR PARAMETRIZED GSPECTRA
Theorem 12.1.10. The forgetful functor U from spectra over B to prespectra
over B has a left adjoint P such that (P, U) is a Quillen equivalence.
12.2.Some Quillen adjoint pairs relating level model structures
This section gives the analogues for the level model structure of some of the
exspace level results in xx7.27.4. These results are also analogues of resul*
*ts in
[61, III.x2], which in turn have nonequivariant precursors in [62, x6]. They a*
*dmit
essentially the same proofs as in Chapter 7 or in the cited references. The lev*
*el qf
model structure is understood throughout. More precisely, where a qf(C )model
structure was used in Chapter 7, we must use the corresponding level qf(C )mod*
*el
structure here. Since we want our model structures to be Gtopological, we only
use generating sets C that are closed under finite products.
Our first observation is immediate from the fact that equivalences and fibra
tions are defined levelwise, the next follows directly from its exspace analog*
*ue
Proposition 7.3.1, and the third and fourth are proven in the same way as their
exspace analogues 7.3.2 and Corollary 7.3.3. All apply to the level qf(C )mod*
*el
structures for any choice of C .
Proposition 12.2.1. The pair of adjoint functors (FV , EvV) between GKB
and GSB is a Quillen adjoint pair.
Proposition 12.2.2. For a based GCW complex T , (() ^B T, FB (T, )) is
a Quillen adjoint pair of endofunctors of GSB .
Proposition 12.2.3. If i: X  ! Y and j :W  ! Z are scofibrations of
spectra over base spaces A and B, then
i j :(Y Z W ) [XZW (X Z Z) ! Y Z Z
is an scofibration over A x B which is level acyclic if either i or j is acycl*
*ic.
As in x7.2, we cannot expect this result to hold for internal smash products*
* over
B. The case A = *, which relates spectra to spectra over B, is particularly imp*
*or
tant. As we explain in x14.1, it leads to a fully satisfactory theory of parame*
*trized
module spectra over nonparametrized ring spectra.
Corollary 12.2.4. If Y is scofibrant over B, then the functor () Z Y from
GSA to GSAxB is a Quillen left adjoint with Quillen right adjoint ~F(Y, ).
Again the next result is a direct consequence of its exspace analogue Propo*
*si
tion 7.3.4 and applies with any choice of C .
Proposition 12.2.5. Let f :A ! B be a Gmap. Then (f!, f*) is a Quillen
adjoint pair. The functor f! preserves level qequivalences between wellsecti*
*oned
Gspectra over B. If f is a qffibration, then f* preserves all level qequival*
*ences.
Proposition 12.2.6. If f :A ! B is a qequivalence, then (f!, f*) is a Quil*
*len
equivalence.
Proof. We mimic the proof of Proposition 7.3.5, but with X and Y taken to
be an scofibrant Gspectrum over A and a level qffibrant Gspectrum over B. It
is clear that f*Y  ! Y is a level qequivalence since A ! B is a qequivalenc*
*e.
Since X is scofibrant, *A ! X is a level hcofibration. Note that it is essen*
*tial for
this statement that we start from the qf and not the qmodel structure on exsp*
*aces.
12.3. THE STABLE MODEL STRUCTURE ON GSB 153
Since pushouts along level hcofibrations preserve level qequivalences, X ! f*
*!X
is a level qequivalence. The conclusion follows as in Proposition 7.3.5.
Proposition 12.2.7. Let f :A ! B be a Gbundle whose fibers Ab are Gb
CW complexes. Then f* preserves level qequivalences and scofibrations. There
fore (f*, f*) is a Quillen adjoint pair.
Proof. Here we must use a generating set C (f) as specified in Proposi
tion 7.3.8. The proof that f* preserves scofibrations reduces to showing that
the maps f*FV i ~=FV f*i are scofibrations for generating scofibrations i. Si*
*nce
FV is a Quillen left adjoint it takes qfcofibrations to scofibrations, so we *
*are
reduced to the exspace level, where f*i is shown to be a qfcofibration in Pro*
*po
sition 7.3.8.
Now consider the change of groups functors of x11.5. The following result sh*
*ows
that the equivalence of Proposition 11.5.3 descends to homotopy categories. It *
*is
proven by levelwise application of its exspace analogue Proposition 7.4.6, tog*
*ether
with change of universe considerations that are deferred until x14.2 and x14.3.
Proposition 12.2.8. Let ': H ! G be the inclusion of a subgroup. The pair
of functors ('!, *'*) relating HSA and GS'!Agive a Quillen equivalence. If A is
completely regular, then '!is also a Quillen right adjoint.
For a point b in B, we combine the special case "b:G=Gb ! B of Propo
sition 12.2.5 with Proposition 12.2.8, where ': Gb ! G and :* ! G=Gb, to
obtain the following analogue of Proposition 7.5.1. Recall from Example 11.5.5 *
*that
the fiber functor ()b:GSB ! GbS is given by *'*"b*= b*'*. Its left adjoint
()b therefore agrees with "b!'!.
Proposition 12.2.9. For b 2 B, the pair of functors (()b, ()b) relating
GbS* and GSB is a Quillen adjoint pair.
12.3. The stable model structure on GSB
The essential point in the construction of the stable model structure is to *
*define
the appropriate (stable) homotopy groups. The weak equivalences will then be the
maps of parametrized spectra that induce isomorphisms on all homotopy groups.
We refer to them as the ss*isomorphisms or sequivalences, using these terms i*
*n
terchangeably. There are several motivating observations for our definitions. We
return the group G to the notations for the moment.
First, a Gspectrum X over B is level qffibrant if and only if each projec
tion X(V ) ! *B (V ) = B is a qffibration of exGspaces. It is equivalent th*
*at
each fixed point map X(V )H  ! BH be a nonequivariant qffibration, and, by
Proposition 6.5.1, we have resulting long exact sequences of homotopy groups
(12.3.1) . ..! ssHq+1(B) ! ssHq(Xb(V )) ! ssHq(X(V )) ! ssHq(B) ! . . .
for each b 2 BH . Here, for a Gspace T , ssHq(T ) denotes ssq(T H).
Second, as we have already discussed in x11.4, the fibers Xb of a Gspectrum*
* X
are Gbspectra, and our guiding principle is to use these nonparametrized spect*
*ra
to encode the homotopical information about our parametrized spectra. Proposi
tion 12.2.9 allows us to encode levelwise information in the level homotopy gro*
*ups
of fibers, and it is plausible that we can similarly encode the full structure *
*of our
154 12. MODEL STRUCTURES FOR PARAMETRIZED GSPECTRA
parametrized Gspectrum X in the spectrum level homotopy groups of the fiber
Gbspectra Xb. However, we can only expect to do so when X is level qffibrant
and we have the long exact sequences (12.3.1).
Recall that the homotopy groups ssHq(Y ) of a nonparametrized Gspectrum
Y are defined in [61, III.3.2] as the colimits of the groups ssHq( V Y (V )), *
*where
the maps of the colimit system are induced in the evident way by the adjoint
structure maps "oe:Y (V ) ! WV Y (W ) of Y . The functor V on based G
spaces preserves qfibrations and the functor VB= FB (SV , ) on Gspectra ove*
*r B
preserves level qffibrations. Formally, these hold since SV is a qcofibrant G*
*space
and the relevant model structures are Gtopological. This leads to two families*
* of
long exact sequences relating the homotopy groups ssHq( V Xb(W ) of fibers to t*
*he
homotopy groups of the base space B and of the total spaces X(W ). First, if X *
*is
a level qfibrant Gspectrum over B, then, using basepoints determined by a poi*
*nt
b 2 BH for any H Gb, the qfibrations V X(W ) ! V B of based Gspaces
with fibers V Xb(W ) induce long exact sequences
(12.3.2)
. ..! ssHq+1( V B) ! ssHq( V Xb(W )) ! ssHq( V X(W )) ! ssHq( V B) ! . ...
Second, if X is level qffibrant, then the qffibrations ( VBX)(W ) ! *B of ex*
*G
spectra over B with fibers V Xb(W ) induce long exact sequences
(12.3.3)
. ..! ssHq+1(B) ! ssHq( V Xb(W )) ! ssHq(( VBX)(W )) ! ssHq(B) ! . ...
The first allows us to relate the homotopy groups of the Xb to the homotopy gro*
*ups
of the ordinary loops V X(W ) on total spaces. The second allows us to relate *
*the
homotopy groups of the Xb to the homotopy groups of the parametrized loop ex
spaces ( VBX)(W ). It is the second that is most relevant to our work.
Definition 12.3.4. The homotopy groups of a level qffibrant Gspectrum
over B, or of a level qffibrant Gprespectrum X, are all of the homotopy groups
ssHq(Xb) of all of the fibers Xb, where H Gb. The homotopy groups of a general
Gspectrum, or Gprespectrum, X over B are the homotopy groups ssHq((RX)b) of a
level qffibrant approximation RX to X. We still denote these homotopy groups by
ssHq(Xb). In either category, a map f :X ! Y is said to be a ss*isomorphism o*
*r,
synonymously, an sequivalence, if, after level qffibrant approximation, it in*
*duces
an isomorphism on all homotopy groups.
There are also homotopy groups specified in terms of maps out of sphere spec*
*tra
over B, but we choose to ignore them in setting up our model theoretic foundati*
*ons.
Our choice captures the intuitive idea that spectra over B should be parametriz*
*ed
spectra: the fiber spectra should carry all of the homotopy theoretical informa*
*tion.
With this choice, a good deal of the work needed to set up the stable model str*
*ucture
reduces to work that has already been done in [61]. The following observation i*
*s a
starting point that illustrates the pattern of proof. Now that we have seen how*
* the
equivariance appears in the definition of homotopy groups, we revert to our cus*
*tom
of generally deleting G from the notations.
Lemma 12.3.5. A level qequivalence of spectra over B is a ss*isomorphism.
Proof. A level qffibrant approximation to the given level qequivalence is*
* a
level acyclic qffibration, and it induces a level qequivalence on fibers over*
* points
12.3. THE STABLE MODEL STRUCTURE ON GSB 155
of B by Proposition 12.2.9. This allows us to apply [61, III.3.3], which gives *
*the
same conclusion for nonparametrized spectra, one fiber at a time.
To exploit our definition of homotopy groups, we need the following accompa
nying definition and proposition.
Definition 12.3.6. An prespectrum over B is a level qffibrant prespectrum
X over B such that each of its adjoint structure maps "oe:X(V ) ! WVBX(W )
is a qequivalence of exspaces over B, that is, a qequivalence of total space*
*s. An
(orthogonal) spectrum over B is a level qffibrant spectrum over B such that
each of its adjoint structure maps is a qequivalence; equivalently, its underl*
*ying
prespectrum must be an prespectrum over B.
Since we are omitting the adjective "orthogonal" from "orthogonal spectrum
over B", we must use the term " prespectrum over B" on the prespectrum level
to avoid confusion; the more standard term " spectrum" was used in [61].
Proposition 12.3.7. A level fibrant Gspectrum X over B is an Gspectrum
over B if and only if each fiber Xb is an Gbspectrum. The Gprespectrum ana
logue also holds.
Proof. By the five lemma, this is immediate from a comparison of the long
exact sequences in (12.3.1) and (12.3.3).
This result leads to the following partial converse to Lemma 12.3.5.
Theorem 12.3.8. A ss*isomorphism between spectra over B is a level q
equivalence.
Proof. The analogue for nonparametrized spectra is [61, III.3.4]. In view
of Proposition 12.3.7, we can apply that result on fibers and then use that s*
*pectra
over B are required to be level qffibrant to deduce the claimed level qequiva*
*lence
on total spaces from (12.3.1).
Technically, the real force of our definition of homotopy groups is that this
result describing the ss*isomorphisms between spectra over B is an immediate
consequence of the work in [61]. Given this relationship between spectra and
homotopy groups, many of the arguments of [61] apply fiberwise to allow the de
velopment of the stable model structure. However, as discussed in the next sect*
*ion,
careful use of level fibrant approximation is required. We shall use the terms *
*"stable
model structure" and "smodel structure" interchangeably. The scofibrations are
the same as those of the level qfmodel structure and the sfibrant spectra ove*
*r B
turn out to be the spectra over B.
Definition 12.3.9. A map of spectra or prespectra over B is
(i)an acyclic scofibration if it is a ss*isomorphism and an scofibration,
(ii)an sfibration if it satisfies the RLP with respect to the acyclic scofib*
*rations,
(iii)an acyclic sfibration if it is a ss*isomorphism and an sfibration.
We shall prove the following basic theorem in the next two sections.
Theorem 12.3.10. The categories GSB and GPB are wellgrounded model
categories with respect to the ss*isomorphisms (= sequivalences), sfibration*
*s and
scofibrations. The sfibrant objects are the spectra over B.
156 12. MODEL STRUCTURES FOR PARAMETRIZED GSPECTRA
Remark 12.3.11. Recall Remark 12.1.9. We can define positive prespectra
and positive analogues of our sclasses of maps, starting with the positive lev*
*el qf
model structure. As in [61, IIIx5], the positive analogue of the previous theor*
*em
also holds, with the same proof. The identity functor is the left adjoint of a *
*Quillen
equivalence from GSB or GPB with its positive stable model structure to GSB
or GPB with its stable model structure.
The proof of the following result is virtually the same as the proof of its *
*non
parametrized precursor [61, III.4.16 and III.5.7] and will not be repeated.
Theorem 12.3.12. The adjoint pair (P, U) relating the categories GPB and
GSB of prespectra and spectra over B is a Quillen equivalence with respect to e*
*ither
the stable model structures or the positive stable model structures.
As in [61, III.x6], Theorem 12.3.10 leads to the following definition and th*
*eorem,
whose proof is the same as the proof of [61, III.6.1].
Definition 12.3.13. Let [X, Y ]` denote the morphism sets in the homotopy
category associated to the level qfmodel structure on GPB or GSB . A map
f :X ! Y is a stable equivalence if f* :[Y, E]` ! [X, E]` is an isomorphism f*
*or
all spectra E over B. Define the positive analogues similarly. Let [X, Y ] de*
*note
the morphism sets in the stable homotopy category Ho GSB of spectra over B.
Theorem 12.3.14. The following are equivalent for a map f :X  ! Y of
spectra or prespectra over B.
(i)f is a stable equivalence.
(ii)f is a positive stable equivalence.
(iii)f is a ss*isomorphism.
Moreover [X, E] = [X, E]` if E is an spectrum.
Lemma 12.6.1 below should make it clear why the last statement is true.
12.4.The ss*isomorphisms
In the main, the proof of Theorem 12.3.10 is obtained by applying the result*
*s in
[61] fiberwise. Since total spaces are no longer assumed to be weak Hausdorff, *
*we
have to be a little careful: we are quoting results proven for T and using them*
* for
K*. However, we can just as well interpret [61] in terms of K*. The total spaces
X(V ) of an scofibrant spectrum over B are weak Hausdorff, hence scofibrant
approximation places us in a situation where total spaces are in U and therefore
fibers are in T .
There is a more substantial technical problem to overcome in adapting the
proofs of [61, 62] to the present setting. In the situations encountered in th*
*ose
references, all objects were level qfibrant, and that simplified matters consi*
*derably.
Here, level qffibrant approximation entered into our definition of homotopy gr*
*oups,
and for that reason the results of this section are considerably more subtle th*
*an
their counterparts in the cited sources.
We begin by noting that any level exquasifibrant approximation, not neces
sarily a qffibrant approximation, can be used to calculate the homotopy groups*
* of
parametrized spectra.
Lemma 12.4.1. A zigzag of level qequivalences connecting a spectrum X over
B to a level exquasifibrant spectrum Y over B induces an isomorphism between t*
*he
12.4. THE ss*ISOMORPHISMS 157
homotopy groups of X and of Y , and the latter can be computed directly in terms
of the fibers of Y .
Proof. This follows from Lemma 12.3.5 by applying a level qffibrant approx
imation functor to the zigzag.
Theorem 12.4.2. Let f :X ! Y be a map between Gspectra over B. For
any H G and b 2 BH , there is a natural long exact sequence
. ..! ssHq+1(Yb) ! ssHq((FB f)b) ! ssHq(Xb) ! ssHq(Yb) ! . . .
and, if X is wellsectioned, there is also a natural long exact sequence
. ..! ssHq(Xb) ! ssHq(Yb) ! ssHq((CB f)b) ! ssHq1(Xb) ! . ...
Proof. For the first long exact sequence, let R be a level qffibrant appro*
*xi
mation functor and consider Rf. We claim that the induced map FB f ! FB Rf
is a level qequivalence and that FB Rf is level qffibrant. This means that FB*
* Rf is
a level qffibrant approximation to FB f, so that the homotopy groups of the fi*
*bers
(FB Rf)b ~=F ((Rf)b) are the homotopy groups of FB f. When restricted to fibers
over b, the parametrized fiber sequence RX ! RY ! FB Rf of spectra over
B gives the nonparametrized fiber sequence (RX)b ! (RY )b ! F ((Rf)b), and
the long exact sequence follows from [61, III.3.5]. To prove the claim, observe*
* that
since FB (I, Y ) ! Y is a Hurewicz fibration, it has a pathlifting function w*
*hich
levelwise shows that FB (I, Y ) ! Y is a level fpfibration and therefore a le*
*vel
qffibration (since all qfcofibrations are fpcofibrations in GKB ). The dual *
*gluing
lemma (see Definition 5.4.1(iii)) then gives that the induced map FB f ! FB Rf
is a level qequivalence. Since FB (I, ) preserves level qffibrant objects an*
*d since
pullbacks of level qffibrant objects along a level qffibration are level qff*
*ibrant,
FB Rf is level qffibrant.
Since the maps X ! CB X and RX ! CB RX are cylcofibrations between
wellsectioned spectra and therefore level hcofibrations by Lemma 12.1.3, the *
*glu
ing lemma gives that CB f ! CB Rf is a level qequivalence. Since RX and RY
are level wellsectioned and level qffibrant, they are level exquasifibration*
*s. It
follows from Proposition 8.5.3 that CB Rf is a level exquasifibration. We cann*
*ot
conclude that CB Rf is level qffibrant, but by Lemma 12.4.1 we can nevertheless
use CB Rf to calculate the homotopy groups of CB f. On fibers over b, the cofib*
*er
sequence of Rf is just the cofiber sequence of (Rf)b, and the long exact sequen*
*ce
follows from [61, III.3.5].
Recall Proposition 12.1.4, which specifies the ground structure in GSB and
shows that the level qequivalences give a wellgrounded subcategory of weak eq*
*uiv
alences; the gcofibrations are just the level hcofibrations. The following r*
*esult
shows that the same is true for the ss*isomorphisms. However, in contrast to
Proposition 12.1.4, it is crucial to assume that the relevant maps in the gluing
and colimit lemmas are both cylcofibrations and gcofibrations, as prescribed *
*in
Definition 5.4.1.
Theorem 12.4.3. The ss*isomorphisms in GSB give a wellgrounded subcat
egory of weak equivalences. In detail, the following statements hold.
(i)A homotopy equivalence is a ss*isomorphism.
(ii)The homotopy groups of a wedge of wellgrounded spectra over B are the dir*
*ect
sums of the homotopy groups of the wedge summands.
158 12. MODEL STRUCTURES FOR PARAMETRIZED GSPECTRA
(iii)The ss*isomorphisms are preserved under pushouts along maps that are both
cyl and gcofibrations.
(iv)Let X be the colimit of a sequence in :Xn ! Xn+1 of maps that are both
cyl and gcofibrations and assume that X=BX0 is wellgrounded. Then the
homotopy groups of X are the colimits of the homotopy groups of the Xn.
(v)For a map i: X ! Y of wellgrounded spectra over B and a map j :K ! L
of wellbased spaces, i j is a ss*isomorphism if either i is a ss*isomorp*
*hism
or j is a qequivalence.
Proof. The conclusion that the ss*isomorphisms give a wellgrounded subcat
egory of weak equivalences, as prescribed in Definition 5.4.1, follows directly*
* from
the listed properties, using Lemma 5.4.2 to derive the gluing lemma. Since leve*
*l q
equivalences are ss*isomorphisms, scofibrant approximation in the level qfmo*
*del
structure gives the factorization hypothesis Lemma 5.4.2(ii).
A homotopy equivalence of spectra is a level fpequivalence and hence a level
qequivalence, so (i) follows from Lemma 12.3.5. For finite wedges, (ii) is im*
*me
diate from the evident split cofiber sequences and Theorem 12.4.2. For arbitrary
wedges of wellgrounded spectra over B, _B Xi ! _B RXi is a level qequivalence
since the level qequivalences are wellgrounded and _B RXiis level quasifibran*
*t by
Proposition 8.5.3. By Lemma 12.4.1 we can use _B RXito calculate the homotopy
groups of _B Xi. Over a point b in B, _B RXiis just _(RXi)b and the result foll*
*ows
from the nonparametrized analogue [61, III.3.5].
Now consider (iii). Let i: X ! Y be both a cylcofibration and a gcofibrat*
*ion
and let f :X ! Z be a ss*isomorphism. Since i and its scofibrant approxima
tion Qi are both cyl and gcofibrations and since the level qequivalences give*
* a
wellgrounded subcategory of weak equivalences, the gluing lemma shows that we
may approximate our given pushout diagram by one in which all objects are well
sectioned. Let j :Z ! Y [X Z be the pushout of i along f. Since i and j are
cylcofibrations and j is the pushout of i, their cofibers are homotopy equival*
*ent.
Comparing the long exact sequences of homotopy groups associated to the cofiber
sequences of i and j gives that the pushout Y  ! Y [X Z of f along i is a ss*
isomorphism.
For (iv), we may use scofibrant approximation in the level model structure
to replace our given tower by one in which all objects are wellsectioned. We
note as in the proof of Lemma 5.6.5 that the natural map TelXn ! colimXn is
a level qequivalence and therefore a ss*isomorphism. Relating the telescope *
*to
a classical homotopy coequalizer as in the cited proof, we reduce the calculati*
*on
of the homotopy groups of the telescope to an algebraic inspection based on (ii*
*).
Alternatively, one can commute double colimits to reduce the verification to its
space level analogue.
For (v), it suffices to show that the tensor X ^B T preserves ss*isomorphis*
*ms in
either variable, by Lemma 5.4.4. That follows from Proposition 12.4.4 below.
Proposition 12.4.4. Let f :X ! Y be a map between wellgrounded spectra
over B.
(i)If f is a level qequivalence and g :T ! T 0is a qequivalence of wellbas*
*ed
spaces, then
id^B g :X ^B T ! X ^B T 0
is a level qequivalence and therefore a ss*isomorphism.
12.5. PROOFS OF THE MODEL AXIOMS 159
(ii)If f is a ss*isomorphism, then
f ^B id:X ^B T ! Y ^B T
is a ss*isomorphism for any wellbased space T and
FB (id, f): FB (T, X) ! FB (T, Y )
is a ss*isomorphism for any finite based CW complex T .
(iii)For a representation V in V , the map f is a ss*isomorphism if and only if
VBf is a ss*isomorphism.
Proof. Part (i) holds since the level qequivalences are wellgrounded. The*
*re
fore, for the first part of (ii), we may assume by qcofibrant approximation in*
* the
space variable that T is a based CW complex. Using Proposition 8.5.3, it also
implies that  ^B T preserves approximations of wellgrounded spectra over B
by level exquasifibrations. Now the first part of (ii) follows fiberwise from*
* its
nonparametrized analogue [61, III.3.11] and (iii) follows fiberwise from its no*
*n
parametrized analogue [61, III.3.6]. Since FB (, X) takes cofiber sequences of
based spaces to fiber sequences of spectra over B, the second part of (iii) fol*
*lows
from the first exact sequence in Theorem 12.4.2, as in the proof of [61, III.3.*
*9].
This leads to the following result, which shows that we are in a stable situ*
*ation.
Proposition 12.4.5. For all wellgrounded spectra X over B and all repre
sentations V in IG , the unit j :X ! VB VBX and counit ": VB VBX ! X of
the ( VB, VB) adjunction are ss*isomorphisms. Therefore, if f :X ! Y is a map
between wellgrounded spectra over B, then the natural maps j :FB f ! B CB f
and ffl: B FB f ! CB f are ss*isomorphism.
Proof. For j, after approximation of X by an exquasifibration, the conclus*
*ion
follows fiberwise from its nonparametrized analogue [61, III.3.6]. Using the tw*
*o out
of three property and the triangle equality for the adjunction, it follows that*
* VB"
is a ss*isomorphism, hence so is ". For the last statement, the maps j and " a*
*re
the parametrized analogues of the maps defined for ordinary loops and suspensio*
*ns
in [71, p. 61], and they fit into diagrams relating fiber and cofiber sequences*
* like
those displayed there. Now the last statement follows from the five lemma and t*
*he
exact sequences in Theorem 12.4.2.
12.5. Proofs of the model axioms
We need some Gspectrum level recollections from [61] and their analogues for
Gspectra over B to describe the generating acyclic scofibrations. Let (SG , G*
*S )
denote the Gcategory of Gspectra. To keep track of enrichments, we return G to
the notations for the moment.
We have a shift desuspension functor FV from based Gspaces to Gspectra
given by FV T = V *^ T , where V *(W ) = JG (V, W ) [61, III.4.6]. It is left a*
*djoint
to evaluation at V . For Gspectra X, the adjoint structure Gmap
"oe:X(V ) ! W X(V W )
may be viewed by adjunction as a Gmap
"oe:SG (FV S0, X) ! SG (FV WSW , X).
160 12. MODEL STRUCTURES FOR PARAMETRIZED GSPECTRA
Passing to Gfixed points and taking X = FV S0, the image of the identity map
gives a map of Gspectra
~V,W :FV WSW ! FV S0.
(The notation ~V,W was used in [61], but we need room for a subscript). A Yoneda
lemma argument then shows that the map of Gspaces
SG (~V,W, id): SG (FV S0, X) ! SG (FV WSW , X)
can be identified with "oe:X(V ) ! W X(V W ).
We need the analogue for Gspectra over B. Recall from Definition 11.3.5 tha*
*t,
for an exGspace K over B, (FV K)(W ) = V *(W ) ^B K, where
V *(W ) = JG,B(V, W ) = JG (V, W )B = (FV S0)(W ) ^B S0B.
It follows that we can identify FV K with the evident external tensor FV S0^B K*
* of
the Gspectrum FV S0 and the exGspace K over B; compare Remark 11.1.7. We
have used the notation ^B for this generalized tensor, but viewing it as a spec*
*ial
case of the external smash product of spectra over * and over B would suggest t*
*he
alternative notation Z.
Definition 12.5.1. For exGspaces K over B, we define a natural map
~V,WB:FV W WBK ! FV K.
Namely, identifying the source and target with external tensor products, define
~V,WB= ~V,W ^B id:(FV WSW ) ^B K ! (FV S0) ^B K.
We can describe the adjoint structure maps of Gspectra over B in terms of
these maps ~V,WB.
Lemma 12.5.2. Under the adjunctions
PB (FV S0B, X) ~=FB (S0B, X(V )) ~=X(V )
and
PB (FV WSWB, X) ~=FB (S0B, WBX(V W )) ~= WBX(V W ),
the map
PB (~V,WB, id): PB (FV S0B, X) ! PB (FV WSWB, X)
corresponds to
"oe:X(V ) ! WBX(V W ).
Proof. When X = FV S0B, the conclusion holds by comparison with the case
of Gspectra. The general case follows from the Yoneda lemma of enriched catego*
*ry
theory. See, for example, [12, 6.3.5].
We could have started off by defining ~V,WBin a conceptual manner analogous
to our definition of ~V,W, but we want the explicit description of ~V,WBin term*
*s of
~V,W in order to deduce homotopical properties in the parametrized context from
homotopical properties in the nonparametrized context. For that and other pur
poses, we need the following observation. We return to our convention of deleti*
*ng
G from the notations, on the understanding that everything is equivariant.
Lemma 12.5.3. If OE: X ! Y is an sequivalence of level wellbased nonpara
metrized spectra and K is a wellgrounded exspace with total space of the homo*
*topy
type of a GCW complex, then OE ^B id:X ^B K ! Y ^B K is an sequivalence.
12.5. PROOFS OF THE MODEL AXIOMS 161
Proof. We use the exfibrant approximation functor P of Definition 8.3.4. We
have a natural zigzag of hequivalences between K and P K. By Proposition 8.2.*
*6,
it induces a zigzag of level hequivalences between X ^B K and X ^B P K and, by
Corollary 8.2.5, X ^B P K is a level exfibration. Therefore, by Lemma 12.4.1, *
*it
suffices to consider the case when K is an exfibration. Since (X ^B K)b = X ^ *
*Kb
and Kb is of the homotopy type of a GbCW complex, by Theorem 3.4.2, each
(OE ^B id)b is an sequivalence by [61, III.3.11].
The following result is crucial.
Proposition 12.5.4. Let K be a wellgrounded exspace with total space of the
homotopy type of a CW complex. Then
~V,WB:FV W WBK ! FV K
and
~V,W Z id:FV WSW Z FZK ! FV S0 Z FZK
are ss*isomorphisms of spectra over B.
Proof. Since ~V,WB= ~V,W ^B id, Lemma 12.5.3 and the corresponding non
parametrized statement [61, III.4.5] imply the first statement. For the second
statement, observe that for spectra X we have the associativity relation
X Z FZK ~=X Z (FZS0 ^B K) ~=(X ^ FZS0) ^B K.
Taking X = FV T for a based space T and using Lemma 11.3.6, we see that
FV T Z FZK ~=FV Z(T ^B K).
Using equivalences of this form and checking definitions, we conclude that the *
*map
~V,W Z idof the statement can be identified with the map
~V Z,W^B id:(FV Z W SW ) ^B K ! (FV ZS0) ^B K.
Thus the second ss*isomorphism is a special case of the first.
From here, the proof of Theorem 12.3.10 closely parallels arguments in [61,
III.x4], but simplified a little by Theorem 5.5.1. The generating set of scofi*
*brations
is again F IfB. The generating set F KfBof acyclic scofibrations is given by a*
* variant
of the definition in the nonparametrized case [61, III.4.6].
Definition 12.5.5. Recall the factorization of ~V,W through the mapping
cylinder (in the category of spectra) as
V,W rV,W
~V,W :FV WSW k___//_M~V,W____//_FV S0.
Here kV,W is an scofibration and rV,W is a deformation retraction. For i: C !*
* D
in IfB, the map
i kV,W :C ^B M~V,W [C^BFV WSW D ^B FV WSW ! D ^B M~V,W
is an scofibration in GSB by Proposition 12.2.3, and it is therefore also a cy*
*l
cofibration by Theorem 12.1.7. It is a ss*isomorphism by Proposition 12.5.4 and
inspection of definitions. The scofibrations in F JfBare level acyclic and are*
* there
fore also ss*isomorphisms. Restricting to V and W in skIG , define the generat*
*ing
set F KfBof acyclic scofibrations to be the union of F JfBand the set of all m*
*aps
of the form i kV,W with i 2 IfB.
162 12. MODEL STRUCTURES FOR PARAMETRIZED GSPECTRA
A fortiori, the following result identifies the sfibrations, but it must be*
* proven
a priori as a first step towards the verification of the model axioms.
Proposition 12.5.6. A map f :X ! Y satisfies the RLP with respect to
F KfBif and only if f is a level qffibration and the diagrams
(12.5.7) X(V ) _o"e//_ WBX(V W )
f(V ) WBf(V W)
fflffl fflffl
Y (V )_o"e//_ WBY (V W )
are homotopy pullbacks for all V and W .
Proof. As in [61, III.4.7], the homotopy pullback property must be inter
preted as requiring a qequivalence from X(V ) into the pullback in the display*
*ed
diagram. Recall that F JfBis contained in F KfBand that a map has the RLP with
respect to F JfBif and only if it is a level qffibration. This gives part of *
*both
implications. It remains to show that a level qffibration f has the RLP with r*
*e
spect to i kV,W for all i 2 IfBif and only if the displayed diagram is a homoto*
*py
pullback. This is a formal but not altogether trivial exericise from the fact *
*that
the level qfmodel structure is Gtopological in the sense characterized in Pro*
*po
sition 10.3.18. Notice that the map i kV,W is isomorphic to the map i kV,WB,
where kV,WB= kV,W ^B S0B. With notation as in (10.3.6), f has the RLP with
respect to i kV,WBfor all i 2 IfBif and only if the pair (i, PB (kV,WB, f)) has*
* the
lifting property for all i 2 IfB, which holds if and only if the map PB (kV,WB,*
* f) of
exspaces over B is an acyclic qffibration. This map is a qffibration since,*
* for
j 2 JfB, the map j kV,W ~=j kV,WBis a level acyclic scofibration of spectra ov*
*er
B by Proposition 12.2.3. Since f is a level qffibration, (j kV,WB, f) has the *
*lift
ing property, hence, by adjunction, so does (j, PB (kV,WB, f)). Finally, PB (kV*
*,WB, f)
is homotopy equivalent to PB (~V,WB, f) so one is a qequivalence if and only i*
*f the
other is. Under the isomorphisms in Lemma 12.5.2, the map PB (~V,WB, f) coincid*
*es
with the map from X(V ) into the pullback in the displayed diagram and is thus a
qequivalence if and only if that diagram is a homotopy pullback.
Let *B be the terminal spectrum over B, so that each *B (V ) is the terminal
exspace *B . Observe that *B is an spectrum with trivial homotopy groups.
Corollary 12.5.8. The terminal map F ! *B satisfies the RLP with respect
to F KB if and only if F is an spectrum over B.
Corollary 12.5.9. If f :X ! Y is a ss*isomorphism that satisfies the RLP
with respect to F KB , then f is a level acyclic qffibration.
Proof. Since f is a level qffibration by Proposition 12.5.6, the dual of t*
*he
gluing lemma applied to the diagram
*B _______//_Yofo_X
  
  
fflffl  
FB (I, Y_)___//Yoof__X
12.6. SOME QUILLEN ADJOINT PAIRS RELATING STABLE MODEL STRUCTURES 163
gives that the induced map F ! FB f of pullbacks is a level qequivalence. Sin*
*ce
f has the RLP with respect to F KB , so does its pullback F  ! *B . By the
previous corollary, F is thus an spectrum over B. In particular, it is level *
*qf
fibrant. We conclude that F is a level qffibrant approximation for FB f. Since*
* f
is a ss*isomorphism, Theorem 12.4.2 gives that F is acyclic. By Theorem 12.3.8,
this implies that F  ! *B is a level qequivalence. Thus the fibers F (V )b a*
*ll
have trivial homotopy groups. We conclude (with a bit of extra argument as in
[62, 9.8] to handle ss0) that each map of fibers f(V )b induces an isomorphism *
*on
homotopy groups. Therefore, since each f(V ) is a qffibration, each f(V ) indu*
*ces
an isomorphism on homotopy groups.
The proof of the model axioms for the stable model structure is now immediat*
*e.
Proof of Theorem 12.3.10. The ss*isomorphisms give a wellgrounded sub
category of weak equivalences, by Theorem 12.4.3. Conditions (i), (iii), and (*
*iv)
in Theorem 5.5.1 are clear from our specification of the generating acyclic s
cofibrations and the result for the level qfmodel structure. For condition (i*
*i),
a ss*isomorphism that satisfies the RLP with respect to F KB has the RLP with
respect to F IB by Corollary 12.5.9. Conversely, a map that has the RLP with re
spect to F IB is a level acyclic qffibration and therefore has the RLP with re*
*spect to
F KB by Proposition 12.5.6. It is a ss*isomorphism since it is level acyclic. *
*Since
all sfibrations are level qffibrations, right properness follows from the sli*
*ghtly
stronger observation in the following result.
Proposition 12.5.10. The ss*isomorphisms in GSB are preserved under pull
backs along level qffibrations.
Proof. Let g be the pullback of a level qffibration f along a ss*isomorph*
*ism.
Then g is a level qffibration and the fibers of g(V ) are isomorphic to the fi*
*bers of
f(V ). Therefore the homotopy fibers FB g are level qequivalent to the homotopy
fibers FB f. The result follows by comparison of the first long exact sequence*
* in
Theorem 12.4.2 for f and g.
12.6. Some Quillen adjoint pairs relating stable model structures
We prove here that all of the adjoint pairs that were shown to be Quillen
adjoints with respect to the level model structure in x12.2 are still Quillen a*
*djoints
with respect to the stable model structure. In view of the role played by level
qffibrant approximation in our definition of homotopy groups, it is helpful to
first understand the relationship between sfibrant approximation and level qf
fibrant approximation. Now that the model structures have been established, we
henceforward use the term sequivalence rather than the synonymous term ss*
isomorphism.
Lemma 12.6.1. Let :X  ! RX and `:X  ! R`X be an sfibrant ap
proximation of X and a level qffibrant approximation of X. Then there is an
sequivalence , :R`X ! RX under X.
Proof. Since ` is a level acyclic scofibration, it is an acyclic scofibr*
*ation by
Lemma 12.3.5. Since RX is sfibrant, the RLP gives a map , under X, and it is
an sequivalence since and ` are sequivalences.
164 12. MODEL STRUCTURES FOR PARAMETRIZED GSPECTRA
We have the following relationship between the homotopy categories of ex
spaces over B and of spectra over B.
Proposition 12.6.2. The pair ( 1B, 1B) is a Quillen adjunction relating GSB
and GKB . More generally, ( 1V, 1V) = (FV , EvV ) is a Quillen adjunction for *
*any
representation V 2 V .
Proof. The maps 1Vi, where i 2 IfBis a generating cofibration for the qf
model structure on GKB , are among the generating cofibrations of the smodel
structure on GSB , and it follows that 1V preserves cofibrations. Since 1V ta*
*kes
acyclic qfcofibrations to level acyclic qfcofibrations, and these are acyclic*
* by
Lemma 12.3.5, 1V also preserves acyclic cofibrations.
Now consider an adjoint pair (F, V ) between categories of parametrized spec*
*tra
that is a Quillen adjunction with respect to the level model structures. Since *
*the
cofibrations are the same in the level model structure and in the stable model
structure, the left adjoint F certainly preserves cofibrations. Thus, to show t*
*hat
(F, V ) is also a Quillen adjunction with respect to the stable model structure*
*s, we
need only show that F carries acyclic scofibrations to sequivalences. When F
preserves all sequivalences, this is obvious; otherwise, by Lemma 5.4.6, it su*
*ffices
to verify this for the generating acyclic scofibrations. The cited result appl*
*ies in
general to subcategories of wellgrounded weak equivalences, and in our context
it applies to both the level qequivalences and the sequivalences. Recall tha*
*t a
Quillen left adjoint in any model structure preserves weak equivalences between
cofibrant objects, by Ken Brown's lemma [44, 1.1.12]. The following parenthetic*
*al
observation applies to give a stronger conclusion for the Quillen left adjoints*
* that
we shall encounter. It will play a crucial role in exploiting the equivalence *
*of
homotopy categories that we will establish in the next chapter. Note that the s
cofibrant spectra are the cofibrant objects in both the level and the stable mo*
*del
structures, and they are wellgrounded.
Proposition 12.6.3. Let F be a Quillen left adjoint between categories of
parametrized spectra with their stable model structures and suppose that F pres*
*erves
level qequivalences between wellgrounded spectra. Then F preserves sequivale*
*nces
between wellgrounded spectra.
Proof. If g :X ! Y is an sequivalence, where X and Y are wellgrounded,
factor g in the level model structure as
g0 g00
X _____//W_____//Y,
where g0 is an scofibration and g00is a level acyclic qffibration. Then W is *
*well
grounded and F g00is a level qequivalence by assumption. Since F is a Quillen *
*left
adjoint in the smodel structures, F g0is an sequivalence. Since level qequiv*
*alences
are sequivalences it follows that F g = F g00O F g0 is an sequivalence.
The following sequence of results consists of analogues for the stable model
structures of results proven for the level model structures in x12.2. Recall th*
*at we
actually have wellgrounded stable model structures s(C ) for any closed genera*
*ting
set C . As in x12.2, wherever a qf(C )model structure was used in Chapter 7
for some particularly well chosen C , we must use the corresponding s(C )model
structure here.
12.6. SOME QUILLEN ADJOINT PAIRS RELATING STABLE MODEL STRUCTURES 165
Proposition 12.6.4. Let T be a based GCW complex. Then (^B T, FB (T, ))
is a Quillen adjunction on GSB . When T = SV , it is a Quillen equivalence.
Proof. This is immediate from the fact that the stable model structure is
Gtopological, together with Propositions 12.4.4 and 12.4.5.
Proposition 12.6.5. If i: X  ! Y and j :W  ! Z are scofibrations of
spectra over base spaces A and B, then
i j :(Y Z W ) [XZW (X Z Z) ! Y Z Z
is an scofibration over A x B which is sacyclic if either i or j is sacyclic.
Proof. The statement about scofibrations is part of the analogue, Proposi
tion 12.2.3, for the level model structure. As usual, it suffices to show that*
* i j
is an sequivalence if i 2 F IfBand j 2 F KfB, where F KfBis the set of generat*
*ing
acyclic scofibrations specified in Definition 12.5.5. Arguing as in Lemma 5.4.*
*4 and
using properness, this will hold if smashing the source and the target of i wit*
*h j
give sequivalences. The reduction so far would work just as well for internal *
*smash
products. The required last step reduces via inspection of Definition 12.5.5 to*
* an
application of Proposition 12.5.4, with base space taken to be A x B. The reason
that this last step works for external smash products but fails for internal sm*
*ash
products is made clear in Warning 6.1.7.
Corollary 12.6.6. If Y is an scofibrant spectrum over B, then the functor
() Z Y from GSA to GSAxB is a Quillen left adjoint with Quillen right adjoint
~F(Y, ).
Proposition 12.6.7. Let f :A ! B be a Gmap. Then (f!, f*) is a Quillen
adjoint pair. If f is a qequivalence, then (f!, f*) is a Quillen equivalence.
Proof. We must show that f! takes acyclic scofibrations to sequivalences.
Since f! preserves wellgrounded objects and level qequivalences between well
grounded objects by Proposition 12.2.5, it suffices by Lemma 5.4.6 to prove tha*
*t f!k
is an sequivalence for each map k in F KfA. This follows from the corresponding
Quillen adjunction with respect to the level model structure if k 2 F JfA, so a*
*ssume
that k is of the form i kV,W ~=i kV,WA. We claim that f!k is a map in F KfBand *
*is
therefore an sequivalence. Observe that kV,WA~=f*kV,WB. Using (11.4.5) and the
fact that f! preserves pushouts, we see from the definition of the product th*
*at
f!(i f*kV,WB) ~=(f!i) kV,WB. Since i is obtained from a map over A by adjoining*
* a
disjoint section, f!i is obtained from a map over B by adjoining a disjoint sec*
*tion
and is thus in IfB.
Now assume that f is a qequivalence. By [44, 1.3.16], (f!, f*) is a Quillen
equivalence if and only if f* reflects sequivalences between sfibrant objects*
* and
the composite X  ! f*f!X  ! f*Rf!X given by the unit of the adjunction
and sfibrant approximation is an sequivalence for all scofibrant X. Since th*
*e s
fibrant objects are the spectra over B and the sequivalences between spect*
*ra
over B are the level qequivalences, the reflection property follows directly f*
*rom
the corresponding Quillen equivalence with respect to the level model structure.
That result also gives that the composite X ! f*f!X ! f*R`f!X is a level q
equivalence and hence an sequivalence. Applying Lemma 12.6.1 with X replaced
by f!X and observing that f* preserves sequivalences between level qffibrant
166 12. MODEL STRUCTURES FOR PARAMETRIZED GSPECTRA
Gspectra over B since (f*Y )a ~= Yf(a), a little diagram chase shows that the
composite X ! f*f!X ! f*Rf!X is an sequivalence.
Observe that Proposition 12.6.3 applies to f!.
Proposition 12.6.8. Let f :A ! B be a Gbundle whose fibers Ab are Gb
CW complexes. Then (f*, f*) is a Quillen adjoint pair.
Proof. We must show that f* preserves acyclic scofibrations. Again it suf
fices by Lemma 5.4.6 to prove that f*k is an sequivalence between wellgrounded
spectra for each map k 2 F KfB. That f*k is a map between wellgrounded spectra
follows from the fact that if K q B is a space over B with a disjoint section, *
*then
f*FV (KqB) = FV f*KqA is wellgrounded. To see that f*k is an sequivalence, it
is enough, as in the proof of Proposition 12.6.7, to consider k = i kV,WBwith i*
* 2 IfB.
We have that f*kV,WB= kV,WAand, since f* preserves pushouts, smash products,
and factorizations through mapping cylinders, we see as in the cited proof that
f*k ~=f*i kV,WA, which is an acyclic scofibration.
Proposition 12.6.9. Let ': H ! G be the inclusion of a subgroup. The pair
of functors ('!, *'*) relating HSA and GS'!Agives a Quillen equivalence. If A *
*is
completely regular, then '!is also a Quillen right adjoint.
Proof. By Proposition 14.3.1 below, ('!, *'*) is a Quillen adjoint pair. T*
*he
proof that it is a Quillen equivalence is the same as the proof of the exspace
level analogue in Proposition 7.4.6. The last statement is less obvious. As in
the proof of the corresponding statement in Proposition 7.4.6, it follows from *
*the
spectrum level analogue of Remark 2.4.4, which in turn requires the spectrum le*
*vel
analogue of Proposition 2.4.3, and the analogue in the stable model structure of
Proposition 7.4.3. The required analogues are proven in x14.4 below.
We shall see that ('!, *'*) descends to a closed symmetric monoidal equival*
*ence
of homotopy categories in Proposition 13.7.9 below.
Corollary 12.6.10. The functor '*: HoGSB ! HoHS'*B is the composite
*
Ho GSB __~__//HoGK'!'*B' HoHK'*B
Using Example 11.5.5 as in Proposition 12.2.9, the following result is now a
special case of Propositions 12.6.9 and 12.6.7.
Proposition 12.6.11. For b 2 B, the pair of functors (()b, ()b) relating
GbS and GSB is a Quillen adjoint pair.
CHAPTER 13
Adjunctions and compatibility relations
Introduction
The utility of the stable homotopy category Ho GSB depends on the fact that
the usual functors and adjunctions descend to it and still satisfy appropriate *
*com
mutation relations. We consider such matters in this chapter. Many of our basic
adjunctions are Quillen adjunctions in the stable model structure. We recorded
those in x12.6. The crucial adjunction missing from x12.6 is (f*, f*) for a gen*
*eral
map f of base spaces. This cannot be a Quillen adjoint pair by the argument in
Counterexample 0.0.1. We used Brown representability to construct the right ad
joint f* between homotopy categories of exspaces in Theorem 9.3.2. Analogously,
in x13.1 we use Brown representability to construct f* between homotopy categor*
*ies
of parametrized spectra, and we use base change along diagonal maps to internal*
*ize
smash products and function spectra. There is an interesting twist here. It is *
*not
easy to verify the MayerVietoris axiom directly. Rather, we use the triangulat*
*ed
category variant of the Brown representability theorem, whose hypotheses turn o*
*ut
to be easier to check.
In x13.7, we complete the proof that our stable homotopy categories are sym
metric monoidal and prove some basic compatibility relations among smash prod
ucts and base change functors. These results involve commutation of Quillen
left and right adjoints, and we would not know how to prove them using only
model theoretic fibrant and cofibrant replacement functors. Rather, their proo*
*fs
depend on an equivalence between our model theoretic stable homotopy category
of parametrized Gprespectra and a classical homotopy category of what we call
"excellent" parametrized Gprespectra. We used an analogous, but more elemen
tary, equivalence of categories in Chapter 9. It is essential to use parametri*
*zed
Gprespectra rather than parametrized Gspectra to make the comparison since
the relevant constructions do not all preserve functoriality on linear isometri*
*es;
that is, they do not preserve IG spaces. Results proven using the comparison a*
*re
then translated to parametrized Gspectra along the Quillen equivalence between
parametrized Gprespectra and parametrized Gspectra.
These equivalences of categories allow us to use a prespectrum level analogue
T of the exfibrant approximation functor P to study derived functors. We define
excellent parametrized Gprespectra in x13.2. We lift the exfibrant approximat*
*ion
functor P from exGspaces to parametrized Gspectra in x13.3. There are several
further twists here. First, the functor P on exGspaces does not behave well w*
*ith
respect to tensors, so extending it to a functor on parametrized Gprespectra is
subtle. Second, with the extension, the zigzag of hequivalences connecting P *
*to
the identity functor is no longer given by honest maps of parametrized Gprespe*
*ctra,
only weak maps. Third, the functor P does not take parametrized Gprespectra to
167
168 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS
excellent ones. To remedy this, we introduce two auxiliary functors K and E in
x13.4. The composite T = KEP does land in excellent parametrized Gprespectra,
and K converts weak maps to honest maps. In xx13.5 and 13.6 we use T to prove
the promised equivalence of homotopy categories and show how to study derived
funtors in this context.
There are few issues of equivariance in this chapter, and we generally conti*
*nue
to omit the (compact Lie) group G from the notations. We adopt the convention
of calling isomorphisms in homotopy categories equivalences and we denote them
by ' rather than ~=.
13.1. Brown representability and the functors f* and FB
We need some preliminaries about the two versions of Brown representability
that are applicable in stable situations. Recall Example 11.5.5.
Definition 13.1.1. For n 2 Z and H G, we have an scofibrant sphere
Gspectrum SnHsuch that ssHn(X) = [SnH, X]G for all Gspectra X. Explicitly,
( 1 n
SnH= (G=H+ ^ S ) if n 0,
Fn (G=H+ ^ S0) if n <,0
as in [61, II.4.7], where Fn is the shift desuspension by Rn. We may allow the
ambient group to vary. Replacing G by Gb for b 2 B and letting H Gb, define
Sn,bHto be the Gspectrum (SnH)b over B. Note that Sn,bHis scofibrant, by Prop*
*o
sition 12.2.9. By adjunction, for Gspectra X over B, ssHn(Xb) is isomorphic to
[Sn,bH, X]G,B. Let DB be the set of all such Gspectra Sn,bHover B.
From here, the following three results work in exactly the same way as their
exspace analogues in x7.4. Observe that the category Ho GKB has coproducts and
homotopy pushouts, hence homotopy colimits of directed sequences.
Lemma 13.1.2. Each X 2 DB is compact, in the sense that
colim[X, Yn]G,B ~=[X, hocolimYn]G,B
for any sequence of maps Yn ! Yn+1 in GSB .
Proposition 13.1.3. A map , :Y  ! Z in GSB is an sequivalence if and
only if the induced map ,*: [X, Y ]G,B ! [X, Z]G,B is a bijection for all X 2 *
*DB .
Proof. This is a tautology since as X ranges through the Sn,bH, [X, Y ]G,B
ranges through the homotopy groups ssHn(Yb) that define the sequivalences.
Theorem 13.1.4 (Brown). A contravariant setvalued functor on the category
Ho GSB is representable if and only if it satisfies the wedge and MayerVietoris
axioms.
Since we have the Quillen adjoint pair (f!, f*), we have the right derived f*
*unc
tor f* :HoGSB ! Ho GSA . As in the proof of the analogous result on the level
of exspaces, Theorem 9.3.2, we can obtain the desired right adjoint f* to f* b*
*y use
of Brown's theorem provided that we can show that f* preserves the relevant ho
motopy colimits. However, since f* :GSB ! GSA does not preserve scofibrant
objects, this is not obvious. We will later give results that would allow us to*
* carry
13.1. BROWN REPRESENTABILITY AND THE FUNCTORS f* AND FB 169
out the proof in a manner analogous to the proof of Theorem 9.3.2, but it is in
structive to switch gears and give a more direct proof. It is based on the use*
* of
triangulated categories and would not have applied on the exspace level.
Lemma 13.1.5. The category Ho GSB is triangulated.
Proof. The treatment of triangulated categories in [74] gives a general pat*
*tern
of proof for showing that homotopy categories associated to appropriate model
categories are triangulated. It applies here. The distinguished triangles are t*
*hose
equivalent in HoGSB to cofiber sequences that start with a wellgrounded spectr*
*um
or, equivalently by Proposition 12.4.5, those equivalent to the negatives of fi*
*ber
sequences. Note that, by the proof of Theorem 12.4.2, every cofiber sequence is
equivalent in Ho GSB to a cofiber sequence of level exquasifibrations.
In triangulated categories, there is an alternative version of Brown's repre
sentability theorem due to Neeman [80]. It requires a "detecting set of compact
objects". In triangulatedLcategoriesLwith coproducts (or sums), an object X is *
*said
to be compact if [X, Yi] ~=[X, Yi] for any set of objects Yi. In our topolo*
*gi
cal situations, this reduces to the compactness of spheres, exactly as the proo*
*f of
Lemma 13.1.2. A detecting set of objects is one that detects equivalences, in t*
*he
sense suggested by Proposition 13.1.3. We have the following result.
Lemma 13.1.6. DB is a detecting set of compact objects in Ho GSB .
Recall that an additive functor between triangulated categories is said to be
exact if it commutes with up to a natural equivalence and preserves distingui*
*shed
triangles. The following theorems are proven in [80, 3.1, 4.1]; they are discus*
*sed
with an eye to applications such as ours in [40, x8].
Theorem 13.1.7. Let A be a compactly detected triangulated category. A
functor H :A op! A b that takes distinguished triangles to long exact sequences
and converts coproducts to products is representable.
Theorem 13.1.8. Let A be a compactly detected triangulated category and B be
any triangulated category. An exact functor F :A ! B that preserves coproducts
has a right adjoint G.
Theorem 13.1.9. For any Gmap f :A ! B, there is a right adjoint f* to
the functor f* :HoGSB ! Ho GSA , so that
[f*Y, X]G,A ~=[Y, f*X]G,B
for X in GSA and Y in GSB .
Proof. The left adjoint f!commutes with and preserves cofiber sequences,
and this remains true after passage to derived homotopy categories. Therefore t*
*he
derived functor f! is exact. Since f* is Quillen right adjoint to f!, the deri*
*ved
functor f* is right adjoint to f!and is therefore also exact; see, for example,*
* [79,
3.9]. If X is in DA , then f!X is compact in Ho GSB , as we see from commutation
relations between relevant Quillen left adjoints given in Remark 11.4.7. It fol*
*lows
formally that f* preserves coproducts, by [80, 5.1] or [40, 7.4].
Remark 13.1.10. For composable maps f and g, there is a natural equivalence
g* O f* ' (g O f)* on homotopy categories since f* O g* ' (g O f)*.
170 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS
Exactly as for exspaces in Theorem 9.3.4, we apply change of base along the
diagonal map : B ! B x B to obtain internal smash product and function
spectra functors in Ho GSB .
Theorem 13.1.11. Define ^B and FB on HoGSB to be the composite (derived)
functors
X ^B Y = *(X Z Y ) and FB (X, Y ) = ~F(X, *Y ).
Then
[X ^B Y, Z]G,B ~=[X, FB (Y, Z)]G,B
for X, Y and Z in Ho GSB .
Proof. The displayed adjunction is the composite of the adjunction for the
external smash product and function spectra functors given by Corollary 12.6.6 *
*and
the adjunction ( *, *).
13.2. The category GEB of excellent prespectra over B
We must still prove that Ho GSB is a closed symmetric monoidal category un
der the derived internal smash product, that the derived functor f* is closed s*
*ym
metric monoidal, and that various compatibility relations that hold on the poin*
*tset
level descend to homotopy categories. In particular, since our right adjoints f*
**, *,
and therefore FB come from Brown's representability theorem, it is not at all o*
*bvi
ous how to prove that they are wellbehaved homotopically. In Chapter 9, we sol*
*ved
the corresponding exspace level problems by proving that Ho GKB is equivalent
to the more classical and elementary homotopy category hGWB . Here GWB is the
category of exfibrations over B whose total spaces are compactly generated and*
* of
the homotopy types of GCW complexes, and hGWB is obtained from GWB simply
by passage to homotopy classes of maps. This equivalence allowed us to exploit *
*the
exfibrant approximation functor P of x8.3 to resolve the cited problems.
We shall resolve our spectrum level problems similarly, and the following de*
*fi
nitions give the appropriate analogues of GWB and hGWB . However, to keep closer
to the exspace level, it is essential to work with parametrized prespectra rat*
*her
than parametrized spectra. It is safe to do so in view of the Quillen equivalen*
*ce
(P, U) of Theorem 12.3.12 relating GPB and GSB .
Definition 13.2.1. Let X be a Gprespectrum over B.
(i)X is wellstructured if each level X(V ) is in GWB .
(ii)X is cofibrant if it is wellgrounded and each structure map
oe : WBX(V ) ! X(V W )
is an fpcofibration.
We can now give the definition of excellent Gprespectra over B and of the
associated classical homotopy category. Working with classical nonequivariant a*
*nd
nonparametrized coordinatized prespectra {En}, it has been known since the 1960*
*'s
that the following definition gives the simplest quick and dirty rigorous const*
*ruction
of the stable homotopy category.
Definition 13.2.2. The category GEB of excellent Gprespectra over B is the
full subcategory of GPB whose objects are the wellstructured cofibrant G
prespectra over B. Let hGEB denote the classical homotopy category obtained
from GEB by passage to homotopy classes of maps.
13.2. THE CATEGORY GEB OF EXCELLENT PRESPECTRA OVER B 171
We comment on the conditions we require of excellent prespectra over B. We
require that they be wellstructured so that we can exploit levelwise our equiv*
*alence
of homotopy categories on the exspace level. We require that they be cofibra*
*nt
since that provides "homotopical glue" that is necessary for the transition from
the known equivalence on the exspace level to the desired equivalence on the p*
*re
spectrum level. We shall make this idea precise shortly, in Proposition 13.2.5.*
* We
require that they be prespectra over B since it is clearly sensible to restri*
*ct at
tention to sfibrant objects in GSB if we hope to compare homotopy categories.
Recall that X is an prespectrum if it is a level qffibrant prespectrum over B
whose adjoint structure maps
"oe:X(V ) ! WVBX(W )
are qequivalences. Since excellent prespectra over B are required to be level *
*ex
fibrations, they are automatically level qffibrant. The condition on the adjo*
*int
structure maps is stronger than it appears on the surface.
Lemma 13.2.3. For excellent Gprespectra X over B, the adjoint structure
maps
"oe:X(V ) ! WBX(V W )
are fpequivalences.
Proof. The "oeare qequivalences between GCW homotopy types and are
therefore hequivalences. Since they are maps between exfibrations, they are f*
*p
equivalences by Proposition 5.2.2.
This implies, for example, that homotopypreserving functors GEB ! GPB
that may not preserve level qequivalences nevertheless do preserve the equival*
*ence
property required of the adjoint structure maps.
Remark 13.2.4. Our definition of excellent parametrized prespectra is close *
*to
that used by Clapp and Puppe [18, 19], who in turn were influenced by definitio*
*ns
in [66]. Curiously, while Clapp [18] focuses on exfibrations, Clapp and Puppe
[19] never mention fibration conditions. These papers are nonequivariant, but t*
*he
second is written in terms of what the authors call "coordinatefree spectra" o*
*ver
B. These are the same as our nonequivariant prespectra over B, except that their
adjoint structure maps "oeare required to be closed inclusions, which holds aut*
*omat
ically for cofibrant prespectra. Clapp and Puppe [19] use the term "cofibrant"
for our notion of cofibrant.
A crucial result of Clapp and Puppe makes the idea of homotopical glue preci*
*se.
It is stated nonequivariantly in [19, 6.1], but it works just as well equivaria*
*ntly.
Translated to our language, it reads as follows.
Proposition 13.2.5 (ClappPuppe). If f :X ! Y is a level fpequivalence
between cofibrant prespectra over B, then f is a homotopy equivalence of pre
spectra over B. Therefore, if f :X ! Y is a level hequivalence between well
structured cofibrant prespectra over B, then f is a homotopy equivalence of p*
*re
spectra over B.
Sketch proof. The proof is analogous to the proof that a ladder of homotopy
equivalences connecting sequences of cofibrations induces a homotopy equivalence
on passage to colimits. The point is that, for cofibrant parametrized prespec*
*tra
172 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS
Y , we can carry out inductive arguments just as if Y were just such a colimit.*
* Us
ing standard cofibration arguments, carried over to the parametrized case, we c*
*an
extend an fphomotopy inverse of WiBX(Vi) ! WiBY (Vi) to an fphomotopy
inverse of X(Vi+1) ! Y (Vi+1) and proceed inductively. The last statement fol
lows by Corollary 5.2.6(i), which shows that a level hequivalence between well
structured prespectra over B is a level fpequivalence.
13.3.The level exfibrant approximation functor P on prespectra
We seek an approximation functor to play the role on the parametrized pre
spectrum level that the functor P played on the exspace level functor. We shall
introduce three approximation functors, P , E and K, that successively build in*
* the
properties of being wellstructured, being an prespectrum, and being cofibr*
*ant,
each preserving the properties already obtained. We define P in this section and
E and K in the next.
Lifting the exspace level functor P of x8.3 to the prespectrum level requir*
*es
care. Recall that P is the composite of the whiskering functor W and the Moore
mapping path space functor L, together with the natural zigzag of hequivalenc*
*es
(13.3.1) K ooae_W K __W'_//W LK = P K
of Definition 8.3.4 for exspaces K over B. The functors W and L do not com
mute with tensors with based spaces, hence cannot be enriched over GKB , by
Lemma 11.3.4. There is therefore no canonical way of inducing structure maps
after applying P levelwise to a prespectrum, as one might at first hope. We sha*
*ll
resolve this by constructing by hand certain noncanonical but natural maps
(13.3.2) ffV :W K ^B SV ! W (K ^B SV )
and
(13.3.3) fiV :LK ^B SV ! L(K ^B SV )
such that ff0 = id, fi0 = id and the following associativity diagram commutes,
where (F, fV ) stands for either (W, ffV ) or (L, fiV ).
(13.3.4)
fV^id fV00 0
F K ^B SV ^B SV 0____//_F (K ^B SV ) ^B SV___//F (K ^B SV ^B SV )
~= ~=
fflffl0 fV V 0 fflffl 0
F K ^B SV V ________________________________//F (K ^B SV V )
The definitions of these maps and the proofs that these diagrams commute
depend on chosen decompositions of V and V 0as direct sums of indecomposable
representations, and we cannot choose compatible decompositions for all represe*
*n
tations V and V 0at once. For this reason, and for other reasons that will beco*
*me
apparent later, we must switch gears and work with sequentially indexed prespec*
*tra.
Thus, to be precise about the constructions in this section and the next, we
restrict our original collection V of indexing representations to a countable c*
*ofinal
sequence W of expanding representations in our given universe U. More precisely,
W consists of representations Vi for i 0 such that V0 = 0 and Vi Vi+1. We set
Wi= Vi+1Vi. Such a sequence can be chosen in any universe. We could just as we*
*ll
start with representations Wiand define Viinductively by Vi+1= Vi Wi. There is
13.3. THE LEVEL EXFIBRANT APPROXIMATION FUNCTOR P ON PRESPECTRA 173
no need to use orthogonal complements. We shall write in terms of complements,
but on the understanding that that is just a notational convenience.
Remark 13.3.5. There is a small quibble here since we originally defined our
categories of parametrized prespectra only on collections of representations th*
*at
are closed under finite direct sums, which W clearly is not. However, if we l*
*et
W 0consist of all finite sums of the Wi, then0we recover such a collection. As*
* in
x11.3 (or [62, x2]), we can interpret0GPWB as a diagram category indexed on a
certain small category, say DWG , with object set W 0, and we can0interpret GPWB
as a diagram category indexed on the full subcategory0DWG of DWG whose object
set is W . This gives a restriction functor U: GPWB ! GPWB that is right adjoi*
*nt
to a prolongation functor P [62, x3], and (P, U) induces an adjoint equivalence
of homotopy categories. We shall study such "change of universe" adjunctions in
x14.2. They allow us to lift all results we prove about the categories of param*
*etrized
prespectra indexed on cofinal sequences to our usual ones indexed on collection*
*s of
representations closed under direct sums.
Definition 13.3.6. Let X be a prespectrum over B indexed on the countable
cofinal sequence W = {Vi}, where V0 = 0 and Vi+1= Vi Wi. Let X have structure
maps oei: WiBX(Vi) ! X(Vi+1). Then the maps
W oeiO ff: W X(Vi) ^B SWi ! W X(Vi+1)
and
LoeiO fi :LX(Vi) ^B SWi ! LX(Vi+1)
specify structure maps for prespectra W X and LX over B. Therefore P X = W LX
is a prespectrum over B.
Unfortunately, as will be clear from the following construction, the maps in*
* the
zigzag (13.3.1) do not lift to the prespectrum level. They only induce weak ma*
*ps
of prespectra, that is, levelwise maps that only commute with the structure maps
up to (canonical) fphomotopy. Fortunately, the last approximation functor K,
which arranges cofibrancy and will be discussed in the next section, turns we*
*ak
maps into honest ones.
Construction 13.3.7. We define ffV and fiV . Fix a decomposition of V into
irreducible representations and let PV be the set of the projections from V to *
*the
irreducible subrepresentations in this fixed decomposition. Define three equiva*
*riant
maps from V to the real numbers by setting
Y Y
kvkV = maxss2Pssv, ~V (v) = (1  ssv), V (v) = max(1, ssv).
V ss2PV ss2PV
Applying the same definitions to another representation V 0and to V V 0with
its induced decomposition as a sum of irreducible representations, we see that *
*the
following equations hold.
kv v0kV V 0= max{kvkV , kv0kV 0},
~V V 0(v v0) = ~V (v)~V 0(v0),
V V 0(v v0) = V (v) V 0(v0).
Define a natural map
hV :W K ^B SV ^B [1, 1)+ ! W (K ^B SV ),
174 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS
by setting
( 1 1
hV (x ^ v ^ t) = x ^ ~(t v) . v if kvk t,
(p(x), 1  (t1v)1)if kvk t,
(
hV ((b, s) ^ v ^ t) = (b, s) if kvk t,
(b, 1  (1  s) (t1v)1)if kvk t.
At time t = 1 this specifies ffV and it is easy to verify that the associativi*
*ty
diagram (13.3.4) commutes. Further, the map ae O hV extends to t = 1 to give an
fphomotopy from aeOffV to ae^B id. It follows that ae induces levelwise a weak*
* map
of prespectra W X ! X.
Similarly define
kV :LK ^B SV ^B [1, 1)+ ! L(K ^B SV ),
by setting (
kV ((x, ~) ^ v ^ t) = (x ^ v, ~) if kvk t,
(x ^ v, (t1v)~if kvk t.
Here, if 1 a < 1, and ~ 2 B, then a~ denotes the Moore path of length l~=a
given by a~(t) = ~(at). At time t = 1 this specifies fiV , and it is again easy*
* to check
the required associativity. The map kV O (' ^ id) extends to an fphomotopy from
fiV O (' ^B id) to ', hence ' induces levelwise a weak map of prespectra X ! L*
*X,
to which we can apply W to obtain a weak map W X ! W LX = P X.
In view of Definition 8.3.4, naturality arguments from Definition 13.3.6 and
Construction 13.3.7 prove the following theorem.
Theorem 13.3.8. There are functors L, W , and P = W L on GPB that are
given levelwise by the functors L, W , and P on GKB . There are natural weak ma*
*ps
ae: W X ! X and ': X ! LX that are given levelwise by the exspace maps ae
and '. Therefore, there is a natural zigzag of weak maps OE = (ae, W ') as dis*
*played
in the diagram
ae W'
X oo___W X _____//W LX = P X.
These maps are level hequivalences, and P converts level hequivalences to lev*
*el
fpequivalences. If each X(V ) is compactly generated and of the homotopy type
of a GCW complex, then P X is wellstructured. If X is wellstructured, then t*
*he
weak maps in the above display are level fpequivalences between wellstructure*
*d G
prespectra over B. If, further, the adjoint structure maps of X are hequivalen*
*ces
or qequivalences, then so are the adjoint structure maps of LX, W X, and P X.
Proof. The only point that may need elaboration is the last clause. For a
weak map f :X ! Y , we have a homotopy commutative diagram
X(V )__"oe//_ WBX(V W )
f WBf
fflffl fflffl
Y (V )_"oe//_ WBY (V W ).
The functor WB preserves fpequivalences. Therefore, if f is an fpequivalenc*
*e,
then the "oefor X are hequivalences or qequivalences if and only if the "oefo*
*r Y are
so. We apply this to f = ae and f = W '.
13.4. THE AUXILIARY APPROXIMATION FUNCTORS K AND E 175
13.4. The auxiliary approximation functors K and E
We begin with the parametrized prespectrum approximation functor E. This
is a folklore construction when B is a point. In the parametrized context, the *
*proof
of the following result makes essential use of Stasheff's theorem, Theorem 3.4.*
*2,
and therefore depends on our standing assumption that G acts properly on B.
Proposition 13.4.1. There is a functor E :GPB  ! GPB and a natural
map ff: X ! EX with the following properties.
(i)The functor E preserves level fpequivalences and wellgrounded prespectra.
(ii)If X is wellstructured, then EX is a wellstructured prespectrum and the
map ff: X ! EX is an sequivalence.
Proof. Define EX by letting EX(Vi) be the telescope over j i of the ex
spaces VjViBX(Vj) with respect to the adjoint structure maps
VjViB"oe: VjViBX(Vj) ! VjViB WjBX(Vj+1) ~= Vj+1ViBX(Vj+1).
Since the functor WiBcommutes with telescopes, WiBEX(Vi+1) is isomorphic to t*
*he
telescope over j i+1 of the exspaces VjVi+1BX(Vj). The adjoint structure m*
*ap
EX(Vi) ! WiBEX(Vi+1) is induced by the maps VjViB"oejfor j i. The map
ff: X ! EX is given by the inclusion of the bases of the telescopes. If f :X *
*! Y
is a level fpequivalence, then Ef :EX ! EY is a level fpequivalence since a
standard inductive argument (applicable in any topologically bicomplete categor*
*y)
shows that the telescope of a ladder of fpequivalences is an fpequivalence.
If X is wellgrounded or level exfibrant, then so is EX since the construct*
*ion
clearly stays in the category of compactly generated spaces and since it preser*
*ves
the conditions of being wellsectioned or level exfibrant by results in x8.2. *
* To
show that E preserves wellstructured prespectra, it remains to show that if X *
*has
total spaces of the homotopy types of GCW complexes, then so does EX. By
Stasheff's theorem (Theorem 3.4.2), the fibers X(V )b = Xb(V ) have the homotopy
types of GbCW complexes. We have the analogous construction E in the category
of Gbprespectra and, by Milnor's theorem (Theorem 3.3.5) and standard facts
about telescopes, the (E(Xb))(V ) have the homotopy types of GbCW complexes.
It is clear from the definition of E that (E(Xb))(V ) = ((EX)(V ))b. That is, t*
*he
Gbprespectrum E(Xb) is the fiber (EX)b of the Gprespectrum EX over B. By
Stasheff's theorem again, it follows that the (EX)(V ) have the homotopy types *
*of
GCW complexes.
To check that the adjoint structure maps are qequivalences when X is well
structured, it suffices to check that they induce qequivalences on the fibers *
*over
b for all b 2 B. That holds by inspection of the homotopy groups of the colimits
that define (EX)b ~=E(Xb). Similarly, we see that ff is a ss*equivalence when X
is wellstructured by fiberwise comparison of the colimits of homotopy groups of
fibers that define the homotopy groups of X and EX.
To approximate parametrized prespectra by level fpequivalent cofibrant pr*
*e
spectra, we use the elementary cylinder construction K that was first defined in
[63] and has been used in various papers since. We recall the construction and *
*its
main properties from [59, 6.8], which carries over verbatim to the parametrized
context. A more sophisticated but less convenient treatment is given in [39].
176 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS
Proposition 13.4.2. There is a functor K :GPB  ! GPB and a natural
level fpequivalence ss :KX ! X. Therefore K preserves level fpequivalences.
If X is wellgrounded, then KX is cofibrant. If X is wellstructured, then KX*
* is
wellstructured. If X is a wellstructured prespectrum, then so is KX and thus
KX is excellent. There is a natural weak map ': X ! KX that is a right inverse
of ss, and K takes weak maps f to honest maps Kf such that ' O f = Kf O '.
Proof. Define KX, a level inclusion ': X ! KX, and a level fpdeformation
retraction ss :KX ! X right inverse to ' as follows. Let KX(0) = X(0) and
'(0) = ss(0) = id. Inductively, suppose given KX(Vi), an inclusion '(Vi): X(Vi)*
* !
KX(Vi) and an inverse fpdeformation retraction ss(Vi): KX(Vi) ! X(Vi). Let
KX(Vi+1) be the double mapping cylinder in GKB of the pair of maps
WiB'(Vi) oe
WiBKX(Vi) oo______ WiBX(Vi)________//X(Vi+1)
in GKB . Let oe : WiBKX(Vi) ! KX(Vi+1) be the inclusion of the left base of the
double mapping cylinder, which is an fpcofibration and let '(Vi+1): X(Vi+1) !
KX(Vi+1) be the inclusion of the right base. Let ss(Vi+1): KX(Vi+1) ! X(Vi+1)
be the map obtained by first using the fpequivalence WiBss(Vi) on the left ba*
*se to
map to the mapping cylinder of oe and then using the evident deformation retrac*
*tion
to the right base. There is an equivalent description as a finite telescope. Ce*
*rtainly
ss is a map of prespectra over B and a level fpdeformation retraction with lev*
*el
inverse the weak map '. The functoriality of the construction is clear.
If X is wellgrounded, then KX is clearly also wellgrounded and thus KX is
cofibrant. If X is wellstructured, then so is KX by Propositions 8.2.1 and 8*
*.2.3.
If, further, the adjoint structure maps of X are qequivalences, then they are *
*fp
equivalences since X is wellstructured. Since K preserves fphomotopies, it fo*
*llows
that KX is also an prespectrum. Alternatively, since VBis a Quillen right ad*
*joint
in the qfmodel structure, it preserves qequivalences between qffibrant exsp*
*aces.
In particular, the maps WBss(Vi) are qequivalences.
If f :X ! Y is a weak map with fphomotopies
hi: WiBX(Vi) ^B I+ ! Y (Vi+1)
from oeY O Wif(Vi) to f(Vi+1) O oeX , define Kf inductively by setting Kf(0) =
f(0) and letting Kf(Vi+1) be WjBKf(Vi) on the left end of the mapping cylinder,
f(Vi+1) on the right end and as follows on the cylinder itself:
( Wi 1
, _
Kf(Vi+1)[x, t] = [ B f(Vi)(x), 2t]if10 t 2
hi(x, 2t  1) if _2 t 1.
Then Kf is a map of prespectra over B and ' O f = Kf O '.
The composite approximation functor T = KEP has various good preservation
properties. The exspace level properties of P recorded in x8.4 are inherited o*
*n the
prespectrum level, and we have the following sample result for E and K.
Lemma 13.4.3. For a Gmap f :A ! B, a prespectrum Y over B and a
prespectrum X over A, there are natural isomorphisms
f*EY ~=Ef*Y, f*KY ~=Kf*Y and Kf!X ~=f!KX.
13.5. THE EQUIVALENCE BETWEEN HoGPB AND hGEB 177
Proof. The relevant telescopes commute with f* since it is a symmetric
monoidal left adjoint and with f!since it is a left adjoint and the projection *
*formula
(2.2.5) holds.
13.5. The equivalence between Ho GPB and hGEB
We can now extend the results of x9.1 to parametrized prespectra. As in the
previous section, our parametrized prespectra are indexed on a countable cofinal
sequence of expanding representations in our given universe. We begin by collat*
*ing
the results of the previous two sections.
Theorem 13.5.1. Let X be a wellgrounded Gprespectrum over B whose total
spaces are of the homotopy types of GCW complexes and define T X = KEP X.
(i)T X is an excellent Gprespectrum.
(ii)T takes level qequivalences between Gprespectra over B that satisfy the *
*hy
potheses on X to homotopy equivalences of Gprespectra.
(iii)There is a zigzag of sequivalences between X and T X.
(iv)If X is an excellent Gprespectrum over B, then the zigzag consists of lev*
*el
fpequivalences, and it gives rise to a zigzag of homotopy equivalences of
Gprespectra over B connecting X and T X.
Proof. We have that P X is wellstructured by Theorem 13.3.8, EP X is a
wellstructured prespectrum by Proposition 13.4.1, and T X is excellent by Pr*
*opo
sition 13.4.2. In (ii), a level qequivalence is a level hequivalence. By the *
*results
just quoted, P takes level hequivalences to level fpequivalences, which are p*
*re
served by E, and K takes level fpequivalences to homotopy equivalences. Since
K converts weak maps to genuine maps, we have the following diagram of maps of
Gpresepectra over B.
(13.5.2) KX ooKaeKW_X W KX _WK'_//W KLX KEP X
ss ss Wss  Wss ss
fflffl fflffl fflffl fflffl fflffl
X W X ________W X W LX __ff__//_EP X
The vertical maps ss, hence also the vertical maps W ss, are level fpequivalen*
*ces.
The map ae is a level fequivalence. The map ' is a level hequivalence, hence *
*so is
W K'. The map ff is an sequivalence because P X is wellstructured. Since leve*
*l q
equivalences are also sequivalences, the diagram displays a zigzag of sequiv*
*alences
between X and T X.
For the last statement, observe that all prespectra in the diagram are well
structured prespectra over B. Moreover, ff is a level qequivalence by Theo
rem 12.3.8. It is therefore a level hequivalence since our total spaces have *
*the
homotopy types of GCW complexes. Since all prespectra in our diagram are well
structured, our level hequivalences are level fpequivalences, by Proposition *
*8.3.2.
Applying K where needed, we can expand the diagram to a zigzag of level fp
equivalences between cofibrant prespectra. By Proposition 13.2.5, this gives*
* a
zigzag of homotopy equivalences connecting X and T X.
We introduce a category that is intermediate between GPB and GEB .
Definition 13.5.3. Define GQB to be the full subcategory of GPB consisting
of the wellgrounded prespectra over B whose total spaces are of the homotopy
178 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS
types of GCW complexes. Define HoGQB to be the homotopy category obtained
by inverting the sequivalences in GQB ; by the proof of the next theorem, ther*
*e are
no settheoretic problems in defining HoGQB . Define T = KEP :GQB ! GEB .
Since the prespectra over B are the sfibrant prespectra over B and since
scofibrant spectra are wellgrounded and have total spaces of the homotopy typ*
*es
of GCW complexes, all Gprespectra over B that are scofibrant and sfibrant a*
*re
in GQB . We prove that Ho GPB is equivalent to hGEB by proving that these
categories are both equivalent to HoGQB .
Theorem 13.5.4. The canonical scofibrant and sfibrant approximation func
tor RQ and the composite approximation functor T = KEP , together with the
forgetful functors, induce the following equivalences of homotopy categories.
_RQ__// __T__//
Ho GPB oo___Ho GQB oo___hGEB
I J
Proof. For X in GPB , we have a natural zigzag of sequivalences in GPB
X oo___QX _____//RQX.
Therefore X and IRQX are naturally sequivalent in GPB . If X is in GQB , then
it is sfibrant and therefore so is QX. Then the above zigzag is in GQB so X a*
*nd
RQIX are naturally sequivalent in GQB .
By Theorem 12.3.8, sequivalences in GQB are level qequivalences, and T tak*
*es
level qequivalences to homotopy equivalences by Theorem 13.5.1. Conversely, si*
*nce
homotopy equivalences are sequivalences, the forgetful functor J induces a fun*
*ctor
in the other direction.
For X in GQB we have the natural zigzag of sequivalences displayed in
(13.5.2). Applying sfibrant approximation, we get a natural zigzag of sequiv*
*alences
in GQB so X and JT X are naturally sequivalent in GQB . Starting with X in
GEB , the last statement of Theorem 13.5.1 shows that X and T JX are naturally
homotopy equivalent in GEB .
13.6.Derived functors on homotopy categories
With P replaced by T , the discussion of derived functors in x9.2 carries ov*
*er
from the level of exspaces to the level of parametrized prespectra indexed on *
*cofinal
sequences. In x13.7 and x14.2 we will discuss how to pass from there to conclus*
*ions
on the level of parametrized spectra indexed on our usual collections of repres*
*en
tations closed under direct sums. We must show that if V is a Quillen left or
right adjoint, then its model theoretic left or right derived functor agrees un*
*der our
equivalences of categories with the functor obtained simply by passing to homot*
*opy
classes of maps from the composite T V . As on the exspace level, we need some
mild good behavior for this to work.
Definition 13.6.1. A functor V :GPA  ! GPB is good if it is continu
ous, preserves wellgrounded parametrized prespectra, and takes prespectra over
A whose levelwise total spaces are of the homotopy types of GCW complexes to
prespectra over B with that property. Since V is continuous, it preserves homo
topies. There are evident variants for functors V with source or target GK*: V
must be continuous, preserve wellgrounded objects, and preserve GCW homotopy
type conditions on objects.
13.6. DERIVED FUNCTORS ON HOMOTOPY CATEGORIES 179
Note that a good functor V need not take Gprespectra to Gprespectra
and recall that a Quillen right adjoint must preserve fibrant objects and thus,*
* in
our context, must preserve Gprespectra.
Proposition 13.6.2. Let V :GPA ! GPB be a good functor that is a part
of a Quillen adjoint pair. If V is a Quillen left adjoint, assume further that *
*it pre
serves level qequivalences between wellgrounded objects. Then the derived fu*
*nc
tor Ho GPA  ! Ho GPB , induced by V Q or V R, is equivalent to the functor
T V J :hGEA ! hGEB under the equivalence of categories in Theorem 13.5.4
Proof. If V is a Quillen right adjoint, then it preserves sequivalences be*
*tween
sfibrant objects. If V is a Quillen left adjoint, then it preserves sequivale*
*nces be
tween wellgrounded objects by Proposition 12.6.3. Therefore, since GQA consists
of wellsectioned sfibrant objects, the functor V :GQA ! GPB passes straight
to homotopy categories to give V :HoGQA ! HoGPB in both cases.
If V is a Quillen right adjoint, then it takes an sequivalence f in GQA to
an sequivalence since the objects of GQA are sfibrant. Then V f is a level q
equivalence by Theorem 12.3.8 and, since V is good, it is a level hequivalence*
*. On
the other hand, if V is a Quillen left adjoint, then Theorem 12.3.8 gives that *
*f is
a level qequivalence and, by assumption, V f is then a level qequivalence. Si*
*nce
V is good, V f is actually a level hequivalence. In both cases it follows tha*
*t V
takes sequivalences to level hequivalences and therefore T V passes to a func*
*tor
Ho GQA ! hGEB .
To show that T V J and either V Q or V R agree under the equivalence of cate
gories, it suffices to verify that the following diagram commutes.
V Q orV R
Ho GPA ___________//HoGPB
RQ  TRQ
fflffl fflffl
Ho GQA ____TV______//hGEB
We have functorial scofibrant and sfibrant approximation functors Q and R,
with natural acyclic sfibrations QX ! X and acyclic scofibrations X ! RX.
Clearly Q and R preserve sequivalences. If V is a Quillen left adjoint, then *
*we
have a zigzag of natural sequivalences
RQV Q _____//RV Qoo___V Q ____//_V RQ
because V preserves acyclic scofibrations. If V is a Quillen right adjoint, th*
*en we
have a zigzag of natural sequivalences
RQV R oo___ RQV RQ ____//_RV RQoo___V RQ
because V preserves sequivalences between sfibrant objects. In both cases, *
*all
objects have total spaces of the homotopy types of GCW complexes, hence we
have zigzags of level hequivalences. Applying T , we obtain a zigzag of homo*
*topy
equivalences in GEB by Theorem 13.5.1.
Remark 13.6.3. If V preserves excellent parametrized prespectra, then T V
is naturally homotopy equivalent to V on excellent parametrized prespectra. The
derived functor of V can then be obtained directly by applying V and passing to
homotopy classes of maps.
180 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS
13.7.Compatibility relations for smash products and base change
This section is parallel to x9.3. The main change is just that we must repla*
*ce
the functor P used there with the functor T = KEP that we have here. This
gives us results for the categories GPWB of parametrized prespectra indexed on
a collection W consisting of a cofinal sequence in some universe U. In order *
*to
obtain statements about GSBV, where V = V (U), we have two pairs of Quillen
equivalences, both of which can be viewed as consisting of a prolongation funct*
*or
left adjoint to a forgetful functor that creates the weak equivalences; see [61*
*, 1.2].
__j*_// __P__//
GPWB oo___GPVB oo___GSBV
j* U
We postpone until x14.2 consideration of the pair (j*, j*) and the extension fr*
*om
GPWB to GPVBand deal with the extension from GPVBto GSBV in this section.
One general remark is in order, though. The forgetful functors j* and U crea*
*te
weak equivalences and therefore pass directly to homotopy categories. If they c*
*om
mute on the point set level with a functor V which is a part of a Quillen adjoi*
*nt
pair, then they will also commute with its derived functor on the level of homo*
*topy
categories. It follows formally that the derived prolongation functors P and j**
* then
also commute with the derived functor V and its adjoints. This holds in particu*
*lar
for the base change functor V = f*. Extending commutation results for such func
tors from GPWB to GSBV is therefore easy. However, some of the functors V that
we need to consider only exist on some of the categories in the above display, *
*and
such functors require special care. These include the change of universe functo*
*rs
that we discuss in x14.2, which don't exist on the level of GPWB, and the smash
product ^B , which we have only specified on the spectrum level and which we now
discuss on the prespectrum level.
Remark 13.7.1. Because the domain category for the diagram category of
(equivariant and parametrized) prespectra is only monoidal, not symmetric mon
oidal, we cannot use left Kan extension to internalize "external" smash product*
*s of
prespectra; see [62, 4.1]. Here "external" is understood in the sense of indexi*
*ng on
pairs of representations. Therefore, on the equivariant parametrized prespectrum
level, when we write X Z Y for prespectra X over A and Y over B, we should
understand the external external smash product, in the sense of Remark 11.1.7.
When passing from prespectrum level arguments to spectrum level conclusions usi*
*ng
(P, U), we are implicitly using composites of the general form PV U, and simila*
*rly
for functors of several variables involving smash products. We can carry out the
several variable arguments externally on the prespectrum level, only internaliz*
*ing
with left Kan extension after passage to spectra, where we have good homotopical
control by Corollary 12.6.6.
Alternatively, we can make use of classical "handicrafted smash products" of
prespectra, which are defined by use of arbitrary choices of sequences of repre*
*senta
tions. As discussed on the nonequivariant nonparametrized level in [62, x11], h*
*and
icrafted smash products of prespectra agree under the adjoint equivalence (P, U)
with the internalized smash products. Provided that we use external parametrized
handicrafted smash products over varying base spaces, only internalizing along
diagonal maps at the end, the discussion there adapts readily to give the same *
*con
clusion for homotopy categories of equivariant parametrized prespectra and spec*
*tra.
13.7. COMPATIBILITY RELATIONS FOR SMASH PRODUCTS AND BASE CHANGE 181
The advantage of handicrafted smash products is that their definition involves *
*only
direct use of exspace level constructions that enjoy good preservation propert*
*ies
with respect to exfibrations. This often allows direct transposition of exspa*
*ce level
arguments in hGWB to parametrized prespectrum level arguments in hGEB .
We state the following results in terms of parametrized spectra, and we indi*
*cate
which parts of the proofs require the use of hGEB and which parts work directly*
* in
the stable homotopy category HoGSB .
Proposition 13.7.2. Let f :A ! B and g :A0 ! B0 be Gmaps. If W and
X are spectra over A and A0, then
f!W Z g!X ' (f x g)!(W Z X)
in Ho GSBxB0 . If Y and Z are spectra over B and B0, then
f*Y Z g*Z ' (f x g)*(Y Z Z)
in Ho GSAxA0.
Proof. Working directly in Ho GSBxB0 , the first equivalence reduces to its
pointset level analogue by consideration of Quillen left adjoints, as in the c*
*orre
sponding proof of Proposition 9.4.1. We work in hGEAxA0 to prove the second
equivalence. Here f* and Z (understood in the external or handicrafted sense)
are both good, and both preserve excellent prespectra. Indeed, they preserve we*
*ll
structured prespectra by levelwise application of Propositions 8.2.2 and 8.2.3,*
* they
preserve cofibrant prespectra since f* and Z on exspaces preserve fpcofibra*
*tions
because they are left adjoints that commute with fphomotopies, and they preser*
*ve
prespectra by Lemma 13.2.3 since they preserve fphomotopies. Therefore, using
excellent prespectra, we can pass straight to homotopy categories, without use *
*of
T , as in the corresponding proof of Proposition 9.4.1.
Theorem 13.7.3. The category Ho GSB is closed symmetric monoidal under
the functors ^B and FB .
Proof. Working in HoGSB , the associativity, commutativity, and unity of ^B
follow by pullback along diagonal maps from their easily proven external analog*
*ues
and the second equivalence in the previous result, exactly as in Theorem 9.4.4.
We have a commutation relation between change of base and suspension spec
trum functors that is analogous to the relation between change of base and smash
products recorded in Proposition 13.7.2.
Proposition 13.7.4. For a Gmap f :A ! B, there are natural equivalences
1Bf!' f! 1A and 1Af* ' f* 1B
of (derived) functors. The same conclusion holds more generally for the shift d*
*esus
pension functors FV = 1V.
Proof. Working in Ho GSB , the first equivalence is clear since it is a com
parison of Quillen left adjoints that commute on the pointset level. For the s*
*econd
equivalence, we start in hGWB and end in hGEA . For K 2 GWB , the point set lev*
*el
suspension prespectrum 1BK is both cofibrant and wellstructured, by Corol
lary 8.2.5, but of course it is not an prespectrum over B. Since 1B is good *
*and
takes wellgrounded qequivalences to wellgrounded level qequivalences, T 1B*
* is
182 13. ADJUNCTIONS AND COMPATIBILITY RELATIONS
equivalent to the model theoretic left derived functor of the Quillen left adjo*
*int 1B.
Here we may omit P from the composite functor T and, since f* commutes with
both K and E, the conclusion follows on passage to homotopy categories.
Applying this to : B ! B x B and using Proposition 11.4.11, we obtain the
following consequence.
Proposition 13.7.5. For exspaces K and L over B,
1B(K ^B L) ' 1BK ^B 1BL
in Ho GSB .
For f :A ! B, evident properties of the functor f! on exspaces imply that
the functor f!:GPA ! GPB is good, and f! satisfies the other hypotheses of
Proposition 13.6.2 by Proposition 12.2.5. We use this to prove the following ba*
*sic
result.
Theorem 13.7.6. For a Gmap f :A ! B between base spaces, the derived
functor f* :HoGSB ! Ho GSA is closed symmetric monoidal.
Proof. Since SB is not sfibrant, the isomorphism f*SB ~=SA in GSB does
not immediately imply the required equivalence f*SB ' SA in Ho GSA , where
f*SB means f*RSB . However, Proposition 13.7.4 specializes to give this equiva
lence. For the rest, we must show that the isomorphisms (11.4.2) through (11.4.*
*6)
descend to equivalences on homotopy categories. By category theory in [40], it
suffices to consider (11.4.2) and (11.4.5), and the proofs are similar to those*
* in
Theorem 9.4.5. Since Z and * both preserve excellent prespectra, so do the int*
*er
nalized smash products ^A and ^B . For excellent prespectra Y and Z over B, it
follows that both sides of
f*(Y ^B Z) ~=f*Y ^A f*Z
are excellent prespectra over A, hence the pointset level isomorphism descends
directly to the desired equivalence on the homotopy category level. Next consid*
*er
f!(f*Y ^A X) ~=Y ^B f!X,
where X is an excellent prespectrum over A. Here we must replace f! by T f! on
both sides. By Theorem 13.5.1 we have a natural zigzag OE of level hequivalen*
*ces
connecting T to the identity functor which, when applied to excellent parametri*
*zed
prespecra gives rise to a zigzag _ of actual homotopy equivalences. We obtain *
*the
following zigzag.
T(id^BOE) _
T f!(f*Y ^A X) ~=T (Y ^B f!X)_________//_To(Yo^B_T f!X)_//_Yo^BoT_f!X.
Using handicrafted products with their termwise construction in terms of smash
products of exspaces, it follows from Proposition 8.2.6 that id^B  preserves *
*level
hequivalences between wellsectioned spectra. Thus id^B OE is a zigzag of le*
*vel
hequivalences and T (id^B OE) is a zigzag of actual homotopy equivalences.
13.7. COMPATIBILITY RELATIONS FOR SMASH PRODUCTS AND BASE CHANGE 183
Theorem 13.7.7. Suppose given a pullback diagram of Gspaces
g
C _____//D
i j
fflfflfflffl
A __f__//B
in which f (or j) is a qfibration. Then there are natural equivalences of (der*
*ived)
functors on stable homotopy categories
(13.7.8) j*f!' g!i*, f*j* ' i*g*, f*j!' i!g*, j*f* ' g*i*.
Proof. Working in hGEB , the proof is similar to that of Theorem 9.4.6 but
with P replaced by T . Again it suffices to consider the first equivalence, and*
*, as
explained there, since f is a qfibration there is a level fpequivalence ~: P *
*f* !
f*P . Since f* commutes with both K and E, we obtain a level fpequivalence
~: T f* ! f*T between cofibrant prespectra over A so it is in fact a homotopy
equivalence by Proposition 13.2.5. Then f*T j!X ' T f*j!X ~=T i!g*X.
The following observation holds by the same proof as the analogous exspace
level result Proposition 9.4.8.
Proposition 13.7.9. Let ': H ! G be the inclusion of a subgroup and A be
an Hspace. The closed symmetric monoidal Quillen equivalence ('!, *'*) descen*
*ds
to a closed symmetric monoidal equivalence between HoHSA and HoGS'!A.
Combined with Theorem 13.7.6, applied to the inclusion "b:G=Gb ! B, and
Proposition 13.7.4, this last observation gives us the following stable analogu*
*e of
Theorem 9.4.9.
Theorem 13.7.10. The derived fiber functor ()b:Ho GKB ! Ho GbK* is
closed symmetric monoidal and it has both a left adjoint ()b and a right adjoi*
*nt
b(). Moreover, the derived fiber functor commutes with the derived suspension
spectrum functor, ( 1BK)b ' 1 (Kb) as Gbspectra.
For emphasis, we repeat a remark that we made after the analogous exspace
level result. This innocent looking result packages highly nontrivial and impo*
*rtant
information. In particular, it gives that FB (X, Y )b ' F (Xb, Yb) in Ho GbS f*
*or
X, Y 2 Ho GSB , where the fiber and function object functors are understood in
the derived sense. This reassuring consistency result is central to our applica*
*tions
in the last two chapters, where parametrized duality is studied fiberwise.
CHAPTER 14
Module categories, change of universe, and change
of groups
Introduction
We first give a discussion of module categories of parametrized spectra over
nonparametrized ring spectra. Although we shall not go into these applications
here, one basic motivation for our work is to set up the homotopical foundations
for studying the generalized homology and cohomology theories of parametrized
spectra that are represented by such nonparametrized ring spectra. The good
behavior of the external smash product GS x GSB ! GSB makes it easy to do
this. While the mathematics here is evident, it deserves emphasis since the ide*
*as
are likely to be central to future applications.
In the rest of the chapter, we focus on problems that are special to the equ*
*ivari
ant context. We give the parametrized generalization of some of the work in [61]
concerning change of universe, change of groups, and fixed point and orbit spec*
*tra.
As usual, an essential point is to determine which of the standard adjunctions *
*are
given by Quillen adjoint pairs and to prove that other adjunctions and compatib*
*il
ities that are evident on the point set level also descend to homotopy categori*
*es.
We discuss change of universe in x14.2. Here the use of prespectra indexed on
cofinal sequences in the previous chapter introduces some minor difficulties th*
*at
were not studied in the nonparametrized theory of [61, Vx1] and are already rel*
*e
vant nonequivariantly. We study subgroups and fixed point spectra in x14.3. We
study quotient groups and orbit spectra in x14.4. Aside from some analogues for
parametrized spectra of earlier results for parametrized spaces, these sections*
* are
precisely parallel to [61, Vxx2 and 3]. We have not written down the parametriz*
*ed
analogue of [61, Vx4], which gives the theory of geometric fixed point spectra,*
* since
it would be tedious to repeat the constructions given there. It will be apparen*
*t to
the interested reader that, mutatis mutandis, the definitions and results in [6*
*1,
Vx4] generalize to the parametrized context.
14.1. Parametrized module Gspectra
We can define a parametrized (strict) ring Gspectrum R over B to be a monoid
in the symmetric monoidal category GSB , and we can then define parametrized
Rmodules and Ralgebras in the usual way, as has become standard in stable
homotopy theory [39, 46, 61, 62]. However, even though the smash product ^B
in GSB gives a pointset level symmetric monoidal structure, we cannot expect to
obtain Quillen model structures on the categories of such Rmodules or Ralgebr*
*as,
as was done for orthogonal Gspectra in [61, IIIxx7,8]. To do that, we would ne*
*ed
better homotopical behavior than we can prove here. We have only set up adequate
foundations for the classical style theory of up to homotopy parametrized module
184
14.1. PARAMETRIZED MODULE GSPECTRA 185
spectra over up to homotopy parametrized ring spectra. From that point of view,
our homotopical foundations are entirely satisfactory. The source of the proble*
*m is
Warning 6.1.7, which implies that X ^B () in GSB cannot be a Quillen functor.
However, in applications, it is natural to start with a nonparametrized orth*
*og
onal ring Gspectrum R. We are then interested in understanding the Rhomology
and Rcohomology theories of Gspectra over B and their relationships with the
Rhomology and Rcohomology of the fibers. For this study, just as in the non
parametrized work of [39, 46, 61, 62], one is interested in the theory of Rmod*
*ules.
The external smash product Z: GS x GSB  ! GSB has enough of the good
properties of the nonparametrized smash product GS x GS ! GS to give us
homotopical control over parametrized module spectra over nonparametrized ring
spectra. We devote this section to developing the relevant theory, which is par*
*allel
to [61, IIIx7]. Let R be a ring spectrum in GS which is wellgrounded when view*
*ed
as a spectrum, meaning that each R(V ) is wellbased and compactly generated.
Definition 14.1.1. A (left) Rmodule over B is a Gspectrum M over B to
gether with a left action R Z M ! M satisfying the usual associativity and unit
conditions. The category GRMB of left Rmodules over B consists of the Gspectra
M over B and the maps of Gspectra over B that preserve the action by R.
Since (R Z X)b = R ^ Xb, a parametrized Rmodule over B is precisely that:
each Xb is an Rmodule Gbspectrum. More formally, we have the Gcategory
(RMG,B, GRMB ), as discussed in xx1.4 and 12.2, and the following result is cle*
*ar.
Proposition 14.1.2. The Gcategory (RMG,B, GRMB ) is Gtopologically bi
complete in the sense of Definition 10.2.1. All of the required limits, colimi*
*ts,
tensors, and cotensors are constructed in the underlying Gcategory (SG,B, GSB )
and then given induced Rmodule structures in the evident way. A cylcofibration
of Rmodules is a cylcofibration of underlying Gspectra over B.
The last statement holds by the retract of mapping cylinders characterizatio*
*n of
cylcofibrations. This immediately implies that GRMB inherits a ground structure
from GSB , in the sense of Definition 5.3.2. Recall that the wellgrounded Gsp*
*ectra
over B are those that are level wellgrounded (wellsectioned and compactly gen*
*er
ated) and that the gcofibrations of Gspectra over B are the level hcofibrati*
*ons;
see Definition 12.1.2 and Proposition 12.1.4.
Definition 14.1.3. An Rmodule over B is wellgrounded if its underlying G
spectrum over B is wellgrounded. A map of Rmodules over B is a gcofibration,
level qequivalence, or sequivalence if its underlying map of Gspectra over B*
* is
such a map.
Also recall the notion of a subcategory of wellgrounded weak equivalences
from Definition 5.4.1. Since colimits and tensors for Rmodules are defined in
terms of the underlying Gspectra over B, the following theorem is immediate fr*
*om
its counterpart for Gspectra over B, which is given by Proposition 12.1.4 and
Theorem 12.4.3.
Theorem 14.1.4. Definition 14.1.3 specifies a ground structure on GRMB
such that the level qequivalences and the sequivalences both give subcategori*
*es of
wellgrounded weak equivalences.
186 14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS
Finally, recall the definition of a wellgrounded model structure from Defin*
*i
tion 5.5.2. Such model structures are compactly generated, and we must define t*
*he
generators of GRMB . The free Rmodule functor FR = R Z : GSB ! GRMB
is left adjoint to the forgetful functor U: GRMB ! GSB . Adjunction_arguments
from the definitions show that FR preserves cylcofibrations and cylcofibratio*
*ns.
Definition 14.1.5. Define FR F IfB, FR F JfBand FR F KfBby applying the free
Rmodule functor to the maps in the sets specified in Definition 12.1.6 and Def*
*ini
tion 12.5.5. A map of Rmodules over B is
(i)a level qffibration or an sfibration if it is one in GSB ,
(ii)an scofibration if it satisfies the LLP with respect to the level acyclic*
* qf
fibrations,
Theorem 14.1.6. The category GRMB is a wellgrounded model category with
respect to the level qequivalences, the level qffibrations, and the scofibra*
*tions.
The sets FR F IfBand FR F JfBgive the generating scofibrations and generating *
*level
acyclic scofibrations. All scofibrations of Rmodules over B are scofibratio*
*ns of
Gspectra over B.
We omit the proof since it is virtually the same as the proof of the followi*
*ng
theorem, which gives the starting point for serious work on the homology and
cohomology theory of parametrized Gspectra.
Theorem 14.1.7. The category GRMB is a wellgrounded model category with
respect to the sequivalences, the sfibrations, and the scofibrations; FR F K*
*fBgives
the generating acyclic scofibrations.
Proof. The compatibility condition is automatic by adjunction from the para
metrized spectrum level,_and we have already observed that the free Rmodule
functor FR preserves cylcofibrations. It also preserves the relevant produc*
*ts,
and FR FV K = (R ^ FV S0) Z K is wellgrounded if K is a wellgrounded exspace.
Only the acyclicity condition remains. If R is scofibrant as a ring spectrum, *
*then
R is also scofibrant as a spectrum, by [61, III.7.6(iv) and (v)]. In that case*
*, the
functor R Z () = UFR is a Quillen left adjoint by Corollary 12.6.6 and therefo*
*re
preserves level acyclic scofibrations. It follows that the maps in FR KfBare *
*s
equivalences. The case of a general wellgrounded R reduces to the cofibrant ca*
*se
by use of the next result; compare Proposition 14.1.9 below.
Proposition 14.1.8. The following statements hold.
(i)For an scofibrant spectrum X over B, the functor  Z X :GS  ! GSB
preserves sequivalences between wellgrounded spectra in GS .
(ii)If Y is wellgrounded in GS , j :A ! X is an acyclic scofibration in GSB*
* ,
and A is wellgrounded, then Y Z j :Y Z A ! Y Z X is an sequivalence.
Proof. Let OE: Y  ! Z be an sequivalence between wellgrounded spectra.
By parts (ii)(iv) of Definition 5.4.1, it suffices to show that OE Z FV K is a*
*n s
equivalence if K is the source or target of a map in IfB. This map is isomorphi*
*c to
the map (OE ^ FV S0) ^B K, where FV S0 is the shift desuspension in GS , not GS*
*B .
Here OE ^ FV S0 is an sequivalence by the nonparametrized analogue [61, III.7.*
*3],
and the conclusion follows from Lemma 12.5.3. (The comment on the notations Z
and ^B above Definition 12.5.1 is relevant: the former is an external smash pro*
*duct
and the latter is a tensor).
14.1. PARAMETRIZED MODULE GSPECTRA 187
For (ii), we apply an argument from [62, 12.5]. We let Z = X=BA, which is
scofibrant, and we let QY  ! Y be an scofibrant approximation. Since j is an
scofibration, it is a cylcofibration and Cj is homotopy equivalent to Z. Sin*
*ce
A is wellgrounded, we can apply the long exact sequence of homotopy groups of
Theorem 12.4.2 to conclude that Z is sacyclic. The map Z ! *B is then an
sequivalence between scofibrant spectra over B. Since QY Z  is a Quillen left
adjoint, by Proposition 12.2.3, QY ZZ ! QY Z*B ~=*B is an sequivalence. Since
QY Z Z ! Y Z Z is an sequivalence by part (i), Y Z Z is sacyclic. Since the
functor Y Z  preserves cofiber sequences, another application of Theorem 12.4.2
shows that Y Z j is an sequivalence.
Proposition 14.1.9. If OE: Q ! R is an sequivalence of wellgrounded ring
spectra, then the functors
OE* = R ^Q (): GQMB ! GRMB and OE*: GRMB ! GQMB
given by extension of scalars and restriction of action define a Quillen equiva*
*lence
(OE*, OE*) between the categories of Qmodules and of Rmodules over B.
Proof. Since sfibrations and sequivalences are created in the underlying
category of spectra over B, it is clear that they are preserved by OE*, so that*
* we
have a Quillen pair. If M is an scofibrant Qmodule, then, by the previous res*
*ult,
the unit map OE ^ id:M ~= Q ^Q M ! OE*(R ^Q M) of the adjunction is an s
equivalence of spectra over B. Therefore, if N is an sfibrant Rmodule, then a
map M ! OE*N of Qmodules is an sequivalence if and only if its adjoint map
R ^Q M ! N of Rmodules is an sequivalence.
Implicitly, we have been dealing all along with the case when R is the sphere
spectrum S, and we can mimic all of the model theoretic work that we have done
in that case. The results of x12.6 and x13.1 carry over directly. For f :A ! B,
base change preserves Rmodules, (f!, f*) gives a Quillen adjoint pair relating*
* the
categories of Rmodules over A and over B, and we obtain a Quillen equivalence *
*if f
is a qequivalence. If f is a bundle with CW fibres, we obtain a Quillen pair (*
*f*, f*),
and we can apply the triangulated category version of Brown representability to
construct a right adjoint f* in general. However, we do not know how to general*
*ize
the rest of Chapter 13 to the module context since we have not worked out a the*
*ory
of excellent Rmodules with an accompanying excellent Rmodule approximation
functor. In view of the retreat to prespectra with their primitive handicrafted*
* smash
products in that theory, it seems unlikely to us that any such construction can*
* be
expected.
We also have the notion of a right Rmodule over a nonparametrized ring
spectrum R. If M and N are right and left Rmodules over A and B and L is a left
Rmodule over A x B, then we define spectra M ZR N over A x B and ~FR(N, L)
over A by the usual coequalizer
M Z R Z N ____//_//_M_Z_N_//M ZR N
and equalizer
~FR(N, L)____//~F(N, L)___////_~F(R Z N, L).
If R is commmutative, then M ^R N and FR (N, L) are naturally Rmodules.
We have good homotopical control over these external constructions, as in
Propositions 12.2.3 and 12.6.5. For example, if we take A = *, then we have good
188 14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS
homotopical control over the smash product spectrum M ^R N over B and the
nonparametrized function spectrum FR (N, L), where M is a nonparametrized
right Rmodule and N and L are left Rmodules over B. However, if we take
A = B and internalize M ZR N along the diagonal : B ! B x B by setting
M ^R N = *M ZR N and FR (M, N) = ~FR(M, *N), we lose homotopical control.
Similarly, when R is commutative, RMB has the structure of a closed sym
metric monoidal category, and that allows us to define (commutative) Ralgebras
over B to be (commutative) monoids in RMB . However, because of the lack of
homotopical control, in the absence of the theory of Chapter 13, we cannot give*
* the
categories of Ralgebras and of commutative Ralgebras over B model structures.
Remark 14.1.10. Although we have not pursued the idea, it seems highly likely
that there are interesting examples of rings and modules that allow varying base
spaces and are defined in terms of the external smash product. For example, one
might consider Gspectra Rn over Bn with products Rm Z Rn ! Rm+n , or one
might consider "globally defined" parametrized ring spectra R consisting of spe*
*ctra
RB over B for all B together with appropriate products RA Z RB ! RAxB . The
RB would in particular be module spectra over the nonparametrized ring spectrum
R*. As in the nonparametrized theory, one must use the positive stable model
structures to study such ring objects model theoretically when R* is commutativ*
*e.
The essential point is that the external smash product is sufficiently wellbeh*
*aved
homotopically that there is no obstacle to passage from pointset level constru*
*ctions
to homotopy category level conclusions.
14.2. Change of universe
Recall that Gspectra over B are defined in terms of a chosen collection V of
representations of G. As usual in equivariant stable homotopy theory, we must
introduce functors that allow us to change the collection V . Such functors are
usually referred to as "change of universe" functors, since V is often given as*
* the
collection V (U) of all representations that embed up to isomorphism in a given
Guniverse U. It is however often convenient to restrict V to be a cofinal subc*
*ol
lection of V (U) that is closed under direct sums, and when we dealt with excel*
*lent
prespectra it became essential to restrict V further to a countable cofinal seq*
*uence
of expanding representations in U. In both cases it is usual to insist that the*
* trivial
representation R is included in V . In order to deal with the change functors *
*in
all of the above cases at once, we adopt a slightly different approach from the*
* one
that was used in [61, V.x1]. We then explain how it specializes to the more exp*
*licit
approach given there.
Let GSBV denote the category of Gspectra over B indexed on V . If V is
not closed under direct sums, then we are thinking0of GSBV as the restriction of
the diagram category corresponding to GSBV, where V 0is the closure of V under
sums, as discussed in Remark 13.3.5.
Let i: V V 0be the inclusion of one collection of representations in anoth*
*er.
Thinking of parametrized spectra as diagram exspaces, we see that the evident
forgetful functor 0
i*: GS V ! GS V
has a left adjoint i* given by the prolongation, or expansion of universe, func*
*tor
0 0
(i*X)(V 0) = JGV(, V ) JGV X.
14.2. CHANGE OF UNIVERSE 189
Such prolongation functors are discussed in detail in [62, Ix3] and [61, Ix2]. *
* By
[61, I.2.4], the unit Id! i*i* of the adjunction is a natural isomorphism.
We have more concrete descriptions of the functor i* when V consists of a
cofinal sequence of representations in some universe U. Recall that JGV(V, V ) *
*is
the orthogonal group O(V ) with a disjoint base point.
Lemma 14.2.1. If V = {Vi} V 0is a countable expanding sequence in some
Guniverse U, then
0 0
(i*X)(V 0) ~=JGV(Vi, V ) ^O(Vi)X(Vi)
where i is the largest natural number such that there is a linear isometry Vi *
*! V 0.
Proof. The forgetful functor i* is restriction along a functor ': JGV ! JG*
*V 0
and (i*X)(V 0) is constructed as the coequalizer of the pair of parallel maps
W V 0 0 V ____//_W V 0 0
j,kJG (Vj, V ) ^B JG (Vk, Vj) ^B X(Vk)__//_jJG (Vj, V ) ^B X(Vj)
given by composition in JGV 0and by the evaluation maps associated to the diagr*
*am
X. A cofinality argument that0is easily made precise by use of the explicit des*
*crip
tion of the category JGV given in [61, II.x4] shows that the above coequalizer
agrees with the coequalizer of the subdiagram
JGV(0Vi, V 0) ^B JGV(Vi, Vi) ^B X(Vi)//_//_JGV(0Vi, V 0) ^B X(Vi).
This coequalizer is the space that we have denoted by JGV(0Vi, V 0)^O(Vi)X(Vi).
Remark 14.2.2. The argument above works in the same way for prespectra.
It gives the conclusion that, for parametrized prespectra X in GPVB,
(i*X)(V ) ~= VBViX(Vi).
Remark 14.2.3. Assume that V and V 0are closed under finite sums and con
tain the trivial representation. We can then define the change of universe func*
*tors
0 V
IVV 0= i*i0*:GSBV ! GSB
where i: {Rn} V and i0:{Rn} V 0. Explicitly
(IVVX0)(V ) ~=JGV(Rn, V ) ^O(n)X(Rn).
This is the definition given in [61, V.1.2]. These change of universe functors *
*IVV 0
are exceptionally well behaved on the point set level and agree with those we a*
*re
using when V V 0. They are symmetric monoidal equivalences of categories. For
collections V , V 0and V 00, they satisfy
0 V V V 0 V V
IVV 0O VB ~= B , IV 0O IV 00~=IV 00, IV ~=Id.
Moreover, IVV 0is continuous and commutes with smash products with exspaces.
In particular, it is homotopy preserving and therefore induces equivalences of *
*the
classical homotopy categories. Unfortunately, however, the functors IVV 0are as
poorly behaved on the homotopy level as they are well behaved on the point set *
*level.
They do not preserve either level qequivalences or sequivalences in general a*
*nd
the point set level relations above do not descend to the model theoretic homot*
*opy
categories that we are interested in. Furthermore, these functors IVV 0do not e*
*xist if
V is a cofinal expanding sequence. We shall therefore not make much use of them.
190 14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS
Returning to our full generality, let i: V V 0. The adjoint pair (i*, i*)*
* has
good homotopical properties.
Theorem 14.2.4. Let i: V V 0. Then i* preserves level qequivalences, level
qffibrations, sfibrations, and sacyclic sfibrations. Therefore (i*, i*) is *
*a Quillen
adjoint pair in the level qfmodel structure and in the smodel structure. More*
*over,
i* on homotopy categories is symmetric monoidal. If V is cofinal in V 0, then *
*i*
creates the weak equivalences and (i*, i*) is a Quillen equivalence.
Proof. It is clear from its levelwise definition that i* preserves level q*
*equi
valences and level qffibrations. It follows that its left adjoint i* preserve*
*s s
cofibrations and level acyclic scofibrations. This in turn implies that i* pre*
*serves
sacyclic sfibrations, since those are the maps that satisfy the RLP with resp*
*ect
to the scofibrations. The levelwise description of sfibrations in Proposition*
* 12.5.6
implies that i* preserves sfibrations. The last statement follows from the def*
*ini
tion of homotopy groups and the fact that the unit id! i*i* is an isomorphism.
The functor i* commutes with Z on the point set level, by [61, I.2.14], and this
commutation relation descends directly to homotopy categories. Applying Propo
sition 14.2.8 below to the diagonal map of B, it follows that the derived funct*
*or i*
is symmetric monoidal.
We have constructed the change of universe functors on both the spectrum and
prespectrum level and they are compatible with the restriction functors U. Howe*
*ver,
in order to make use of excellent parametrized prespectra, we must restrict to
parametrized prespectra indexed on cofinal sequencess j :W V and j0:W 0 V 0
of indexing representations in the given universes U U0. But then there need *
*not
be an induced inclusion i: W W 0. We therefore also define change of universe
functors for prespectra indexed on cofinal sequences.
Definition 14.2.5. Let i: V V 0and choose cofinal sequences W = {Vi}
and W 0= {Vi0} in V and V 0such that Vi+1 = Vi Wi and Vi0= Vi Zi, where
Zi+1= Zi Wi0and thus Vi0+1= Vi0 Wi Wi0. Define a pair of adjoint functors
__~*//_ 0
GPWB oo___GPWB
~*
by setting
(~*X)(Vi0) = ZiBX(Vi) and (~*Y )(Vi) = ZiBY (Vi0).
The structure maps are induced from the given structure maps in the evident way.
Proposition 14.2.6. The pair (~*,~*) is a Quillen adjoint pair with respect
to both the level qfmodel structure and the stable model structure. The follow*
*ing
diagram commutes when the vertical arrows point in the same direction.
j*
HoGPWBOOoo____Ho GPVBOO
~*~* i*i*
fflffl0 fflffl0
Ho GPWB o(j0)*HoGPVBo_
14.2. CHANGE OF UNIVERSE 191
Proof. This is clearly a Quillen adjunction in the level qfmodel structure,
and to show that it is a Quillen adjunction in the stable model structure it th*
*erefore
suffices to verify the condition of Proposition 12.5.6. The homotopy pullback 1*
*2.5.7
associated to the pair (Vi, Wi) and an sfibration f :X ! Y is still a homotopy
pullback after we apply ZiBto it and displays the required diagram 12.5.7 for *
*the
map ~*f. We have that
0Vi
(~*j*X)(Vi0) = ZiBX(Vi) ~= ViB X(Vi) = ((j0)*i*X)(Vi0)
and this point set level isomorphism descends to homotopy categories since the
functors0j* and (j0)* preserve all sequivalences. The adjoint structure maps *
*of
X 2 GPVB induce maps
(j*i*X)(Vi) = X(Vi) ! ZiBX(Vi0) = (~*(j0)*X)(Vi).
When X is sfibrant, its adjoint structure maps are level qequivalences, and we
thus obtain an equivalence j*i* ' ~*(j0)* on homotopy categories.
On the pointset level, we have the following commutation relations between
change of universe functors and change of base functors.
Lemma 14.2.7. Let i: V V 0and let f :A ! B be a Gmap. Then i*
commutes up to natural isomorphism with the change of base functors f!, f*, and
f*, and i* commutes up to natural isomorphism with f!and f*.
Proof. The first statement is clear from the levelwise constructions of the
base change functors, and the second statement follows by conjugation since i*,*
* f!,
and f* are left adjoints of i*, f*, and f*.
The missing relation, i*f* ~=f*i*, would hold with the alternative pointset
level definitions of Remark 14.2.3, where i* and i* are inverse equivalences. H*
*ow
ever, these are pointset level relationships that need not descend to model th*
*eoretic
homotopy categories. With our preferred definition of i* in terms of prolongati*
*on,
the following result shows that i*f* ' f*i* on homotopy categories even though
we need not have an isomorphism on the pointset level.
Proposition 14.2.8. Let i: V V 0and let f :A ! B be a Gmap. Then
there are natural equivalences of derived functors
i*f* ' f*i*, i*f!' f!i*, i*f* ' f*i*, i*f* ' f*i*, i*f!' f!i*
in the relevant homotopy categories.
Proof. The first two equivalences are clear since we are commuting Quillen
right adjoints and their corresponding Quillen left adjoints. The fourth will f*
*ollow
by adjunction from the third. If f is a homotopy equivalence, then f* ' (f!)1 *
*and
in this case the third follows from the second and the fifth from the first. Fa*
*ctoring
f as the composite of an hfibration and a homotopy equivalence, we see that the
third will hold in general if it holds when f is an hfibration. Similarly, fac*
*toring f
as the composite of an hcofibration and a homotopy equivalence, we see that the
fifth will hold in general if it holds when f is an hcofibration.
Further, for the third equivalence, it suffices to show that ~*f* ' f*~*si*
*nce
Proposition 14.2.6 then gives that
i*f* ' i*j*j*f* ' (j0)*~*f*j* ' (j0)*f*~*j* ' f*(j0)*(j0)*i* ' f*i*.
192 14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS
Similarly, for the fifth equivalence, it suffices to show that ~*f!' f!~*, fo*
*r then
i*f!' i*(j0)*(j0)*f!' j*~*f!(j0)* ' j*f!~*(j0)* ' f!(j0)*(j0)*i* ' f!i*.
We have reduced the proof of the third equivalence to the situation when f i*
*s an
hfibration and i* is replaced by ~*. The functor f* preserves excellent presp*
*ectra
over B, but we must apply T to ~*before passing to homotopy categories. As in *
*the
proof of Theorem 13.7.7, since f is assumed to be an hfibration we have a natu*
*ral
homotopy equivalence ~: T f* ! f*T in our categories indexed on W or on W 0.
Therefore
T~*f* ~=T f*~*' f*T~*.
Similarly, we have reduced the proof of the fifth equivalence to the situati*
*on
when f is an hcofibration and i* is replaced by ~*. Then f! preserves level *
*h
equivalences, and so does ~*since it preserves level qequivalences and preser*
*ves
objects whose total spaces are of the homotopy types of GCW complexes. Since
T takes zigzags of level hequivalences to homotopy equivalences,
T f!T~*oo'//_T f!~*~=T~*f!oo'//_T~*T f!
displays a zigzag of homotopy equivalences showing that f!~*' ~*f!.
14.3.Restriction to subgroups
Let ` :G0 ! G be a homomorphism and let `*V be the collection of G0
representations `*V for V 2 V , where V is our chosen collection of indexing G
representations. We have implicitly used the following result in our earlier re*
*sults
on change of groups.
Proposition 14.3.1. The functor `*: GSB ! G0S``*V*Bpreserves level q
equivalences, level qffibrations, sfibrations, and sequivalences provided th*
*at the
model structures are defined with respect to generating sets CG and CG0 of Gce*
*ll
complexes and G0cell complexes such that `!C = G xG0C 2 CG for C 2 CG0.
Proof. Since (`*A)H = A`*(H)for a Gspace A and a subgroup H of G0, this
is clear from the definitions of homotopy groups and from the characterizations*
* of
fibrations given in Definition 7.2.7 and Proposition 12.5.6. Note in particular*
* that
`* preserves the level qffibrant approximations that are used in the definitio*
*n of
the stable homotopy groups.
For the remainder of this section fix a subgroup H of G and consider the
inclusion ': H G. For an Hspace A, we simplify notation by letting HSAV
denote the category of Hspectra over A indexed on '*V . Clearly, we then have
the restriction of action functor
'*: GSBV ! HS'V*B.
For i: V V 0, we have '*i* = i*'* since with either composite we are just re
stricting from the representations in V 0to the representations in V and viewin*
*g all
Gspaces in sight as Hspaces.
When V = V (U) for a Guniverse U, there is a quibble here (as was discussed
in [61, V.10]). We are using '*V as the corresponding indexing collection for *
*H.
However, if V is an irreducible representation of G, '*V is generally not an ir*
*re
ducible representation of H and we should expand '*V to include all representat*
*ions
14.3. RESTRICTION TO SUBGROUPS 193
that embed up to isomorphism in '*U to fit the definitions into our usual frame
work. However, there is a change of universe functor associated to the inclusi*
*on
i: '*V (U) V ('*U) that fixes this. The functor i* preserves all sequivalenc*
*es and
descends to an equivalence on homotopy categories. We can and should use these
rectifications when restricting to Hspectra over '*B for a fixed chosen H.
Remark 14.3.2. Consider passage to fibers and recall Proposition 12.6.11.
(i)Applied to inclusions of orbits, Proposition 14.2.8 implies that the functo*
*rs i*
for i: V V 0are compatible with passage to fibers, in the sense that
(i*X)b ~=i*(Xb) forb 2 B,
where i* on the right is the change of universe functor on Gbspectra.
(ii)When V = V (U), we can view the fiber functor
()b:GSB ! GbS
as landing in spectra indexed on V ('*U), ': Gb ! G by composing with i*
for i: '*V (U) V ('*U). However, these change of universe functors must be
used with caution since they are not compatible as b and therefore Gb vary.
Recall from Propositions 12.6.9 and 13.7.9 that the equivalence of categories
('!, *'*) between HSA and GS'!Ainduces a closed symmetric monoidal equiva
lence of categories between HoHSA and HoGS'!A. By Corollary 12.6.10, we can
interpret the restriction functor '*: HoGSB  ! Ho HS'*B as the composite of
base change ~* along ~: '!'*B ! B and this equivalence applied to A = '*B. The
following spectrum level analogue of Proposition 2.3.11 gives compatibility rel*
*ations
between change of base functors and these results on change of groups.
Proposition 14.3.3. Let f :A ! '*B be a map of Hspaces and "f:'!A ! B
be its adjoint map of Gspaces. Then the following diagrams commute up to natur*
*al
isomorphism, where ~: '!'*B ! B and :A ! '*'!A are the counit and unit of
the adjunction ('!, '*).
"f! "f*
GS'!AO_____//_GSBOOO GSB ______//GS'!A
'! ~!O'! '* *O'*
  fflffl fflffl
HSA __f!_//_HS'*B HS'*B __f*_//_HSA
These diagrams descend to natural equivalences of composites of derived functors
on homotopy categories.
Proof. The point set level diagrams commute by Proposition 2.3.11, ap
plied levelwise. The left diagram is one of Quillen left adjoints and the right
diagram is one of Quillen right adjoints, by Propositions 12.6.7 and 12.6.9 and
Corollary 12.6.10.
We now define a parametrized fixed point functor associated to the inclusion
': H ! G. Its target is a category of nonequivariant parametrized spectra. In *
*the
next section we will consider a fixed point functor that takes values in a cate*
*gory
of parametrized W Hspectra, where W H = NH=H is the Weyl group.
Write GSBtrivfor Gspectra over B indexed on trivial representations. These
are "naive" parametrized Gspectra. As usual, to define fixed point spectra, we
194 14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS
must change to the trivial universe before taking fixed points levelwise. Thus *
*let
V G = {V G  V 2 V }. It is contained in V if V = V (U) for some universe U.
Definition 14.3.4. The Gfixed point functor ()G :GSB  ! SBG is the
composite of i*, i: V G V , and levelwise passage to fixed points. For a subg*
*roup
H of G the Hfixed point functor ()H :GSB  ! SBH is the composite of '*,
': H G, and ()H .
Since the homotopy groups of a level qffibrant Gspectrum X over B are the
homotopy groups ssHq(Xb), we see from the nonparametrized analogue [61, V.3.2]
that these are then the homotopy groups of XH . Recall in particular that the
sfibrant Gspectra over B are the Gspectra over B, which are level qffibra*
*nt.
Therefore, for all subgroups H of G, the homotopy groups of a parametrized G
spectrum X are the nonequivariant homotopy groups of the nonequivariant spectra
XH , provided that ()H is understood to mean the derived fixed point functor.
On the pointset level, the functor ()G is a right adjoint. Thinking of the
homomorphism ": G ! e to the trivial group, let "*: SA  ! GS"triv*Abe the
functor that sends spectra over a space A to Gtrivial Gspectra over A regarded
as a Gtrivial Gspace. The following result is immediate by passage to fibers *
*from
its nonparametrized special case [61, V.3.4]. Let A`` denote the collection of *
*all
representations of G.
Proposition 14.3.5. Let A be a space. Let Y be a naive Gspectrum over "*A
and X be a spectrum over A. There is a natural isomorphism
GS"triv*A("*X, Y ) ~=SA (X, Y G).
For (genuine) Gspectra Y over "*A, there is a natural isomorphism
GS"*A(i*"*X, Y ) ~=SA (X, (i*Y )G ),
where i: triv A``. Both of these adjunctions are given by Quillen adjoint pairs
relating the respective level and stable model structures.
Returning to Gspaces B and comparing Definition 11.3.5 with the proof of
[61, V.3.53.6], we obtain the following curious results.
Proposition 14.3.6. For a representation V and an exGspace K, we have
that (FV K)G = *BG unless G acts trivially on V , when (FV K)G ~=FV (KG ) as a
nonequivariant spectrum over BG . The functor ()G preserves scofibrations, but
it does not preserve acyclic scofibrations.
Corollary 14.3.7. For exGspaces K,
( 1BK)G ~= 1B(KG ).
This isomorphism of spectra over BG does not descend to the homotopy cate
gory Ho GSBG . The reader is warned to consult [61, Vx3] for the meaning of the*
*se
results. There is also an analogue of the comparison between Gfixed points and
smash products in [61, V.3.8], but only when B = BG and only with good behavior
with respect to cofibrant objects when external smash products are used. We sha*
*ll
not state the result formally.
14.4. NORMAL SUBGROUPS AND QUOTIENT GROUPS 195
14.4.Normal subgroups and quotient groups
We now turn to quotient homomorphisms and associated orbit and fixed point
functors. The material of this section generalizes a number of results from x2*
*.4,
x7.3, and x9.5 to the level of parametrized spectra.
Just as we have been using ' generically for inclusions of subgroups, we sha*
*ll use
" generically for quotient homomorphisms. In particular, for an inclusion ': H *
* G,
we let W H = NH=H, where NH is the normalizer of H in G, and we have the
quotient homomorphism ": NH ! W H. We can study this situation by first
restricting from G to NH, thus changing the ambient group. Therefore, there is *
*no
loss of generality if we focus attention on a normal subgroup N of G with quoti*
*ent
group J = G=N, as we do throughout this section.
Definition 14.4.1. Let GSBNtrivbe the category of Gspectra over B indexed
on the Ntrivial representations of G. Regard representations of J as Ntrivial
representations of G by pullback along ": G ! J. For a Jspace A, define
"*: JSA ! GS"Ntriv*A
levelwise by regarding exJspaces over A as Ntrivial Gspaces over "*A. For a
Gspace B, define
()=N :GSBNtriv! JSB=N and ()N :GSBNtriv! JSBN
by levelwise passage to orbits over N and to Nfixed points.
Lemma 14.4.2. The Nfixed point functor ()N preserves level qequivalences,
level qffibrations, sfibrations, and sequivalences, provided that the model *
*struc
tures are defined with respect to generating sets CG and CJ of Gcell complexes*
* and
Jcell complexes such that C=N 2 CJ for C 2 CG .
Proof. This is a special case of Proposition 14.3.1; it also follows direct*
*ly from
the exspace level analogue in Proposition 7.4.3, the characterization of sfib*
*rations
in Proposition 12.5.6, and inspection of the definition of the sequivalences.
Proposition 14.4.3. Let j :BN  ! B be the inclusion and p: B ! B=N be
the quotient map. Then the following factorization diagrams commute
()=N ()N
GSBNtriv____//JSB=N99 and GSBNtriv____//_JSBN
sss ss99s
p! ssss j* ssss N
fflffl()=Nsss fflffl()sss
GSBNtriv=N GSBNtrivN
and they descend to natural equivalences on homotopy categories
(p!X)=N ' X=N and (j*X)N ' XN
for X in Ho GSBNtriv. The following adjunction isomorphisms follow.
(i)For Y 2 GSBNtrivand X 2 JSB=N ,
JSB=N (Y=N, X) ~=GSBNtriv(Y, p*"*X).
(ii)For Y 2 GSBNtrivand X 2 JSBN ,
GSBNtriv(j!"*X, Y ) ~=JSBN (X, Y N).
196 14. MODULE CATEGORIES, CHANGE OF UNIVERSE, AND CHANGE OF GROUPS
(iii)For (genuine) Gspectra Y 2 GSB and X 2 JSBN ,
GSB (i*j!"*X, Y ) ~=JSBN (X, (i*Y )N ),
where i: triv A``.
All of these adjunctions are Quillen adjoint pairs with respect to both the lev*
*el and
the stable model structures and so descend to homotopy categories.
Proof. The factorizations follow from the exspace level analogue Proposi
tion 2.4.1. The statement about Quillen adjunctions holds since ()N , ffl* and*
* i*
preserve level qequivalences, level fibrations, sequivalences and level sfib*
*rations,
by Lemma 14.4.2, Proposition 14.3.1 and Theorem 14.2.4.
The behavior of the orbit and fixed point functors with respect to base chan*
*ge
is recorded in the following result.
Proposition 14.4.4. Let f :A ! B be a map of Gspaces. Then the following
diagrams commute up to natural isomorphism
f! Ntriv Ntrivf* Ntriv Ntriv f! Ntriv
GSANtriv___//GSB GSB _____//GSA GSA ____//_GSB
()=N ()=N()N ()N()N ()N
fflffl fflffl fflffl fflffl fflffl fflffl
JSA=N (f=N)_//_JSB=N JSBN _______//JSAN JSAN ______//_JSBN
! (fN )* (fN )!
and they descend to the following natural equivalences on homotopy categories
(f!X)=N ' (f=N)!(X=N), (f*X)N ' (fN )*(Y N), (f!X)N ' (fN )!(X=N)
for X 2 Ho GSANtrivand Y 2 Ho GSBNtriv.
Proof. The first statement follows levelwise from the exspace level analog*
*ue
Proposition 2.4.2. The proof that it descends to equivalences on homotopy cate
gories is the same as for the exspace level analogue Proposition 7.4.5.
Specializing to Nfree Gspaces, we obtain a factorization result that is an*
*al
ogous to those in Proposition 14.4.3, but is less obvious. It is a precursor of*
* the
Adams isomorphism.
Proposition 14.4.5. Let E be an Nfree Gspace, let B = E=N, and let
p: E ! B be the quotient map. Then the diagram
()=N
GSENtriv_____//JSB::
ttt
p* tttt N
fflffl()ttt
GSBNtriv
commutes up to a natural isomorphism, and it descends to a natural equivalence
X=N ' (p*X)N
in GSENtrivfor X 2 Ho JSB . Therefore the left adjoint ()=N of the functor p**
*"*
is also its right adjoint.
Proof. The point set level result follows levelwise from the exspace level
result Proposition 2.4.3. Since it is an isomorphism between a Quillen left adj*
*oint
on the left hand side and a composite of Quillen right adjoints on the right ha*
*nd
side, it descends directly to homotopy categories.
Part IV
Duality, transfer, and base change
isomorphisms
CHAPTER 15
Fiberwise duality and transfer maps
Introduction
We put the foundations of Part III to use in the two chapters of this last
part. Unless otherwise stated, we work in the derived homotopy categories, and
all functors should be understood in the derived sense. For example, we have the
derived fiber functor
()b:Ho GSB ! Ho GbS .
Since passage to fibers is a Quillen right adjoint, this means that we replace *
*G
spectra X over B by sfibrant approximations before taking pointset level fibe*
*rs.
For emphasis, and to make the notation Xb clear and unambiguous, we may assume
that X is sfibrant, but there is no loss of generality. A map f in Ho GSB is an
equivalence if and only if fb is an equivalence for all b 2 B, and that allows *
*us to
transer information back and forth between the parametrized and unparametrized
homotopy categories with impunity. Here we use the word "equivalence" to mean
an isomorphism in HoGSB , and we use the notation ' for this relation. We reser*
*ve
the symbol ~=to mean an isomorphism on the point set level.
We have proven that the basic structure enjoyed by the category GSB of
parametrized spectra descends coherently to the homotopy category Ho GSB . In
particular, Ho GSB is closed symmetric monoidal, and the derived fiber functor *
*is
closed symmetric monoidal. In any symmetric monoidal category, we have standard
categorical notions of dualizable and invertible objects. In x15.1, we prove t*
*he
fiberwise duality theorem, which says that a Gspectrum X over B is dualizable *
*or
invertible if and only if each fiber Xb is dualizable or invertible. This allow*
*s us to
recognize dualizable or invertible Gspectra over B when we see them.
In x15.2, we explain how the fiberwise duality theorem leads to a simple and
general conceptual definition of trace and transfer maps with good properties. *
*To
define the transfer, we regard a Hurewicz fibration p: E ! B with stably duali*
*z
able fibers as a space over B. We adjoin a copy of B to obtain a section, and we
suspend to obtain a Gspectrum over B. It is dualizable since its fibers are du*
*aliz
able, hence it has a transfer map defined by categorical nonsense. Pushing down*
* to
Gspectra by base change along the map r :B ! *, we obtain the transfer map
of Gspectra 1 B+ ! 1 E+ . This construction is a generalization of various
earlier constructions of the transfer [2, 3, 15, 18, 97], most of which restric*
*t to
finite dimensional base spaces and are nonequivariant. An essential point is th*
*at
the homotopy category of Gspectra over B is closed symmetric monoidal with a
"compatible triangulation", in the sense specified in [74]. We defer the proof*
* of
the required compatibility relations to x15.6. This point implies that our tra*
*ces
and transfers satisfy additivity relations as well as the more elementary stand*
*ard
properties.
198
15.1. THE FIBERWISE DUALITY THEOREM 199
Some of the classical constructions of the transfer work only for bundles, b*
*ut
have various properties that are inaccessible to the more general construction *
*and
are important in calculations. These transfers also admit a perhaps more satisf*
*ying
construction. Rather than relying on duality on the level of parametrized spect*
*ra,
they are obtained by inserting duality maps for fibers fiberwise into bundles. *
*In the
literature, the construction again usually requires finite dimensional base spa*
*ces
and is nonequivariant. We give a general conceptual version of this alternative
construction in x15.5.
As a first preliminary, in x15.3 we show how to insert parametrized spectra
fiberwise into the standard construction of equivariant bundles associated to p*
*rin
cipal bundles. The general construction is of considerable interest nonequivari*
*antly.
The construction on the exspace level is easy enough, but even here many of the
properties that we describe seem to be new. The construction is likely to have
many further applications. The idea is to generalize the standard construction *
*of
the bundle of tangents along the fibers of a bundle by replacing the tangent bu*
*ndle
of the fiber by any spectrum over the fiber. In more detail, we consider Gbund*
*les
p: E ! B with fibers F . We allow the structure group and ambient group G
to be related by an extension 1 ! ! ! G ! 1, and we take F to be a 
space. The bundle p has an associated principal ( ; )bundle ss :P ! B, where
P is a free space and B = P= . We show how to construct a Gspectrum
P x X over E from a spectrum X over F .
As a second preliminary, in x15.4 we develop the theory of free parametriz*
*ed
spectra. This is a direct generalization of the nonparametrized theory and is
important in many contexts. In particular, it will play a role in our proof of *
*the
Adams isomorphism in x16.4.
The application to transfer maps in x15.5 can be described as follows. When
F is dualizable, we have a transfer map o :S ! 1 F+ of spectra. We insert
this into the functor P x () to obtain a map
P x o :P x S ! P x 1 F+
of Gspectra over B. Again pushing down to a map of Gspectra along r :B ! *,
we obtain the transfer Gmap 1GB+ ! 1GE+ . This description hides a subtlety.
The construction involves a change of universe functor, and the key point is th*
*at this
functor is a symmetric monoidal equivalence between categories of parametrized *
* 
free spectra. This makes it transparent from the naturality of transfer maps *
*with
respect to symmetric monoidal functors that the fiberwise transfer map of a bun*
*dle
agrees with its transfer map as a Hurewicz fibration.
We assume throughout that all given groups G are compact Lie groups and all
given base Gspaces are of the homotopy types of GCW complexes.
15.1. The fiberwise duality theorem
We characterize the dualizable and invertible Gspectra over B. A recent ex
position of the general theory of duality in closed symmetric monoidal categori*
*es
appears in [73], to which we refer the reader for discussion of the relevant ca*
*tegori
cal definitions and arguments. The following theorem is a substantial generaliz*
*ation
of various early results of the same nature about exfibrations. These are due,*
* for
example, to Becker and Gottlieb [2, x4], Clapp [18, 3.5], and Waner [97, 4.6].
200 15. FIBERWISE DUALITY AND TRANSFER MAPS
Theorem 15.1.1 (The fiberwise duality theorem). Let X be an (sfibrant) G
spectrum over B. Then X is dualizable (respectively, invertible) if and only if*
* Xb
is dualizable (respectively, invertible) as a Gbspectrum for each b 2 B.
Proof. By definition, X is dualizable if and only if the natural map
:DB X ^B X ! FB (X, X)
in Ho GSB is an equivalence. Passing to (derived) fibers, this holds if and onl*
*y if
the resulting map
DXb^ Xb ' (DB X ^B X)b __b_//_FB (X, X)b ' F (Xb, Xb)
in HoGbS is an equivalence for all b 2 B. By the categorical coherence observat*
*ion
Remark 2.2.8, the latter map is the corresponding natural map in HoGbS . Again
by definition, that map is an equivalence if and only if Xb is dualizable.
Similarly, X is invertible if and only if the evaluation map
ev:DB X ^B X ! SB
in Ho GSB is an equivalence. Passing to (derived) fibers, this holds if and onl*
*y if
the resulting map
DXb^ Xb ' (DB X ^B X)b _evb//_(SB )b ' S
in Ho GbS is an equivalence for all b 2 B. Again by Remark 2.2.8, the latter map
is the evaluation map for Xb in Ho GbS , and that map is an equivalence if and
only if Xb is invertible.
Therefore, to recognize parametrized dualizable and invertible Gspectra, it
suffices to recognize nonparametrized dualizable and invertible Gspectra. As we
now recall from [41], these are wellunderstood.
Recall that a Gspace X is dominated by a Gspace Y if X is a retract up
to homotopy of Y , so that the identity map of X is homotopic to a composite
X ! Y ! X. If Y has the homotopy type of a GCW complex, then so does X.
We say that X is finitely dominated if it is dominated by a finite GCW complex.
This does not imply that X has the homotopy type of a finite GCW complex, even
when X and all of its fixed point spaces XH are simply connected and therefore,
since they are finitely dominated, homotopy equivalent to finite CW complexes.
For example, a Gspace X is a GENR (Euclidean neighborhood retract) if it
can be embedded as a retract of an open subset of some representation V . Such
open subsets are triangulable as GCW complexes, so X has the homotopy type of
a GCW complex. A compact GENR is a retract of a finite GCW complex and is
thus finitely dominated, but it need not have the homotopy type of a finite GCW
complex. Nonsmooth topological Gmanifolds give examples of such nonfinite
compact GENRs.
The following result is [41, 2.1].
Theorem 15.1.2. Up to equivalence, the dualizable Gspectra are the Gspectra
of the form V 1 X where X is a finitely dominated based GCW complex and
V is a representation of G.
Definition 15.1.3. A generalized homotopy representation X is a finitely dom
inated based GCW complex such that, for each subgroup H of G, XH is equivalent
to a sphere Sn(H). A stable homotopy representation is a Gspectrum of the form
15.2. DUALITY AND TRANSFER MAPS 201
V 1 X, where X is a generalized homotopy representation and V is a represen
tation of G.
The following result is [41, 0.5].
Theorem 15.1.4. Up to equivalence, the invertible Gspectra are the stable
homotopy representations.
Combining results, we obtain the following conclusion about exGfibrations.
Theorem 15.1.5. Let E be an exGfibration over B. If each fiber Eb is a
finitely dominated Gbspace, then 1BE is a dualizable Gspectrum over B. If ea*
*ch
Eb is a generalized homotopy representation of Gb, then 1BE is an invertible G
spectrum over B.
Proof. Since the derived suspension spectrum functor commutes with passage
to derived fibers, by Theorem 13.7.10, the derived fiber ( 1BE)b is equivalent *
*to
1 Eb. The conclusion follows directly from Theorems 15.1.1, 15.1.2, and 15.1.4.
In particular, sphere Gbundles and, more generally, spherical Gfibrations *
*over
B, have invertible suspension Gspectra over B.
15.2. Duality and transfer maps
Since the stable homotopy category Ho GSB is closed symmetric monoidal, we
have the following generalized trace maps at our disposal. We state the definit*
*ion
and recall its properties in full generality, and we then specialize to show ho*
*w it
gives a simple conceptual definition of the transfer maps associated to equivar*
*iant
Hurewicz fibrations.
Definition 15.2.1. Let C be any closed symmetric monoidal category with
unit object S. For a dualizable object X of C with a "coaction" map X :X !
X ^ CX for some object CX 2 C , define the trace o(f) of a self map f of X by t*
*he
diagram
j fl
S _____//_X ^ DX_______//_DX ^ X
o(f) Df^X
fflffl fflffl
CX oo~=_S_^ CX ooffl^DX1^_X ^ CX .
Remark 15.2.2. Such a categorical description of generalized trace maps was
first given by Dold and Puppe [35], where they showed that it gives the right
framework for trace maps in algebra, the transfer maps of Becker and Gottlieb [*
*2, 3],
and the fixed point theory of Dold [34]. These early constructions of transfer *
*maps
had finiteness conditions that were first eliminated by Clapp [18, 19]. Indeed,
she gave an early construction of a parametrized stable homotopy category and
proved a precursor of our fiberwise duality theorem. The equivariant analogue of
the attractive space level treatment of SpanierWhitehead duality given by Dold
and Puppe was worked out in [59], and a recent categorical exposition of duality
has been given in [73].
Two cases are of particular interest. The first is when CX = S and X is the
unit isomorphism. Then o(f) is called the Lefschetz constant of f and is denoted
202 15. FIBERWISE DUALITY AND TRANSFER MAPS
by O(f); in the special case when f = idit is called the Euler characteristic o*
*f X
and denoted by O(X). The second is when CX = X. We then think of X as a
diagonal map, and oX = o(id) is called the transfer map of X.
Remark 15.2.3. If CX comes with a "counit" map , :CX ! S such that the
composite
id^,
X _____//X ^ CX____//X
is the identity, then O(f) = , O o(f) by a little diagram chase. The reason for
the terminology "coaction" and "counit" for the maps X and , is that in many
situations CX is a comonoid and X is a coaction of CX on X.
The following basic properties of the trace are proven in [59, IIIx7] and in*
* [74],
where more detailed statements are given. Define a map
(f, ff): (X, X ) ! (Y, Y )
to be a pair of maps f :X  ! Y and ff: CX ! CY such that the following
diagram commutes.
X
X _____//X ^ CX
f f^ff
fflffl fflffl
Y __Y__//Y ^ CY
Proposition 15.2.4. The trace satisfies the following properties, where X and
Y are dualizable and X and Y are given.
(i)(Naturality) If C and D are closed symmetric monoidal categories and
F :C ! D is a lax symmetric monoidal functor such that F SC ~=SD ,
then
o(F f) = F o(f),
where CFX = F CX and FX = F X .
(ii)(Unit property) If f is a self map of the unit object, then O(f) = f.
(iii)(Fixed point property) If (f, ff) is a self map of (X, X ), then
ff O o(f) = o(f).
(iv)(Invariance under retracts) If X i! Y r! X is a retract, f is a self map *
*of
X, and (i, ff) is a map (X, X ) ! (Y, Y ), then
ff O o(f) = o(ifr).
(v)(Commutation with ^) If f and g are self maps of X and Y , then
o(f ^ g) = o(f) ^ o(g),
where X^Y = (id^ fl ^ id) O ( X ^ Y ) with fl the transposition.
(vi)(Commutation with _) If C is additive and h: X _ Y ! X _ Y induces
f :X ! X and g :Y ! Y by inclusion and retraction, then
o(h) = o(f) + o(g),
where CX = CY = CX_Y and X_Y = X _ Y .
(vii)(Anticommutation with suspension) If C is triangulated, then
o( f) = o(f)
for all self maps f, where X = X .
15.2. DUALITY AND TRANSFER MAPS 203
In the triangulated context, there is another and very much deeper property.
Theorem 15.2.5 (Additivity). Let C be a closed symmetric monoidal category
with a "compatible triangulation". Let X and Y be dualizable and let X and Y
be given, where C = CX = CY . Let (f, id) be a map (X, X ) ! (Y, Y ) and
extend f to a distinguished triangle
f g h
X _____//Y____//Z____//_ X.
Assume given maps OE and _ that make the left square commute in the first of the
following two diagrams.
f g h
X _____//Y____//Z____//_ X
OE _ ! OE
fflfflfflfflfflffl fflffl
X __f__//Y_g__//Z_h__//_ X
f g h
X ________//_Y_______//_Z________//_ X
X  Y Z  X
fflffl fflffl fflffl fflffl
X ^ C _f^id//_Y ^_Cg^id//_Z ^hC^id//_ (X ^ C)
Then there are maps ! and Z such that the diagrams commute and
o(_) = o(!) + o(OE).
The most important case starts with only the distinguished triangle (f, g, h)
and concludes with the fundamental additivity relation
O(Y ) = O(X) + O(Z).
The additivity of traces was studied in [59, IIIx7] in the equivariant stable h*
*omotopy
category, but the proof there is incorrect. A thorough investigation of precise*
*ly what
is needed to prove the additivity of traces is given in [74], where the axioms *
*for
a "compatible triangulation" are formulated. These axioms hold in all situations
previously encountered in algebraic topology and algebraic geometry. However,
the model theoretic method of proof described in [74] assumes the usual model
theoretic compatibilities, such as the pushoutproduct axiom of [86], and these*
* fail
to hold in the present context. Since the proof of the following result only ma*
*kes
sense by close comparison with the proof in [74], we shall defer it to x15.6.
Theorem 15.2.6. The category Ho GSB is a closed symmetric monoidal cat
egory with a compatible triangulation.
With these foundations in place, we can now generalize the classical constru*
*c
tion of transfer maps. The results above specialize to give more information ab*
*out
them than is to be found in the literature. If X is a dualizable Gspectrum over
B with a diagonal map X :X  ! X ^B X, then we have the transfer map
oX :SB ! X. We shall apply this to suspension Gspectra associated to G
fibrations p: E ! B, but we do not assume that p has a section. We need some
notation. It has been the custom since the beginnings of algebraic topology to *
*use
the same letter E for a bundle and for its underlying total space. It seems to *
*us that
this standard abuse of notation seriously obscures the literature of parametriz*
*ed ho
motopy theory, and for that reason we shall be very pedantic at this point.
204 15. FIBERWISE DUALITY AND TRANSFER MAPS
Notation 15.2.7. For a Gspace E over B, let (E, p)+ denote the exGspace
E qB over B, with section at the disjoint copy of B. The usual notation is E+ ,*
* but
we shall reserve that notation for the union of the total Gspace E with a disj*
*oint
basepoint. Observe that if p is a Hurewicz Gfibration, then (E, p)+ is an ex*
*G
fibration. Except where otherwise indicated, we agree to write r for the unique
map B ! * for any based Gspace B.
Recall the desription of the base change functors associated to r from Exam
ple 2.1.7. The spectrum level versions of these functors are central to the ded*
*uction
of results in classical stable homotopy theory from results in parametrized sta*
*ble
homotopy theory. The following observation is particularly relevant.
Lemma 15.2.8. For a Gmap p: E ! B, thought of as a Gspace over B,
r! 1B(E, p)+ ' 1 E+ ,
where r :B ! *. In particular, r!SB ' 1 B+ .
Proof. We have r! 1B ' 1 r!. This is a commutation relation between
Quillen left adjoints, and the corresponding commutation relation for right adj*
*oints
holds since
r* 1 X = B x X0 ~= 1Br*X
for a Gspectrum X. It therefore suffices to show that r!(E, p)+ is equivalent*
* to
E+ , where r! denotes the functor on derived categories. By Proposition 7.3.4,
r! preserves qequivalences between wellsectioned exspaces and it follows that
r!Q(E, p)+ ' r!(E, p)+ ~=E+ where the first equivalence is induced by qfcofibr*
*ant
approximation of (E, p)+ .
To be precise about diagonal maps on the parametrized level, we consider base
change along : B ! B x B. We have the obvious commutative diagram
E _____//E x E
p  pxp
fflffl fflffl
B _____//B x B.
We consider E as a space over B x B via this composite. The diagonal map of E
then specifies a natural map
!((E, p)+ ) = (E, O p)+ ! (E x E, p x p)+ ~=(E, p)+ Z (E, p)+
of exspaces over B x B. This is a comparison map between Quillen left adjoints
and therefore descends to a natural map in HoGKBxB . Its adjoint is a natural m*
*ap
(E, p)+ ! (E, p)+ ^B (E, p)+ in HoGKB . Apply the (derived) suspension functor
1Bto this map and note that the target is equivalent to 1B(E, p)+ ^B 1B(E, p*
*)+ ,
by Proposition 13.7.5. This gives the required natural diagonal map
(E,p)+: 1B(E, p)+ ! 1B(E, p)+ ^B 1B(E, p)+
in Ho GSB .
Definition 15.2.9 (The transfer map). Let p: E ! B be a Hurewicz G
fibration over B such that each fiber Eb is homotopy equivalent to a retract of
15.3. THE BUNDLE CONSTRUCTION ON PARAMETRIZED SPECTRA 205
a finite GbCWcomplex. Then 1B(E, p)+ is a dualizable Gspectrum over B by
Theorem 15.1.5 and we obtain the transfer map
o(E,p)+:SB ! 1B(E, p)+
in Ho GSB . Define the transfer map of E to be the map
oE = r!o(E,p)+: 1 B+ ~=r!SB ! r! 1B(E, p)+ ~= 1 E+
in Ho GS .
With this definition, all of the standard properties of transfer maps are di
rect consequences of the general categorical results Proposition 15.2.4 and The*
*o
rem 15.2.5 and the properties of r!.
15.3.The bundle construction on parametrized spectra
The construction of the transfer in the previous section works "globally", s*
*tart
ing on the parametrized spectrum level. We now give a fiberwise construction of
"stable bundles" that leads to an alternative fiberwise perspective. However, i*
*t is
natural to work in greater generality than is needed for the construction of the
transfer. The extra generality will be needed in the proof of the Wirthm"uller *
*iso
morphism in x16.3 and will surely find other applications. The relevant bundle
theoretic background was recalled in x3.2.
Let be a normal subgroup of a compact Lie group such that = = G and
let q : ! G be the quotient homomorphism. Let p: E ! B be a ( ; )bundle
with fiber a space F and with associated principal ( ; )bundle ss :P ! B.
Then P is a free space, ss is the quotient map to the orbit Gspace B = P= ,
and p is the associated Gbundle E ~=P x F ! B. To simplify the homotopical
analysis, we assume for the rest of this section that F and P are CW complexes
such that P is free. We let E = P x F and B = P x *. Note that B is a
GCW complex. We are thinking of the cases when F is a point or when F is a
smooth manifold. On the exspace level, application of P x () to retracts g*
*ives
the functor
P~F = P x (): KF ! GKE .
Thus, for an ex space K over F , the exGspace P x K over P x F has section
and projection induced by the section and projection of K. Observe that if F is
a smooth manifold and So is the sphere bundle obtained by fiberwise onepoint
compactification of the tangent bundle of F , then P x So is the Gbundle of
spherical tangents along the fiber associated to p.
We can extend the functor ~PF from exspaces to exspectra. Change of uni
verse must enter since spectra are indexed on representations of and Gspec*
*tra
are indexed on representations of G. We view representations of G as trivial
representations of . This gives i: q*VG ! V . Implicitly applying the functor
i* to spectra, we agree to index both Gspectra and spectra on VG for the r*
*est
of the section. We are interested in spectra indexed on a complete universe, *
*and
we shall return to this point in the next section. Since acts trivially on o*
*ur
representations V , we have
~PFK ^E SV ~=P~F(K ^F SV ).
Therefore, for a spectrum X over F , the exGspaces ~PFX(V )over E inherit
structure maps from X, so that ~PFX is a Gspectrum over E. We have the same
206 15. FIBERWISE DUALITY AND TRANSFER MAPS
definition on the prespectrum level. These functors P~F are exceptionally well
behaved, as the following results show.
Proposition 15.3.1. The functor ~PF: SF triv! GSE is both a left and a
right Quillen adjoint with respect to the level and stable model structures. Mo*
*reover,
the functor ~PF: PFtriv! GPE takes excellent prespectra over F to excellent
Gprespectra over E = P x F .
Proof. Let ss :P x F ! F be the projection. Clearly ~PFis the composite
of ss*: SF ! SPxF and ()= : SPxF ! GSE . By Propositions 12.2.5,
12.2.7, 12.6.7, and 12.6.8, ss* is both a left and a right Quillen adjoint, pro*
*vided we
use appropriate generating sets in our definitions of the model structures. By *
*Propo
sition 14.4.3, the functor ()= is a Quillen left adjoint. By Proposition 14.*
*4.5,
it coincides with the right adjoint () O p*, where p here is the quotient map
P x F  ! P x F = E. Using Lemma 3.2.1, we see that p: E ! B is a G
bundle with CW fibers. Therefore p* is a Quillen right adjoint by Propositions
12.2.7 and 12.6.8, and () is a Quillen right adjoint by Proposition 14.4.3. T*
*he
last statement is easily checked from Definition 13.2.2 and Lemma 13.2.3.
We need an observation about the behavior of ~PFon fibers.
Lemma 15.3.2. Fix b 2 B. Let ': Gb ! G and aeb:Gb ! be the inclusion
and the homomorphism of Lemma 3.2.1. Let b: {b} ! B and ib:Eb ! E denote
the evident inclusions of Gb spaces. The following diagrams commute, and these
commutation relations descend to homotopy categories.
ae*b ae*b
S* triv____//GbSbOOand SF triv____//GbSEbOO
~P* b* P~F i*b
fflffl  fflffl 
GSB ___'*_//_GbSB GSE __'*__//_GbSE
Proof. On the level of exspaces, this is immediate by inspection. The di
agrams extend levelwise to parametrized spectra, and passage to homotopy cate
gories is clear from the previous result.
Writing 1 for suspension spectra functors indexed on complete universes,
we have that i* 1 ,F, where i: q*VG V , is the suspension spectrum functor
indexed on the trivial universe q*VG .
Theorem 15.3.3. There is a natural isomorphism of functors
~PFi* 1 ,F~= 1G,E~PF: KF ! GSE ,
and this isomorphism descends to homotopy categories. The functor
~PF:Ho SF triv! Ho GSE
is closed symmetric monoidal.
Proof. Let K be an ex space over F . Since we are indexing on representa
tions V of G, we have isomorphisms
(P~Fi* 1 ,FK)(V ) = P x (K ^F SVF)~=(P x K) ^E SV = ( 1G,E~PFK)(V ).
This gives a natural isomorphism of Gspectra over E, and it descends to homoto*
*py
categories since it is a comparison of composites of Quillen left adjoints. Not*
*e in
15.3. THE BUNDLE CONSTRUCTION ON PARAMETRIZED SPECTRA 207
particular that ~PFi*S ,Fis isomorphic to SG,E. We must show that the functor ~*
*PF
commutes up to coherent natural isomorphism with smash products and function
objects. For ex spaces K and L over F , it is easy to check that there is a n*
*atural
isomorphism
~PF(K ^F L) ! ~PFK ^E ~PFL
of exGspaces over E. This isomorphism extends levelwise to external smash pro*
*d
ucts (external in the sense of pairs of representations). However, since exter*
*nal
pairings (in the sense of pairs of base spaces) do not naturally come into play*
* here,
to retain homotopical control it seems simplest to just extend levelwise to han*
*di
crafted smash products of prespectra; compare Remark 13.7.1. Using excellent
prespectra to pass to homotopy categories of prespectra and then using the equi*
*v
alence (P, U) to pass to homotopy categories of spectra, we obtain the required
natural equivalence
~PF(X ^F Y )! ~PFX ^E ~PFY
in Ho GSE for spectra X and Y over F . The adjoint of the composite
P~FFF (X, Y^)E~PFX ' ~PF(FF (X, Y ) ^F X)_P~F(ev)_//_~PFY
is a natural map
P~FFF (X, Y )! FE (P~FX , ~PFY)
in Ho GSE , and we must show that it is an equivalence. This will hold if it ho*
*lds
when restricted to fibers over points of E. Since each point is in some Eb, it
suffices to show that the restriction to each Eb is an equivalence. However, us*
*ing
Lemma 15.3.2, we see that the restriction to Eb is the adjoint to the Gbmap
ev:FEb(ae*bX, ae*bY ) ^Eb ae*bX ! ae*bY , and is thus the identity map.
We have the following relations between ~PFand base change functors.
Proposition 15.3.4. Consider r :F ! * and p = P x r :E ! B. For
Y 2 S* trivand X 2 SF triv, there are natural isomorphisms
p!~PFX ! ~P*r!X, ~PFr*Y! p*P~*Y, and ~P*r*X! p*P~FX ,
and these isomorphisms induce natural equivalences on homotopy categories.
Proof. We first work on the exspace level. Let T be a based Gspace and K
be an exGspace over F . Applying the functor P x () to the maps of retracts
that define r!K and r*T (see Definition 2.1.1), we immediately obtain the first*
* two
maps. The first is the natural isomorphism
(P x K) [E B ~=P x (K=F )
in which the section F is collapsed to a point in K on both sides. The second is
the evident natural isomorphism
P x (F x T ) ~=(P x F ) xB (P x T ).
For the third map, recall that r*K = Sec(F, K). The adjoint of
(P x Sec(F, K)) xB E ~=P x (Sec(F, K) x F_)Px_ev__//_P x K
gives a map
P~*r*K = P x Sec(F, K)! Map B(E, P x K).
208 15. FIBERWISE DUALITY AND TRANSFER MAPS
Together with the projection of the source to B, it induces an isomorphism to
p*P~FK , which is the pullback along B ! Map B(E, E) of the projection of the
target induced by the projection P x K ! P x F = E. Applied levelwise,
these pointset level isomorphisms carry over directly to parametrized prespect*
*ra
and spectra. We must show that they descend to equivalences in homotopy cate
gories. Since Proposition 12.6.8 applies to show that both p* and r* are Quillen
right adjoints (and we have no need to use Brown representability here), the fi*
*rst
commutation relation is between composites of left Quillen adjoints, the second
is between functors that are both left and right Quillen adjoints, and the thir*
*d is
between Quillen right adjoints, so descent to homotopy categories is immediate.
15.4. free parametrized spectra
We retain the notations of the previous section in this section and the next.
In the next section, we show that the bundle construction on parametrized spect*
*ra
leads to a fiberwise generalization of the restriction to bundles of the trace *
*and
transfer maps for fibrations that we described in x15.2. The definition depends*
* on
a result that is proven by use of the theory of free spectra that we presen*
*t here.
We first recall what it means to say that a spectrum X (indexed on any
universe) is free. Let F ( ; ) be the family of subgroups of such that
\ = e. A CW complex T is free if and only if the only orbit types
= that appear in its construction have 2 F ( ; ). We then say that T is
an F ( ; )CW complex. We can make the same definitions for CW spectra,
and in general we say that a spectrum is free if it is isomorphic in Ho S *
*to
an F ( ; )CW spectrum. There is a more conceptual homotopical reformulation
that is the one relevant to the parametrized point of view and that does not de*
*pend
on the theory of CW spectra.
Let E( ; ) be the universal free space, so that E( ; ) is contractible
if \ = e and is empty otherwise. We may take E( ; ) to be an F ( ; )CW
complex. Let B( ; ) = E( ; )= and observe that B is a GCW complex and
therefore also a CW complex. We note parenthetically that the quotient map
p : E( ; ) ! B( ; ) is the universal principal ( ; )bundle. That is, pullb*
*ack
along p gives a bijection
[X, B( ; )]G ! B( ; )(X),
where B( ; )(X) denotes the set of equivalence classes of principal ( ; )bun*
*dles
over the Gspace X; see [55] or [68, VIIx2].
Definition 15.4.1. Let r :E( ; ) ! * be the projection and let oe be the
counit of the (derived) adjunction (r!, r*). A spectrum X is said to be fre*
*e if
oe :r!r*X ! X is an equivalence.
The definition should seem reasonable since r!r*T ~=E( ; )+ ^T for a space
T . It is equivalent to the original definition in terms of an equivalence in H*
*o GS to
an F ( ; )CW spectrum; see [59, II.2.12] or [61, VIx4]. The definition genera*
*lizes
readily to the parametrized context.
Definition 15.4.2. Let ss :E( ; ) x F ! F be the projection and let oe be
the counit of the (derived) adjunction (ss!, ss*). An ex space or spectrum*
* X
over a space F is said to be free if oe :ss!ss*X ! X is an equivalence.
15.5. THE FIBERWISE TRANSFER FOR ( ; )BUNDLES 209
Since the fiber (ss!ss*X)f is E( ; )+ ^ Xf, the definition should seem reas*
*on
able. Since equivalences are detected fiberwise, we have the following results.
Lemma 15.4.3. A spectrum X over F is free if and only if each of its fib*
*ers
Xf is a ( \ f)free fspectrum.
Proof. The fiber of E( ; )xF ! F over f 2 F is the space E( ; ) with
the action restricted along ': f ! . It is a model of the universal ( \ f)*
*free
fspace E( \ f, f). Applying ()f to the counit ss!ss*X ! X and using
Theorem 13.7.7 we obtain the counit r!r*Xf ! Xf where r :'*E( ; ) ! *.
Lemma 15.4.4. If P is a free space and X is any ex space or spectrum
over F , then P x X is a free ex space or spectrum over P x F .
A useful slogan asserts that " free spectra live in the trivial univers*
*e".
To explain it, consider the inclusion i: q*VG ! V of the complete Guniverse *
*VG
as the universe of trivial representations in the complete universe V . Th*
*en
the slogan is given meaning by the following result. In the nonparametrized case
F = *, it is proven in [59, IIx2] and is discussed further in [61, VIx4]. Since*
* the
parametrized case presents no complications and the proof is quite easy, we only
give a sketch.
Proposition 15.4.5. The change of universe adjunction (i*, i*) descends to
a symmetric monoidal equivalence between the homotopy categories of free 
spectra over F indexed on trivial representations of on the one hand and in*
*dexed
on all representations of on the other. For free spectra X over F indexed*
* on
V , there is a natural equivalence i*(E( ; )+ ^ i*X) ' X.
Sketch Proof. If \ = e, then the quotient map q : ! G maps
isomorphically onto a subgroup of G. Any representation V of is therefore of
the form q*W for a representation W of q( ). It follows that the restrictions *
*to
of the universes V and q*VG have the same representations. This makes clear
that, on free spectra over F , the unit and counit of the adjunction (i*, i*
**)
must be F ( ; )equivalences, in the sense that they are equivalences for any
in F ( ; ). Smashing the unit and counit with E( ; )+ , which has trivial fix*
*ed
point sets for subgroups not in F ( ; ), we obtain natural equivalences, and it
follows from Definition 15.4.1 that the unit and counit are themselves equivale*
*nces
when applied to free spectra. Alternatively, restricting to sfibrant spe*
*ctra
over F , the conclusion follows fiberwise from its nonparametrized precursor. S*
*ince
i* is symmetric monoidal, by Theorem 14.2.4, so is the equivalence. The last
statement holds since
i*(E( ; )+ ^ i*X) ' E( ; )+ ^ i*i*X ' X.
15.5. The fiberwise transfer for ( ; )bundles
We consider a fixed given principal ( ; )bundle P , where is a normal su*
*b
group of with quotient group G and quotient map q : ! G. We also consider
a space F and the associated ( ; )bundle
p: E = P x F ! P x * = B.
We have the inclusion i: q*VG ! V of the complete Guniverse VG as the univer*
*se
of trivial representations in the complete universe V .
210 15. FIBERWISE DUALITY AND TRANSFER MAPS
The change of universe functor i*: SF ! SF trivis not symmetric mon
oidal, and it does not preserve dualizable objects. For example, with F = * and
= e, the orbit spectrum i* 1 = is not dualizable if is a nontrivial subg*
*roup
of . The bundle theoretic study of transfer maps is based on the following res*
*ult,
whose proof is based on the theory of free spectra given in the previous se*
*ction.
Theorem 15.5.1. The composite functor ~PFi*:Ho SF ! Ho GSE is sym
metric monoidal.
Proof. Let ss :P x F ! F be the projection and note that ss*X = P x X.
The functor ~PF is the composite of the symmetric monoidal Quillen left adjoint
ss* and the Quillen left adjoint ()= . By Theorem 15.3.3, the functor P~F on
homotopy categories is also symmetric monoidal since the space P is free. We
observe first that the composite ss*i* is symmetric monoidal. Indeed, for spe*
*ctra
X and Y over F , we have
ss*i*(X ^F Y')i*ss*(X ^F Y ) by Proposition 14.2.8
' i*(ss*X ^PxF ss*Y )by Theorem 13.7.6
' i*ss*X ^PxF i*ss*Y by Lemma 15.4.4 and Proposition 15.4.5
' ss*i*X ^PxF ss*i*Y by Proposition 14.2.8.
It follows directly that ~PFi*is symmetric monoidal:
~PFi*(X ^F Y )= (ss*i*(X ^F Y ))= by definition
' (ss*i*X ^PxF ss*i*Y )= by the previous display
' (ss*(i*X ^F i*Y ))= by Theorem 13.7.6
= ~PF(i*X ^F i*Y) by definition
' ~PFi*X^E ~PFi*Y by Theorem 15.3.3.
Now Proposition 15.2.4(i) shows that ~PFi*commutes with trace maps.
Theorem 15.5.2. Let X 2 Ho SF be dualizable. Then P~Fi*X 2 Ho GSE
is dualizable. Suppose given a coaction map X : X ! X ^F CX and a self map
OE: X ! X. Then
o(P~Fi*OE) ' ~PFi*o(OE): SE ! ~PFi*CX,
where ~PFi*X is given the coaction map
~PFi*( X ) : ~PFi*X ! ~PFi*(X ^F CX ) ' ~PFi*X ^E ~PFi*CX .
These trace maps are maps of Gspectra over E, rather than over B. We can
apply r!, r :E ! *, to obtain trace maps of nonparametrized spectra. This kind*
* of
trace map can be viewed as a fiberwise generalization of the kind of nonparamet*
*rized
trace map that is defined bundle theoretically in the literature. To connect up*
* with
the latter, we specialize and change our point of view so as to arrive at bundle
theoretic trace maps over B. Specializing further to transfer maps, we obtain t*
*he
promised comparison with the transfer maps of Definition 15.2.9.
With these goals in mind, we now focus on the case F = *, so that E above
becomes B, with p the identity map, and our trace maps are parametrized over B.
We study our original fixed given ( ; )bundle p: E ! B in a different fashio*
*n.
We rename its fiber M to avoid confusion with respect to the role that space is
15.5. THE FIBERWISE TRANSFER FOR ( ; )BUNDLES 211
playing. In the theory above, F was a base space for paramentrized spectra and
there was no need for F to be dualizable. We now consider the case when M is
stably dualizable, so that 1 M+ is dualizable, and we write oM for the transf*
*er
map S ! 1 M+ in S , as defined in and after Definition 15.2.1. We apply
Theorem 15.5.2 with F = * and X = 1 M+ to obtain the following special case.
Here we use the diagonal map induced by the diagonal map of M. Observe that,
by Theorem 15.3.3,
~P*i* 1 M+' 1 ~P*M+ = 1B(E, p)+ .
Theorem 15.5.3. Let M be a compact ENR and let p: E ! B be a ( ; )
bundle with fiber M and associated principal ( ; )bundle P . Let OE be a self*
*map
of 1 M+ . Then
o(P~*i*OE) ' ~P*i*(o(OE)) : SB ! 1B(E, p)+ .
Therefore, taking OE = idand applying r!, r :B ! *,
oE ' r!~P*i*oM : 1 B+ ! 1 E+ .
This result gives a clear and precise comparison between the specialization
to bundles of the globally defined transfer map for Hurewicz fibrations and the
fiberwise transfer map for bundles. Effectively, we have inserted the transfer *
*map
for M+ fiberwise into P x () to obtain an alternative description of the tran*
*sfer
map for the dualizable Gspectrum 1 (E, p)+ over B.
There is a useful reinterpretation of the description of transfer maps given
by Theorem 15.5.3. Consider ss :P  ! *. Observe that, by Proposition 14.4.4,
instead of applying r!, r :B ! *, to orbit spectra under the action of , we c*
*ould
first apply ss! and then pass to orbits. For a spectrum X, we have a natural
isomorphism
ss!ss*i*X ~=P+ ^ i*X
and a natural equivalence
i*(P+ ^ i*X) ' P+ ^ X.
Corollary 15.5.4. let M be a compact ENR and let p: E ! B be a ( ; )
bundle with fiber M and associated principal ( ; )bundle P . Then the transf*
*er
oE : 1 B+ ! 1 E+ is obtained by passage to orbits over from the map
"o= id^ i*oM :P+ ^ i*S ! P+ ^ i* 1 M+ ,
and i*"ocan be identified with
id^ oM :P+ ^ S ! P+ ^ 1 M+ .
Remark 15.5.5. The corollary gives exactly the transfer map as defined by
Lewis and May [59, IV.3.1]. Working in the nonparametrized context, they tried
in vain to obtain a spectrum level transfer map for Hurewicz fibrations over ge*
*neral
base spaces. The comparison here also sheds light on the relationship between
the two constructions of Becker and Gottlieb [2, 3], both of which require fini*
*te
dimensional base spaces. The first is bundle theoretic and is easily seen to be
equivalent to the construction in this section by using Atiyah duality to inter*
*pret
oM for a manifold M. Precisely, by [59, IV.2.3], if M is embedded in V with
normal bundle and o is the tangent bundle of M, then the transfer map oM is
homotopic to the map obtained by applying the functor V 1 to the composite of
the PontryaginThom map SV ! T and the map T ! T ( o) ~=M+ ^ SV
212 15. FIBERWISE DUALITY AND TRANSFER MAPS
induced by the inclusion ! o. The second, which is generalized to the
equivariant setting by Waner [97], is fibration theoretic and is easily seen to*
* be
equivalent to the construction of x15.2. Another approach to the comparison is
to show that suitable Hurewicz fibrations are equivalent to bundles, as is done*
* by
Casson and Gottlieb in [15].
Remark 15.5.6. Since our definition coincides with that of [59, IV.3.1], the
properties of the transfer catalogued in [59, IVxx37] apply verbatim. Many of
these properties generalize directly to the parametrized trace and transfer map*
*s of
Theorem 15.5.2. Actually, the definition of [59, IV.3.1] works more generally w*
*ith
P , or rather i* 1 P+ , replaced by a general free spectrum P indexed on VG*
* .
The constructions here admit similar generalizations. One way to achieve this w*
*ith
minimal work is to use the case P = E( ; ) of the construction already on hand.
Thus, for a free spectrum P over F indexed on VG , we can define
~PFi*X = _______E(F;(P)^F i*X)
and develop parametrized trace and transfer maps from there. We leave the furth*
*er
development of the theory to the interested reader.
15.6. Sketch proofs of the compatible triangulation axioms
We must explain why Ho GSB is a closed symmetric monoidal category with
a compatible triangulation, in the sense specified in [74]. We have the closed
symmetric monoidal structure and the triangulation, the latter by Lemma 13.1.5.
We must prove the compatibility axioms (TC1)(TC5) of [74, x4]. The essential
idea is to verify the axioms using external smash products and function objects
and then pull back along diagonal maps to obtain the conclusions. The axiom
(TC1) only involves suspension, in our case B , and is thus easily checked usi*
*ng
Proposition 12.6.4. For (TC2), we must show that the functors X^B (), FB (X, *
*),
and FB (, Y ) preserve distinquished triangles, where X and Y are Gspectra ov*
*er
B. Either model theoretically or by standard topological arguments with cofiber
sequences and fiber sequences, it is easy to see that these conclusions hold wi*
*th
^B and FB replaced by the external functors Z and ~F. Since * and therefore its
right adjoint * are exact, the conclusion internalizes directly. Similarly, th*
*e braid
axiom (TC3) and additivity axiom (TC4) hold for Z by the arguments explained in
[74, x6], and they pull back along * to give these axioms internally in Ho GSB*
* .
The braid duality axiom (TC5) is more subtle because it involves simultaneous
use of ^B and FB . Externally, we can work over B x B, using Z. Inspecting the
argument in [74, x7], we see that the only internal homs used in the verificati*
*on of
the braid duality axiom are duals of the form F (, T ) for a suitable approxim*
*ation
T of the unit object. In our context, it turns out that we need to use two anal*
*ogues
of this functor, one to mimic the proof of (TC5a) given in [74, pp 6264] and
another to mimic the proof of (TC5b) given in [74, pp 6567]. For the first, l*
*et
T 2 GSBxB be a fibrant model of the derived *SB , so that ~F(X, T ) is a model
for DX = FB (X, SB ) in Ho GSB . With this replacement for F (X, T ), the cited
proof of (TC5a) goes through, first working externally and then internalizing a*
*long
*. The cited proof of (TC5b) relies on a natural pointset level map
(15.6.1) F (X, T ) ^ F (Y, T ) ! F (X ^ Y, T ),
15.6. SKETCH PROOFS OF THE COMPATIBLE TRIANGULATION AXIOMS 213
and this makes no sense in our external context. Working internally, in Ho GSB ,
we have such a map
(15.6.2) FB (X, SB ) ^B F (Y, SB ) ! FB (X ^B Y, SB ),
but we need a pointset level external model for it to carry out the cited argu*
*ment.
Let U be a fibrant model for *SBxB in GSBxBxBxB . Replacing the functor
D0() = F (X, T ) used in [74, pp. 6667] with the functor
D0() = ~F(, U): GSBxB ! GSBxB ,
we find that the cited argument goes through verbatim on the external level, wo*
*rk
ing in the category GSBxB , once we construct a natural map
(15.6.3) ~F(X, T ) Z ~F(Y, T ) ! ~F(X Z Y, U)
in GSBxB to substitute for the pairing (15.6.1). Starting from the (Z, ~F) adj*
*unc
tion, we obtain an external pairing
(15.6.4) ~F(X, T ) Z ~F(Y, T ) ! ~F(X Z Y, T Z T ).
We also have the natural map
*X Z *Y
j
fflffl
* *( *X Z *Y )
~=
fflffl
*( * *X Z * *Y )
*("Z")
fflffl
*(X Z Y ).
Applying this with X = Y = SB and using that SB Z SB is isomorphic to SBxB ,
we obtain a lift , in the diagram
*SB Z *SB _____// *(SB Z SB ) ~= *SBxB______//U22eeeeeeee
eeeee
 ,eeeeeeee 
 eeeeeeee 
fflffleeeeeeeeee fflffl
T Z T ___________________________________//_*BxB
Composing ~F(X Z Y, ,) with the pairing (15.6.4), we obtain the required pairing
(15.6.3). Internalization along * is then a not altogether trivial exercise w*
*hich
shows that, on passage to homotopy categories, application of * to the pairing
(15.6.3) gives a model for the pairing (15.6.2). The latter pairing can be view*
*ed as
a map
*(F~(X, T ) Z ~F(Y, T )) ! ~F( *(X Z Y ), T ),
and the essential point of the exercise is to verify that *F~(X Z Y, U) is equ*
*ivalent
to ~F( *(XZY ), T ). Using that *SBxB ~=SB and looking at represented functor*
*s,
we see that a Yoneda lemma argument reduces the verification to the proof of a
derived analogue of (11.4.5) that is proven in the same way as Theorem 13.7.6.
CHAPTER 16
The Wirthm"uller and Adams isomorphisms
Introduction
This chapter consists of variations on a theme. For a Gmap f :A ! B, the
base change functor f* from Gspectra over B to Gspectra over A has a left adj*
*oint
f!and a right adjoint f*. We study comparisons between f!and f*. As preamble,
we show in x16.1 that there is always a natural map OE: f! ! f* that relates t*
*he
two adjunctions. It is an equivalence when f is a homotopy equivalence, but not*
* in
general. This comparison is largely formal and applies to analogous sheaf theor*
*etic
contexts.
In the rest of the chapter, we use our foundations together with formal ar
guments developed in [40] to obtain a simple proof of a general version of the
Wirthm"uller isomorphism and to reprove the Adams isomorphism as a special case.
This material constitutes a considerably simplified version of work of Po Hu on*
* the
same topic [47]. We consider Gbundles p: E ! B, as in x3.2 and x15.3. We
assume that the fiber M is a smooth closed manifold; manifolds with boundary
work similarly. The generalized Wirthm"uller isomorphism computes the relatively
mysterious right adjoint p* of the functor p* as a suitable shift of the relati*
*vely
familiar left adjoint p!.
We explain the result in the special case when E ! B is M ! * in x16.2,
but we defer the proof to x16.5. We also show how to relate the Wirthm"uller
isomorphisms for M and N when N is smoothly embedded in M. When M = G=H,
the result specializes under the equivalence between the category of Gspectra *
*over
G=H and the category of Hspectra to the Wirthm"uller isomorphism in the form
proven by Lewis and May [59, IIx6]. As explained in [75], the categorical analy*
*sis
in [40] allows considerable simplification of that proof. Our proof for general*
* M
follows the same pattern, but it is quite different in detail since the special*
* case
M = G=H has certain simplifying features. For example, when G is finite, that
case follows formally from Atiyah duality for G=H and the trivial observation t*
*hat
H=K+ is an Hretract of G=K+ for K H G.
In x16.3, we show that the general case of Gbundles p: E  ! B reduces
fiberwise to the special case M ! *. The proof is an immediate application of *
*the
construction P x () on parametrized spectra that was studied in x15.3. This
allows a simple fiberwise construction of the Gspectrum over E by which one mu*
*st
shift p!to obtain the desired isomorphism. With this construction, it is immedi*
*ate
that the map of Gspectra over B that we wish to prove to be an equivalence
coincides on the fiber over b with a map that we know to be an equivalence by t*
*he
case M ! *. Since equivalences are detected fiberwise, that proves the result.
In turn, we prove in x16.4 that the Adams isomorphism relating orbit spectra
and fixed point spectra that was proven by Lewis and May in [59, IIx8] is a vir*
*tually
214
16.1. A NATURAL COMPARISON MAP f!! f* 215
immediate special case of our generalized Wirthm"uller isomorphism. These resul*
*ts
complete the program originated in [61] of reproving conceptually all of the ba*
*sic
foundational results that were first proven in a less satisfactory ad hoc way i*
*n [59].
The pioneering work of Po Hu [47] paved the way but, in the absence of adequate
foundations, the bundle construction of x15.3, and the simplifying framework of
[40], her arguments were very long and difficult. Our work recovers variant ver*
*sions
of all of her results. The basic idea that parametrized Gspectra should clarif*
*y and
simplify the Wirthm"uller and Adams isomorphisms is due to Gaunce Lewis [58].
Again, we assume throughout that all given groups G are compact Lie groups
and all given base Gspaces are of the homotopy types of GCW complexes.
16.1.A natural comparison map f!! f*
The Wirthm"uller isomorphism that is the subject of the next few sections gi*
*ves
an equivalence between f* and a shift of f! for certain equivariant bundles f. *
*In
the course of our work on that, we came upon a curious natural comparison map
f!! f* for any map f whatever. We have no current applications for it, but sin*
*ce
the relationships among base change functors are so central to the theory and i*
*ts
applications, we shall describe that map in this digressive section. It works j*
*ust as
well on the level of exspaces and indeed quite generally in other contexts whe*
*re
one has analogous base change adjunctions.
Theorem 16.1.1. Let f :A ! B be a Gmap and let X be a Gspectrum
over A. Let ": f*f* ! Idand oe :f!f* ! Iddenote the counits of the adjunc
tions (f*, f*) and (f!, f*) relating Ho GSA and Ho GSB . There is a natural map
OE: f!X ! f*X in Ho GSB such that the following diagram commutes:
(16.1.2) f!f*f*XJ
f!"uuuu JJJoeJ
uu JJJ
zzuuu $$J
f!X_________OE_______//f*X.
Proof. Let K = (K, p, s) be an exspace over A. Then f!K = K [A B and
f*f!K = f!K xB A. Here points (f(a), a) in B xB A are identified with points
(s(a), a) in K xB A, and we see that f*f!K can be identified with the pullback
K xB A. The projection to K is then a map _ :f*f!K ! K of exspaces over
A. When K = f*L for an exspace L over B, _ = f*oe :f*f!f*L ! f*L since
f*oe is also given by the projection f*L xB A ! f*L. Passing to spectra over A
levelwise, we obtain a natural map _ :f*f!X ! X of spectra over A such that
_ = f*oe when X = f*Y .
To pass to homotopy categories, we take two steps. Factoring f as a composite
of a homotopy equivalence and an hfibration, we see that we may assume that f
is either a homotopy equivalence or an hfibration. In the former case, f* must
be inverse to the equivalence f* and thus equivalent to f!. Here " and oe are
equivalences and we may as well define OE by the commutativity of (16.1.2). In *
*the
latter case, we may work in GEA . Since f is an hfibration, we have a natural
homotopy equivalence ~: T f*Y ! f*T Y for Y 2 GEB . The derived functor f!is
induced by T f!, and T X is naturally homotopy equivalent to X when X 2 GEA .
The composite
f*T f!X ' T f*f!X_T__//T X ' X
216 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS
gives a natural map _ :f*f!X ! X in hGEA . When X = f*Y , we have the
commutative naturality diagram
*oe
T f*f!f*Y__T_=Tf____//T f*Y
~ ~
fflffl fflffl
f*T f!f*Y___f*Toe__//_f*T Y.
The bottom arrow is the derived version of f*oe and the composite around the
top is the derived version of _. Using the equivalences of categories of x13.5,
we obtain a natural map _ :f*f!X ! X in Ho GSA . Let j :Id ! f*f* be
the unit of the (derived) adjunction (f*, f*) and define OE: f!X  ! f*X to be
the adjoint of _ in Ho GSA , so that OE = f*_ O j. For Y 2 Ho GSB , we have
f*oe = _ :f*f!f*Y ! f*Y . It follows formally that (16.1.2) commutes. Indeed,
" O f*oe = " O _ = _ O f*f!".
The adjoint of " O f*oe is oe since
f*(" O f*oe) O j = f*" O f*f*oe O j = f*" O j O oe = oe,
while the adjoint of _ O f*f!" is OE O f!" since
f*(_ O f*f!") O j = f*_ O f*f*f!" O j = f*_ O j O f!" = OE O f!".
16.2. The Wirthm"uller isomorphism for manifolds
The classical Wirthm"uller isomorphism in the equivariant stable homotopy ca*
*t
egory relates induction and coinduction, the left and right adjoints of the res*
*triction
functor from Gspectra to Hspectra. More precisely, it says that for Hspectra*
* X,
there is a natural equivalence of Gspectra
(16.2.1) FH (G+ , X) ' G+ ^H (X ^ SL ),
where L is the tangent representation at the identity coset in G=H and SL is
the inverse of the invertible Hspectrum 1 SL . Here again, "equivalence" means
isomorphism in the relevant stable homotopy category and is denoted by '.
One can also think of this in terms of base change functors. Recall from Cor*
*ol
lary 11.5.4 that the category of Hspectra is equivalent to the category of Gs*
*pectra
over G=H. The equivalence is given in one direction by applying the functor GxH*
* ,
and in the other by taking the fiber over the identity coset. This equivalence *
*pre
serves all structure in sight, including the symmetric monoidal and model struc
tures. The map r :G=H ! * induces a pullback functor r* from Gspectra to
Gspectra over G=H, and it has left and right adjoints r! and r*. The functor r*
corresponds under the equivalence to the restriction functor and therefore r! a*
*nd
r* correspond to the induction and coinduction functors. In this terminology, t*
*he
Wirthm"uller isomorphism (16.2.1) takes the form
(16.2.2) r*X ' r!(X ^G=H CG=H )
for Gspectra X over G=H, where CG=H = '!SL , ' : H G (see Proposi
tion 11.5.2).
We think of G=H ! * as the simplest kind of a bundle with a compact man
ifold as fiber, and we generalize (16.2.2) to maps p: E ! B that are equivaria*
*nt
bundles with a smooth closed manifold M as a fiber. We discuss the case B = *
16.2. THE WIRTHM"ULLER ISOMORPHISM FOR MANIFOLDS 217
in this section and prove the general case in the next. However, it is convenie*
*nt to
begin by describing the form of the map that gives the equivalence in general. *
*For
that, we require a Gspectrum Cp over E together with an equivalence
(16.2.3) ffp: p!Cp_'__//D(p!SE )
that identifies the dual of p!SE . We call Cp, together with ffp, a Wirthm"ulle*
*r object.
In [40], Fausk, Hu, and May give a categorical discussion of equivalences of
Wirthm"uller type, including a simplifying formal analysis that describes the m*
*in
imal amount of information that is needed to prove such a result. In particula*
*r,
given a Wirthm"uller object Cp, they define a canonical candidate
!p: p*X ! p!(X ^E Cp)
for an equivalence, namely the composite displayed in the commutative diagram
id^BD(oe)
(16.2.4) p*X ' p*X ^B D(SB ) __________//_p*X ^B D(p!SE )
 OO
 ' id^Bffp
 
!p  p*X ^B p!Cp
 OO
 
 '
fflffl 
p!(X ^E Cp)oo___p_________ p!(p*p*X ^E Cp).
!("^Eid)
The maps oe :p!SE ' p!p*SB ! SB and ": p*p*X ! X are given by the counits
of the adjunctions (p!, p*) and (p*, p*). The arrow labelled ' is an equivalen*
*ce
given by the derived version of the projection formula (11.4.5) that is proven *
*in
Theorem 13.7.6.
When M is a smooth closed Gmanifold and r is the map :M ! *, we write
CM for a Wirthm"uller object Cr and we write !M for !r. It is easy to describe *
*CM .
Let o be the tangent Gbundle of M. Embed M in a Grepresentation V and let
be the normal Gbundle of the embedding. By Atiyah duality, the union M+ of M
and a disjoint basepoint is V dual to the Thom Gspace T . A detailed equivar*
*iant
proof is given in [59, IIIx5], but we require little beyond the mere statement.
For a Gvector bundle , over a Gspace B, let S, denote the fiberwise onepo*
*int
compactification of ,, with section given by the points at infinity. This exG*
*space
over B must not be confused with the Thom complex T ,. The latter is obtained
by identifying the section to a point and is precisely r!S,, r :B ! *.
Definition 16.2.5. Define CM to be the Gspectrum VM 1MS over M.
Remark 16.2.6. By Theorem 15.1.5, the suspension Gspectrum 1MSo is in
vertible. Visibly, CM is its inverse.
Lemma 16.2.7. There is an equivalence ffM :r!CM  ! D(r!SM ), r :M ! *.
Proof. Since SM (V ) = MxSV , r!SM = 1 M+ . Since r!S = T and r!com
mutes with shift desuspension functors, r!CM is equivalent to V 1 T . Ther*
*e is
a canonical evaluation map ev:T ^M+ ! SV of a duality [59, p.152]. Explicitl*
*y,
using the diagonal of M and the zero section of we obtain an embedding of M in
x M with trivial normal bundle M x V , and evis composite of the Pontryagin
Thom map associated to this embedding and the projection M+ ^ SV ! SV . We
218 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS
apply the functor V 1 to obtain
V 1 T ^ 1 M+ ' V 1 (T ^ M+ ) ! V 1 SV ' S.
Atiyah duality states that the adjoint of this map is an equivalence from V *
*1 T
to D(M+ ). This is the required map ffM .
We shall prove the following result in x16.5.
Theorem 16.2.8 (The Wirthm"uller isomorphism for manifolds). For Gspectra
X over M and r :M ! *, the map
!M :r*X ! r!(X ^M CM )
is a natural equivalence in the homotopy category Ho GS of Gspectra.
In an earlier draft of this paper, we thought we could reduce the general ca*
*se
of Theorem 16.2.8 to the special case M = G=H. However, instead of leading to a
simplifiction, the argument we had in mind leads to an interesting relative ver*
*sion
of the Wirthm"uller isomorphism. Its starting point is the following observatio*
*n.
Lemma 16.2.9. Let i: N ! M be an embedding of smooth closed Gmanifolds
and let M,N be the normal bundle of i. Then CN is equivalent to S M,N ^N i*CM .
Proof. An embedding of M in a representation V restricts along i to an
embedding of N in V , and i* M M,N ~= N . Commutation relations in Proposi
tion 13.7.4 give that
i* VM 1MS M ' VN 1Ni*S M .
The conclusion follows after smashing with S M,N.
Corollary 16.2.10 (The relative Wirthm"uller isomorphism). Let i: N ! M
be a smooth embedding of closed Gmanifolds. For Gspectra X over N, there is a
natural equivalence
!M,N :r*i*X ! r*i!(X ^N S M,N).
Proof. Here r :M ! *. Write q = r O i: N ! *. Then q* ' r*i* and
q!' r!i!. Define !M,N by commutativity of the diagram of equivalences
q*X _________!N______//q!(X ^N CN )
'  '
fflffl fflffl
r*i*X r!i!(X ^N S M,N ^N i*CM )
!M,N  '
fflffl fflffl
r*i!(X ^N S M,N)_!M__//r!(i!(X ^N S M,N) ^M CM ).
Here the derived version of the projection formula (11.4.5) gives the lower rig*
*ht
equivalence.
We explain the strategy that we have not implemented for deducing the Wirth
m"uller isomorphism for M from the Wirthm"uller isomorphism for orbits.
16.3. THE FIBERWISE WIRTHM"ULLER ISOMORPHISM 219
Remark 16.2.11. One can use relative Atiyah duality to define an intrinsic
map ffM,N and use ffM,N to define a map !M,N directly. One can then obtain
the displayed diagram by a chase. If one could prove directly that !M,N was an
equivalence, then, using the invertibility of 1NS M,N, one could deduce that !M
is an equivalence on all i!X if !N is an equivalence. By Proposition 13.1.3, *
*!M
is an isomorphism for all Y if it is an isomorphism for all Y in the detecting *
*set
DM of Definition 13.1.1. Those Y , namely the Sn,bHfor b 2 M and H Gb, are of
the form "b!X, where "b:G=Gb ! M is the inclusion of the orbit of b and X is a
Gspectrum over G=Gb. Thus the Wirthm"uller isomorphism for orbits would imply
the Wirthm"uller isomorphism for M.
16.3. The fiberwise Wirthm"uller isomorphism
As in x15.3, let G be a quotient = , where is a normal subgroup of a
compact Lie group . Let M be a smooth closed manifold and let p: E ! B be
a ( ; )bundle with fiber M. This means that p has an associated principal ( ;*
* )
bundle ss :P ! B and p is the associated Gbundle E = P x M ! P= = B.
We apply the functor ~PM to the Wirthm"uller object CM to obtain the Wirthm"ull*
*er
object Cp, and we apply ~PM to ffM to obtain the required equivalence ffp. This
means that the Wirthm"uller object for p is obtained by inserting the Wirthm"ul*
*ler
object for M fiberwise into the functor P x ().
Definition 16.3.1. Define Cp to be the Gspectrum ~PMi*CM over E, where
i* is the change of universe functor associated to the inclusion of the trivi*
*al
universe in the complete universe.
Remark 16.3.2. Recall Remark 16.2.6. By Theorem 15.1.5, the suspension
Gspectrum 1E(P x So) is invertible. The Wirthm"uller object Cp is its invers*
*e.
Lemma 16.3.3. There is an equivalence ffp: p!Cp ! D(p!SE ).
Proof. We define ffp to be the composite
(16.3.4)
~P*i*ffM
p!~PMi*CM_____//~P*i*r!CM_________//_~P*i*D(r!SM_)_//_D(p!SE ).
The left arrow is given by the first equivalence of Proposition 15.3.4 and the *
*last
equivalence of Proposition 14.2.8. The middle arrow is an equivalence since ffM*
* is
one. The right arrow is the following composite equivalence,
~P*i*D(r!SM )' ~P*D(i*r!SM )
' ~P*D(r!i*SM) by Propositions 14.2.8 and 15.4.5
' D(P~*r!i*SM) by Theorem 15.3.3
' D(p!~PMi*SM) by Proposition 15.3.4
' D(p!SE ) by Theorem 15.3.3.
For the first displayed equivalence, r!SM ' 1 M+ by Lemma 15.2.8, hence
D(r!SM ) ' D( 1 M+ ) ' F (M+ , S).
For based spaces T , i*F (T, S) ~= F (T, i*S) by inspection. If T is a based*
* 
CW complex, this is an isomorphism of Quillen right adjoints and so descends to
220 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS
homotopy categories. Again by inspection,
i* 1 ~= 1 :GK* ! GS triv.
This isomorphism passes to homotopy categories since both sides take qequiva
lences to level qequivalences. Therefore i*S ' S and F (T, i*S) ' D(i* 1 T ).
Theorem 16.3.5 (The fiberwise Wirthm"uller isomorphism). For Gspectra X
over E, the map
!p: p*X ! p!(X ^E Cp)
is a natural equivalence of Gspectra over B.
Proof. The action of Gb on the fiber Eb ~=ae*bM of b 2 B is smooth, hence
the Wirthm"uller isomorphism for manifolds gives the result for r :Eb ! *. We
claim that the restriction ffb of ffp to the fiber over b is an equivalence
ffEb: r!CEb ! D(r!SEb)
of Gbspectra of the form used to prove Theorem 16.2.8 for r. Indeed, with pb =*
* r,
i: Eb E and ': Gb G, the derived version of Example 11.5.6 and Lemma 15.3.2
give that the source of ffb is
(p!~PMCM )b ' r!i*'*P~MCM ' r!ae*bCM ~= r!CEb.
For the last isomorphism we must view the representation V of that appears in
the definition of CM as a representation of Gb by pullback along aeb. Similarly*
*, using
Theorem 13.7.10 and the derived version of Example 11.5.6, the target of ffb is
D(p!SG,E)b ' D((p!SG,E)b) ' D(r!i*'*SG,E) ' D(r!SGb,Eb).
In view of the role of ffM in the definition of ffp, diagram chases from the de*
*finitions
show that ffb agrees under these equivalences with the Gbequivalence ffEb.
Now, looking at the definition of !p (16.2.4), we see that, aside from the e*
*quiv
alence ffp, its constituent maps are just counits of adjunctions and derived is*
*omor
phisms coming from the closed symmetric monoidal structures. By Theorem 13.7.10
and the derived versions of commutation relations in Example 11.5.5, these maps
restrict on fibers to maps of the same form. Therefore the restriction
!b:(p*X)b ! (p!(X ^E Cp))b
of !p to the fiber over b can be identified with the map of Gbspectra
!Eb: r*Xb ! r!(Xb^Eb CEb).
This map is an equivalence of Gbspectra by Theorem 16.2.8. Since equivalences *
*of
Gspectra over B are detected fiberwise, this implies that !p is an equivalence.
Remark 16.3.6. When = G x and only acts on M, one can think of
p: E ! B as a topological Gbundle with a reduction of its structural group to
a suitably large compact subgroup of the group of diffeomorphisms of M. Our
fiberwise Wirthm"uller isomorphism theorem is a variant of the main theorem, [4*
*7,
4.8], of a paper of Po Hu. She worked with Diff(M) itself as an implicit struct*
*ure
group, without use of an auxiliary group and without an ambient group . That
bundle theoretic framework leads to formidable complications, hence her argumen*
*ts
are very much more difficult than ours. Her result is both more and less general
than the specialization of ours to the case = G x : it allows bundles that m*
*ight
not admit a single compact structure group , but it requires the base spaces to
16.4. THE ADAMS ISOMORPHISM 221
be GCW complexes with countably many cells. It does not handle more general
group extensions.
16.4.The Adams isomorphism
Let N be a normal subgroup of G and let ffl: G ! J be the quotient by N. The
conjugation action of G on N induces an action of G on the tangent space of N at
the identity element, giving us the adjoint representation A = A(N; G). Let (i**
*, i*)
be the change of universe adjunction associated to the inclusion i: q*VJ ! VG
of the complete Juniverse VJ as the universe of Ntrivial representations in t*
*he
complete Guniverse VG .
Recall the discussion of Nfree Gspectra from x15.4, where and played
the roles of N and G.
Theorem 16.4.1 (Adams isomorphism). For Nfree Gspectra X in GS Ntriv,
there is a natural equivalence
X=N ' (i* A i*X)N
in Ho JS Ntriv.
We shall derive this by applying the fiberwise Wirthm"uller isomorphism to t*
*he
quotient Gmap p: E(N; G) ! B(N; G), where E(N; G) is the universal Nfree
Gspace and B(N; G) = E(N; G)=N. To place ourselves in the bundle theoretic
context of the previous section, we give another description of p, following [6*
*8, IIx7].
It is formal and would similarly identify p: E ! E=N for any Nfree Gspace E.
Let = GnN be the semidirect product of G and N, where G acts by conjugation
on N. Write for the normal subgroup {e} n N of . We then have an extension
1 ! ! `!G ! 1,
where `(g, n) = gn. Give N the action (g, n) . m = gnmg1. Then N ~= =G as
spaces, where we view G as the subgroup G n {e} of . The composite
E(N; G) ~=`*E(N; G) x ( =G) ! `*E(N; G) x * ~=B(N; G)
induced by =G ! * is p. Since `*E(N; G) is a free space, we see that p
is a bundle with fiber =G ~=N to which the fiberwise Wirthm"uller isomorphism
applies. We must identify the relevant Wirthm"uller object. We write r for the *
*map
E(N; G) ! *.
Proposition 16.4.2. The Wirthm"uller object Cp is r*SA .
Proof. The tangent bundle of =G ~= N is the trivial bundle N x A [59,
p. 99]. Indeed, let act on A via the projection ffl: ! G, "(n, g) = g. We
obtain a trivialization of the tangent bundle of =G by sending (n, a) 2 N x A
to deLn(a), where deLn is the differential at e of left translation by n. It fo*
*llows
that the tangent bundle along the fibers of p is also trivial:
`*E(N; G) xN ( =G x A) ~=(`*E(N; G) xN =G)) x A ~=E(N; G) x A.
Thus the spherical bundle of tangents along the fiber is E(N; G) x SA = r*SA ,
and the inverse of its suspension Gspectrum over E(N; G) is r*SA . In view of
Remark 16.3.2, this gives the conclusion.
222 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS
Proof of the Adams isomorphism. Let X 2 GS Ntrivbe Nfree. Apply
ing the fiberwise Wirthm"uller isomorphism to the Gspectrum r*i*X over E(N; G)
and using that Cp is r*SA , we obtain a natural equivalence
p*r*i*X ' p!(r*i*X ^E(N;G)r*SA )
of Gspectra over B(N; G). Write ~rfor the map B(N; G) ! *, so that ~rO p = r.
Applying the functor ~r!((i*())N ) to the displayed equivalence, we obtain a n*
*atural
equivalence
~r!((i*p*r*i*X)N ) ' ~r!((i*p!(r*i*X ^E(N;G)r*SA ))N )
in Ho JS Ntriv. We proceed to identify both sides. The source is
~r!((i*p*r*i*X)N')~r!((p*r*i*i*X)N ) by Proposition 14.2.8
' ~r!((p*r*X)N ) by Proposition 15.4.5
' ~r!((p!r*X)=N) by Proposition 14.4.5
' (~r!p!r*X)=N by Proposition 14.4.4
' (r!r*X)=N by functoriality
' X=N. by Definition 15.4.1.
The target is
~r!((i*p!(r*i*X^E(N;G)r*SA ))N )
' ~r!((i*p!r* A i*X)N )by Theorem 13.7.3
' (~r!i*p!r* A i*X)N by Proposition 14.4.4
' (i*~r!p!r* A i*X)N by Propositions 14.2.8 and 15.4.5
' (i*r!r* A i*X)N by functoriality
' (i* A i*X)N by Definition 15.4.1.
Remark 16.4.3. In outline, the proof just given is essentially that indicated
by Po Hu [47, pp 8199]. However, her argument, although more conceptual, is a
good deal longer and more complicated than the original proof in [59, pp 96102*
*].
16.5. Proof of the Wirthm"uller isomorphism for manifolds
We prove Theorem 16.2.8 here. Thus consider r :M ! * for a smooth com
pact Gmanifold M. With CM = VM 1MS , the diagram (16.2.4) displays a
canonical map
! = !M :r*(X) ! r!(X ^M CM )
of Gspectra, where X is a Gspectrum over M. We must show that ! is an equiva
lence. In outline, we follow the pattern of proof explained in [40] and illustr*
*ated in
the case M = G=H in [75], but the details are very different from those applica*
*ble
in that special case.
We first describe a formal reduction implied by the results of [40]. Consider
the set DM of detecting objects in Ho GSM that is specified in Definition 13.*
*1.1.
The objects in DM are compact, by Lemma 13.1.2, and dualizable. We have the
analogous detecting set D* of compact objects in Ho GS . For Y in D*, r*Y is
dualizable and it follows formally, by [45, 2.1.3(d)], that r*Y is compact (in *
*the
sense of Lemma 13.1.2). Therefore r*, as well as r!, preserves coproducts [40, *
*7.4].
16.5. PROOF OF THE WIRTHM"ULLER ISOMORPHISM FOR MANIFOLDS 223
This verifies the hypotheses of the formal Wirthm"uller isomorphism theorem, [4*
*0,
8.1], and that result shows that ! will be an equivalence for all Gspectra X o*
*ver
M if it is an equivalence for those X in DM .
Such X are of the form Sn,mH= em!'!SnH, where n 2 Z, m 2 M, H Gm , and '
is the inclusion of Gm in G. By commutation with suspension, we can assume that
n 0. Then X is of the form 1MK for an exGspace K over M, and X can be
any such Gspectrum over M in the rest of the proof. By [40, 6.3], it suffices *
*to
construct a map ,X: r*r!(X ^M CM ) ! X such that certain diagrams commute.
To be precise, let oe and i be the counit and unit of the (r!, r*) adjunction, *
*note
that r*S ~=SM , and define maps o = oS and , = ,SM by commutativity of the
diagrams
(16.5.1) S ___o__//_r!CM and r*r!CM _________,_________//_SM
'  ffM r*ffM '
fflffl fflffl fflffl fflffl
DS _Doe//_Dr!r*S r*Dr!SM __'__//Dr*r!SM_Di_//_DSM
Then define oY for a general Gspectrum Y to be the composite
(16.5.2) oY :Y ' Y ^ S id^o//_Y ^ r!CM ' r!(r*Y ^M CM )
and define ,r*Y for the Gspectrum r*Y over M to be the composite
(16.5.3) ,r*Y: r*r!(r*Y ^M CM ) ' r*Y ^M r*r!CMid^,//_r*Y ^M SM ' r*Y.
Here the equivalences are given by the derived versions of (11.4.2) and the pro*
*jection
formula (11.4.5). With these notations, we shall prove the following result.
Proposition 16.5.4. For X = 1MK, there is a map
,X :r*r!(X ^M CM ) ! X
such that the composite
(16.5.5) r!(X ^M CM )
or!(X^MCM)
fflffl
r!(r*r!(X ^M CM ) ^M CM )
r!(,X^Mid)
fflffl
r!(X ^M CM )
is the identity map (in Ho GS ) and, for any map ` :r*Y ! X of Gspectra over
M, the following diagram commutes in Ho GSM .
,r*Y
(16.5.6) r*r!(r*Y ^M CM )____//r*Y
r*r!(`^id) `
fflffl fflffl
r*r!(X ^M CM )__,X___//X
This will complete the proof of the theorem by the cited reduction from [40].
224 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS
Corollary 16.5.7. For X in DM , !M :r*(X) ! r!(X ^M CM ) is an equiv
alence with inverse the adjoint of ,X .
Proof. Taking Y to be r*X and ` to be the counit of the (r*, r*) adjunction
in (16.5.6), the conclusion is a direct application of [40, 6.3].
Thus it suffices to prove Proposition 16.5.4. We shall construct the map ,X *
*and
prove that it satisfies the stated properties by reducing to space level consid*
*erations.
We begin with a space level description of the maps o and , displayed in (16.5.*
*1),
and we need some space level notations.
Notations 16.5.8. Recall that denotes the normal bundle of M and that we
have the duality map ev:T ^ M+ ! SV specified in the proof of Lemma 16.2.7.
Also, recall that
r!K = K=s(M), r*T = TM = M x T, and r*K = Sec(M, K)
for any based Gspace T and any exGspace (K, p, s) over M. In particular,
r!S = T , r!S0M= M+ , and r*r*T ~=F (M+ , T ).
Therefore the adjoint eev:T ! F (M+ , SV ) is a map r!S  ! r*SVM. Let
t: SV ! r!S be the PontryaginThom construction and k be the composite
*"ev "
k :r*r!S_r___//r*r*SVM___//_SVM,
where ": r*r* ! idis the counit of the adjunction (r*, r*); note that, in gene*
*ral,
" is just the evaluation map M x Sec(M, K) ! K.
Recall from Propositions 13.7.4 and 13.7.5 that we can commute suspension
spectrum functors past smash products and base change functors.
Lemma 16.5.9. With these definitions of t and k,
o ' V 1 t: S ~= V 1 SV ! V 1 r!S ' r!CM
and
, ' VM 1Mk :r*r!CM ' VM 1Mr*r!S ! VM 1MSVM' SM .
Proof. By [59, III.5.2], the dual of t is the projection ffi :M+ ! S0. This
means that the following diagram is stably homotopy commutative.
SV ^ M+ _t^id//_T ^ M+
id^ffi ev
fflffl fflffl
SV ^ S0 _________SV
Here ffi :M+ = r!r*S0 ! S0 is the counit of the space level adjunction (r!, r**
*),
and we can identify 1 ffi with the counit oe : 1 M+ ~=r!r*S ! S of the spectr*
*um
level adjunction (r!, r*). Applying V 1 to the diagram and passing to adjoin*
*ts,
the right vertical arrow becomes
ffM :r!CM ' V 1 T ! D(M+ ) = Dr!r*S,
by the proof of Lemma 16.2.7. Comparing the resulting diagram with the diagram
that defines o, we conclude that o ' V 1 t.
For the identification of ,, we consider the composite equivalence r*r!CM '
Dr*r!SM in the diagram that defines , to be an identification of the dual of r**
*r!SM .
16.5. PROOF OF THE WIRTHM"ULLER ISOMORPHISM FOR MANIFOLDS 225
To identify the dual of i modulo that identification, we observe that the follo*
*wing
diagram is commutative.
r*r!S ^M S0M___________k^id__________//_SVM^M S0M
id^i 
fflffl 
r*r!S ^M r*r!S0M~=_//_r*(r!S ^ r!S0M)r*ev//_r*SV
Indeed, recalling that k = " O r*e"vand rewriting the diagram in more familiar
notation, it becomes
M x T __idxe"v//_M x Sec(M, M x SV )
idxi "
fflffl fflffl
M x (T ^ M+ )___idxev__//_M x SV ,
and both composites send (m, x) to (m, ev(x ^ m)). Applying V 1 to the first
diagram and comparing with the definition of ,, we conclude that , ' VM 1Mk.
The following space level result will imply Proposition 16.5.4.
Proposition 16.5.10. Let K be a Gspace over M. There is a natural map
uK :r*r!(K ^M S ) ! VMK
in Ho GKM which satisfies the following properties.
(i)When K = S0M, uK ' k :r*r!S ! SVM.
(ii)For a based Gspace T , the following diagram commutes in Ho GKM .
r*r!(TM ^M K ^M S ) __'__//TM ^M r*r!(K ^M S )
uTM ^M K id^uK
fflffl ~= fflffl
VM(TM ^M K) ____________//TM ^M VMK
Here the top equivalence is given by (11.4.2) and (11.4.5) in Ho GKM .
(iii)The following diagram commutes in Ho GK*.
r!(K ^M S ) ^ SV_____id^t____//r!(K ^M S ) ^ r!S
'  '
fflffl fflffl
r!( VMK ^M S )oo_r!(uK_^idr!(r*r!(K)^M_S ) ^M S )
Here the right vertical equivalence is given by (11.4.5) in Ho GKM . The le*
*ft
vertical equivalence is the composite
r!(K ^M S ) ^ SV ' r!(K ^M S ^M SVM) ' r!(K ^M SVM^M S ),
where the second equivalence is obtained by moving the copy of So from SVM~=
So ^M S and amalgamating it with the displayed copy of S .
226 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS
Proof of Proposition 16.5.4. Let X = 1MK. Define ,X to be the map
VM 1MuK V
r*r!(X ^M CM ) ' VM 1Mr*r!(K ^M S )_________//_ M 1M VMK ~=X.
Using Proposition 16.5.10(ii), we see that , VMX can be identified with VM,X ,*
* which
in turn can be identified with 1MuK . To show that the diagram (16.5.6) commut*
*es,
it suffices to show that the diagram obtained from it by applying VM commutes.
We have just identified the lower horizontal arrow of the resulting diagram in *
*space
level terms. Similarly, the definition (16.5.3) of its upper horizontal arrow, *
*together
with Lemma 16.5.9 and Proposition 16.5.10(i), identifies its upper horizontal a*
*rrow,
with Y serving as a dummy variable. More explicitly, using the projection formu*
*la
(11.4.5), we see that the diagram can be rewritten as
id^uS0M
r*r!(r*Y ^M S )_'__//r*Y ^M r*r!S____//_r*Y ^M SVM
r*r!(`^id) `^id
fflffl fflffl
r*r!(X ^M S ) _________1______________//_X ^M SVM
M uK
Consider the dummy variable Y levelwise. We see from the case K = S0M of
Proposition 16.5.10(ii) that, at level V , the top row is the map ur*Y (V.)Ther*
*efore
the diagram commutes levelwise by the naturality of u.
To prove that the composite (16.5.5) is the identity map, we apply V to it.*
* To
abbreviate notation, write Y = r!(X ^M CM ) and consider the following diagram.
V(id^o) V
V Y____________________________________//UUU (Y ^ r!CM )
UUUUU Vo lll
 UUUYUU lllll '
 UUUUU llll' 
 Vr!(,X^M id) **U uull fflffl
V Yoo____________ V r!(r*Y ^MSCM ) Y ^ r!S
SSS 
'  SS'SSSSS '
fflffl SSS)) fflffl
r!(X ^M S ) oo__________r!(,X^M_id)_________r!(r*Y_^M S )
We must prove that the triangle at the upper left commutes. The arrows marked
' are given by (11.4.5) and the evident equivalence VMCM ' 1 S . The upper
triangle commutes by the definition of oY in terms of o, the bottom trapezoid
commutes by naturality, and the triangle at the right commutes by inspection
of projection formula isomorphisms. Thus it suffices to prove that traversal of*
* the
perimeter gives a commutative diagram, and this will hold for X if it holds for*
* VMX.
In that case, we see from Lemma 16.5.9, Proposition 16.5.10(ii), and a diagram
chase that the perimeter agrees with the diagram that is obtained by applying t*
*he
suspension spectrum functor to the diagram in Proposition 16.5.10(iii).
The proof of Proposition 16.5.10 is based on the following construction of a
natural map
wK :r*r!K ! "K^M So
that will give rise to the required map uK . Here K" is a suitably "fattened u*
*p"
version of K.
16.5. PROOF OF THE WIRTHM"ULLER ISOMORPHISM FOR MANIFOLDS 227
Construction 16.5.11. As in [77, 11.5], identify the tangent bundle of M with
the normal bundle of the diagonal embedding M ! M x M. Let U be a tubular
neighborhood of the diagonal. Let pr1and pr2be the projections M x M ! M
and let ssi:U ! M be their restrictions to U. For an exspace (K, pK , sK ) ov*
*er
M, consider the following diagram of retracts, where is the diagonal and ' is*
* the
inclusion of U in M x M. Note that ssi= priO ' and define "K= (ss1)!ss*2K.
pr2
M _______//UUUUUUUUUU'//_GGM_x_M______//_NMF
 UUUUUUUUUUUUUss1GGGNNpr1NN  FFrFF
 UUUUUU##___________N''_ r  ""//
  M ______________M ___________*
      
fflffl fflffl* fflffl  fflffl
K ______//ss2K___E_//_M xMK__________//KD 
EEE MMMM  DD 
  E"fflffl" M&&fflffl !!fflfflD
  "K  r*r!K________//r!K
      
fflffl fflffl fflffl  fflffl
M _______//UUUUUUUU_G//_M x_M___N___//_MF 
UUUUUUUUUUUGGG NNN  FF 
UUUUUUUfflfflUUU##G NN'fflffl' F"fflffl"F
M _______________M __________//_*
The floor and ceiling of the diagram are identical. The back wall is formed by *
*pulling
K back along the maps of base spaces and then the front wall is obtained from t*
*he
back wall by applying lower shriek functors. Here we have used the canonical
isomorphism pr1!(M x K) = pr1!pr*2K ~=r*r!K associated to the pullback square
on the right side of the floor. Since the ssi are homotopy equivalences, the ma*
*ps
(16.5.12) K ! ss*2K ! "K
at the left are equivalences when K is qcofibrant and qfibrant, and we denote*
* the
displayed composite equivalence as ~.
To get a better feeling for the spaces in the diagram, we make the following
schematic picture.
____________________________O_______
fflfflO _________ffflfl__________
fflffl ____________ffflfl____________
fflfflO ________________ffflfl__________
fflO ____________________ffl____fflfflU
fflfflO____________________ffflfl______fflfflUm
fflfflO____________________ffflfl___
fflffl______________________fflffl_____________
 O    
 O    
 O    
   *  
 O  ss2KUm  K 
 O    
 O    
 ffl```` ``` ` ` ``_``_`_ffflfl`_____________
 ffl____________________ffflfl___________________jj
 fflfflffl _______________ffflfl_______________fflffljj*
*jjj
 fflffl___________________fflffl_________fflfflfflfflffl*
*jjjjjj
 ffl_______________________fflffl______fflfflfflfflfflfflj*
*jjjjj
ffl_______________________fflfflfflfflfflffljjjjjM
____________________________fflffl________fflfflfflfflfflff*
*ljjjjjjjj
____________________________fflfflfflffljjjjjM
The cube represents M x K with M in the horizontal direction and K in the other
two directions. It is sitting over M x M with vertical projection and we think *
*of
the top face as being the image of the section. We can view ss*2K as the subspa*
*ce of
228 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS
M x K sitting over the neighborhood U of the diagonal in M x M. Passing to the
front face of the diagram, the fibers of "Kover points m in M are the slices ss*
**2KUm
of ss*2K as displayed with basepoint obtained by identifying the fiber Um = ss*
*11(m)
over m in the bundle U to a single point. We therefore think of "Kas a fattening
of the space K that sits over the diagonal in M x M.
For a sub Gspace A of a Gspace K over M, not necessarily sectioned, passage
to fiberwise quotients gives an exGspace K=M A over M with total space K [A M.
Given two such pairs (K, A) and (L, B), we obtain a product pair by setting
(K, A) xM (L, B) = (K xM L, K xM B [ A xM L).
Its fiberwise quotient is the exGspace (K=M A) ^M (L=M B) over M.
The pair (M x M, M x M  U) is a model for the Thom complex T o, and we
can identify T o with the quotient space (M x M)=(M x M  U). More relevantly
for us, the fiberwise quotient (M x M)=M (M x M  U) is a model for So. View
M xK = pr*2K, M xM, and U as Gspaces over M via projection to the first factor
and embed U in M x K by sending (m, n) to (m, sK (n)). We have the diagonal
map
M x K ! (M x K, U) xM (M x M, M x M  U)
of Gspaces over M that sends (m, k) to ((m, k), (m, pK (k))) for m 2 M and k 2*
* K.
It induces the top map in the following diagram in GKM .
M x K R____//_((M x K)=M U) ^M So
RR OO
 RRRRR 
 RRRR 
fflffl R(( 
r*r!K_____wK____//"K^M So
Here (M x K)=M U is obtained from M x K by identifying all points of the form
(m, sK (n)) such that (m, n) 2 Um to a single basepoint in the fiber over m. *
*It
therefore contains K" as a subspace, and this gives the right vertical inclusio*
*n.
The image of the top arrow lands in the image of the right vertical arrow since*
* if
(m, pK (k)) is not in Um , then (m, k) maps to the basepoint in So and therefor*
*e to
the basepoint in (M x K)=M U ^M So. This gives the diagonal arrow. Note that
r*r!K = M x (K=sK (M)) and the left vertical arrow is the obvious quotient map.
Since the diagonal arrow maps (m, x) with x 2 sK (M) to the base point of the
fiber over m in "K^M So , it is constant on the fibers of the left vertical arr*
*ow. It
therefore factors through a map wK . Explicitly, wK is specified by
(
(16.5.13) wK (m, [x]) = [m, x] ^ [m, pK (x)]if (m, pK (x)) 2 Um ,
* otherwise,
where m 2 M, x 2 K, and the square brackets denote equivalence classes.
Proof of Proposition 16.5.10. Here we are working in homotopy cate
gories, and we may assume that K is qffibrant and qfcofibrant. Let L = K^M S .
We define the map uK in Ho GKM by the natural zigzag
~
r*r!L_wL__//"L^M Sooo~^M_id__L ^M So _____// VMK
of arrows in GKM , where ~ is induced by the isomorphism
~=
S ^M So _twist//_So ^M_S___//SVM
16.5. PROOF OF THE WIRTHM"ULLER ISOMORPHISM FOR MANIFOLDS 229
and ~: L ! "Lis an equivalence as in (kappak).
Proof of (i) We must show that uS0M' k in Ho GKM . Using our zigzag definition
of uK , we see that it suffices to show that the composite
~1 o~^id
M x T __k__//M x SV____//_S ^M S_____//fS^M So
is homotopic to the map wS defined in (16.5.13). As noted in the proof of
Lemma 16.5.9, k(m, [v]) = (m, ev([v] ^ m)), where v 2 S and brackets denote
equivalence classes. Recall that the map evdepends on a choice of a tubular nei*
*gh
borhood of M in x M (as in the proof of Lemma 16.2.7). We use the obvious
choice
{(v, m)  v 2 , m 2 M, and (p (x), m) 2 U}.
Under our identification of the normal bundle of : M ! M x M and thus of
its tubular neighborhood U with o, this tubular neighborhood is identified with
M x V ~= o. When not in the section, we can view points [m, n] 2 So =
(M xM)=M (M xM U) as vectors (m, n) in the tangent space Um ~=om Vm ~=V
of M at m. We then have that (m, ev([v] ^ m)) = (m, (p (v), m) + v). To identify
this point in the image of ~, let u 2 m be such that (p (v), m)+v = (m, p (v))*
*+u
in V and note that u depends continuously on m and v. Since ~(u ^ [m, p (v)]) =
(m, (m, p (v)) + u), the composite displayed above is given by
(
(~ ^M id)~1k(m, [v]) = [m, u] ^ [m, p (v)]if (m, p (v)) 2 Um ,
* otherwise.
A linear homotopy in the fibers of shows that this map is homotopic to wS .
Proof of (ii) Inspection of the construction of w gives the following naturality
diagram for based Gspaces T and exGspaces K over M.
r*r!(TM ^M K)___'____//TM ^M r*r!K .
w  id^w
fflffl ' fflffl
(TM^^M K ) ^M So____//_TM ^M K" ^M So
Here, using r*r!~= (ss1)!ss*2, the bottom equivalence is the following applicat*
*ion of
the projection formula.
(ss1)!ss*2(r*T ^M'K)(ss1)!(ss*2r*T ^M ss*2K)
' (ss1)!(ss*1r*T ^M ss*2K)
' r*T ^M (ss1)!ss*2K
This use of the projection formula is compatible with its use for r! to obtain *
*the
equivalence of the top row. Analogous naturality diagrams for the other two maps
in the definition of uK give the conclusion.
Proof of (iii) Again let L = K ^M S . Expanding the diagram in the statement
of (iii) in terms of the definition of uK , we must prove that the following di*
*agram
commutes in Ho GK*, where the equivalences here are the vertical arrows of the
230 16. THE WIRTHM"ULLER AND ADAMS ISOMORPHISMS
diagram in (iii).
r!( VMKO^MOS ) oo____'________r!L_^ SV_______id^t______//r!L ^ r!S
r!~ '
 fflffl
r!(L ^M So ^M S )__r!(~^id)//_r!("L^M So ^M Sr)!(wL^idr!(r*r!L)^MoSo)_
We chase the diagram starting in r!(L^M So ^M S ) and mapping to r!("L^M So ^M
S ). Let x = k ^ w 2 L = K ^M S , u 2 So, and v 2 S be points in fibers over
a given m 2 M. Using square brackets to denote passage to quotient spaces (the
lower shriek functors), we see that r!(~ ^ id) sends [x ^ u ^ v] to [[m, x] ^ u*
* ^ v]. The
definitions of ~ and of the top left equivalence (which is the left vertical eq*
*uivalence
in the diagram of the statement) are arranged in such a way that the composite *
*of ~
and the inverse of the equivalence sends [x^u^v] to [x^(u+v)]. Let t(u+v) = [z],
z 2 S , and let n = p (z). Chasing [x ^ u ^ v] around the top of the diagram, w*
*hen
we do not arrive at the basepoint we arrive at the point [[n, x] ^ [n, m] ^ z],*
* where
[n, m] is an element of U ~=o. We can identify the target space with r!("L) ^ SV
using the identification of So ^ S with M x SV and the projection formula. Then
our two maps are homotopic by a homotopy h that can be written in the form
h([x ^ u ^ v], s) = [m + s[m, n], x] ^ (u + v + 2s[n, m]).
Here m + s[m, n] denotes a point on the path from m to n in M that is the image
under the exponential map of the line segment from 0 to [m, n] in the tangent
space at m. The Thom map takes u + v in SV (which in M x SV is based at
m) to the point z in S based at n. In SV , we have u + v = [m, n] + z. Since
[n, m] = [m, n] under the identification of o with U (as in [77, 11.5]), we se*
*e that
the homotopy ends at the composite around the top of the diagram, and it clearly
begins at r!(~ ^ id).
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