Equivariant orthogonal spectra and Smodules
M.A. Mandell
J.P. May
Author address:
Department of Mathematics, The University of Chicago, Chicago,
IL 60637
Email address: mandell@math.uchicago.edu, may@math.uchicago.edu
1991 Mathematics Subject Classification. Primary 55P42, 55P43, 55P91;
Secondary 18A25, 18E30, 55P48, 55U35.
The authors were partially supported by the NSF.
Abstract.The last few years have seen a revolution in our understanding
of the foundations of stable homotopy theory. Many symmetric monoidal
model categories of spectra whose homotopy categories are equivalent to the
stable homotopy category are now known, whereas no such categories were
known before 1993. The most wellknown examples are the category of S
modules and the category of symmetric spectra. We focus on the category
of orthogonal spectra, which enjoys some of the best features of Smodules
and symmetric spectra and which is particularly wellsuited to equivariant
generalization. We first complete the nonequivariant theory by comparing
orthogonal spectra to Smodules. We then develop the equivariant theory. For
a compact Lie group G, we construct a symmetric monoidal model category of
orthogonal Gspectra whose homotopy category is equivalent to the classical
stable homotopy category of Gspectra. We also complete the theory of SG
modules and compare the categories of orthogonal Gspectra and SGmodules.
A key feature is the analysis of change of universe, change of group, fixed *
*point,
and orbit functors in these two highly structured categories for the study of
equivariant stable homotopy theory.
Contents
Introduction 1
Chapter I. Orthogonal spectra and Smodules 3
1. Introduction and statements of results 3
2. Right exact functors on categories of diagram spaces 4
3. The proofs of the comparison theorems 7
4. Model structures and homotopical proofs 11
5. The construction of the functor N* 14
6. The functor M and its comparison with N 19
7. A revisionist view of infinite loop space theory 22
Chapter II. Equivariant orthogonal spectra 26
1. Preliminaries on equivariant categories 26
2. The definition of orthogonal Gspectra 28
3. The smash product of orthogonal Gspectra 30
4. A description of orthogonal Gspectra as diagram Gspaces 31
Chapter III. Model categories of orthogonal Gspectra 34
1. The model structure on Gspaces 34
2. The level model structure on orthogonal Gspectra 37
3. The homotopy groups of Gprespectra 40
4. The stable model structure on orthogonal Gspectra 43
5. The positive stable model structure 47
6. Stable equivalences of orthogonal Gspectra 48
7. Model categories of ring and module Gspectra 49
8. The model category of commutative ring Gspectra 51
9. Level equivalences and ss*isomorphisms of Gspectra 52
Chapter IV. Orthogonal Gspectra and SG modules 55
1. Introduction and statements of results 55
2. Model structures on the category of SG modules 57
3. The construction of the functors N and N# 60
4. The proofs of the comparison theorems 62
5. The functor M and its comparison with N 63
6. Families, cofamilies, and Bousfield localization 64
Chapter V. "Change" functors for orthogonal Gspectra 69
1. Change of universe 69
2. Change of groups 72
3. Fixed point and orbit spectra 74
4. Geometric fixed point spectra 77
iii
iv CONTENTS
Chapter VI. "Change" functors for SG modules and comparisons 83
1. Comparisons of change of group functors 83
2. Comparisons of change of universe functors 86
3. Comparisons of fixed point and orbit Gspectra functors 91
4. Nfree Gspectra and the Adams isomorphism 94
5. The geometric fixed point functor and quotient groups 95
6. Technical results on the unit map : JE ! E 96
Bibliography 98
Index of Notation 100
Index 102
Introduction
There are two general approaches to the construction of symmetric monoidal
categories of spectra, one based on an encoding of operadic structure in the de*
*fini
tion of the smash product and the other based on the categorical observation th*
*at
categories of diagrams with symmetric monoidal domain are symmetric monoidal.
The first was worked out by Elmendorf, Kriz, and the authors in the theory of
"Smodules" [5]. The second was worked out in the case of symmetric spectra
by Hovey, Shipley, and Smith [15] and, in a general topological setting that al*
*so
includes orthogonal spectra, by Schwede, Shipley, and the authors [18].
Philosophically, orthogonal spectra are intermediate between Smodules and
symmetric spectra, enjoying some of the best features of both. They are defined*
* in
the same diagrammatic fashion as symmetric spectra, but with orthogonal groups
rather than symmetric groups building in the symmetries required to define an
associative and commutative smash product. They were first defined in [22].
We prove in Chapter I that the categories of orthogonal spectra and Smodules
are Quillen equivalent and that this equivalence induces Quillen equivalences b*
*e
tween the respective categories of ring spectra, of modules over ring spectra, *
*and of
commutative ring spectra. Combined with the analogous comparison between sym
metric spectra and orthogonal spectra of [18], this reproves and improves Schwe*
*de's
comparison between symmetric spectra and Smodules [29]. We refer the reader to
Ix1 for further discussion. We reinterpret the second author's opproach to infi*
*nite
loop space theory in terms of symmetric and orthogonal spectra in Ix7, where we
recall the purposes for which orthogonal spectra were first introduced [22].
With this understanding of the nonequivariant foundations of stable homotopy
theory in place, we develop new foundations for equivariant stable homotopy the
ory in the rest of this monograph. We let G be a compact Lie group throughout,
and we understand subgroups of G to be closed. There is no treatment of diagram
Gspectra in the literature, and we shall provide one. Just as in the nonequiva*
*ri
ant case, a major advantage of such a treatment is the simplicity of the result*
*ing
definitional framework.
Orthogonal spectra are far more suitable than symmetric spectra for this pur
pose. They are defined just as simply as symmetric spectra but, unlike symmetric
spectra, they share two of the essential features of the spectra of [17] that f*
*acili
tate equivariant generalization. First, they are defined in a coordinatefree f*
*ashion.
This makes it simple and natural to build in spheres associated to representati*
*ons,
which play a central role in the theory. Second, their weak equivalences are ju*
*st the
maps that induce isomorphisms of homotopy groups. This simplifies the equivaria*
*nt
generalization of the relevant homotopical analysis.
We define orthogonal Gspectrum and show that the category of orthogonal
Gspectra is a closed symmetric monoidal category in Chapter II. We prove that
1
2 INTRODUCTION
this category has a proper Quillen model structure whose homotopy category is
equivalent to the classical homotopy category of Gspectra in Chapter III. More*
*over,
we show that the various categories of orthogonal ring and module Gspectra have
induced model structures.
The original construction of the equivariant stable homotopy category, due to
Lewis and the second author, was in terms of Gspectra, which are equivariant v*
*er
sions of coordinatefree spectra. These are much more highly structured and much
less elementary objects than orthogonal Gspectra. The LewisMay construction
was modernized to a symmetric monoidal category of structured Gspectra, called
SG modules, by Elmendorf, Kriz, and the authors [5, 25]. Those monographs did
not consider model structures on SG spectra and SG modules, and we rectify th*
*at
omission in Chapter IV.
In fact there are two stable model structures on the categories of Gspectra*
* and
SG modules, and the difference between them is fundamental to the understanding
of equivariant stable homotopy theory. One has cofibrant objects defined in ter*
*ms
of spheres of representations and is essential to the comparison with orthogona*
*l G
spectra. The other has cofibrant objects defined in terms of integer spheres an*
*d is
essential for the equivariant versions of classical arguments in terms of CW sp*
*ectra.
We refer the reader to IVx1 for further discussion of this point.
Generalizing our nonequivariant comparison between orthogonal spectra and
Smodules, we prove in Chapter IV that the categories of orthogonal Gspectra
and SG modules are Quillen equivalent and that this equivalence induces Quillen
equivalences between the respective categories of ring Gspectra, of modules ov*
*er
ring Gspectra, and of commutative ring Gspectra. We also generalize the model
theoretic framework to deal with families and cofamilies of subgroups of G.
We discuss change of universe functors, change of group functors, orbit func
tors, and categorical and geometric fixed point functors on orthogonal Gspectr*
*a in
Chapter V. We discuss the analogous functors on SG modules in Chapter VI, and
we prove there that the equivalences among Gspectra, SG modules, and orthog
onal Gspectra are compatible with all of these functors interrelating equivari*
*ant
and nonequivariant stable homotopy categories. We conclude that all homotopical
results proven in the original stable homotopy category of Gspectra apply verb*
*atim
to the new stable homotopy categories of SG modules and orthogonal Gspectra.
Implicitly, equivariant orthogonal spectra have already been applied. A glob*
*al
form of the definition, with orthogonal Gspectra varying functorially in G, was
exploited in the proof of the completion theorem for complex cobordism of Green*
*lees
and May [12]. In retrospect, orthogonal S1spectra are pivotal to the construct*
*ion
of topological cyclic homology given by Hesselholt and Madsen [13].
CHAPTER I
Orthogonal spectra and Smodules
1. Introduction and statements of results
It is proven in [18] that the categories of symmetric spectra and of orthogo*
*nal
spectra are Quillen equivalent. It is proven by Schwede in [29] that the catego*
*ries
of symmetric spectra and Smodules are Quillen equivalent. However, this does n*
*ot
give a satisfactory Quillen equivalence between the categories of orthogonal sp*
*ectra
and Smodules since the resulting functor from orthogonal spectra to Smodules *
*is
the composite of a right adjoint (to symmetric spectra) and a left adjoint.
We shall give a Quillen equivalence between the categories of orthogonal spe*
*c
tra and Smodules such that the Quillen equivalence of [29] is the composite of
the Quillen equivalence between symmetric spectra and orthogonal spectra of [18]
and our new Quillen equivalence. Thus orthogonal spectra are mathematically as
well as philosophically intermediate between symmetric spectra and Smodules.
Our proofs will give a concrete Thom space level understanding of the relation
ship between orthogonal spectra and Smodules. To complete the picture, we also
point out Quillen equivalences relating coordinatized prespectra, coordinatefr*
*ee
prespectra, and spectra to Smodules and orthogonal spectra (in x4).
To separate formalities from substance, we begin in x2 by establishing a for*
*mal
framework for constructing symmetric monoidal left adjoint functors whose domain
is a category of diagram spaces. In fact, this elementary category theory sheds*
* new
light on the basic constructions that are studied in all work on diagram spectr*
*a.
In x3, we explain in outline how this formal theory combines with model theory *
*to
prove the following comparison theorems. We recall the relevant model structures
and give the homotopical parts of the proofs in x4. We defer the basic construc*
*tion
that gives substance to the theory to x5.
Theorem 1.1. There is a strong symmetric monoidal functor N : I S ! M
and a lax symmetric monoidal functor N# : M ! I S such that (N; N# ) is a
Quillen equivalence between I S and M . The induced equivalence of homotopy
categories preserves smash products.
Theorem 1.2. The pair (N; N# ) induces a Quillen equivalence between the ca*
*t
egories of orthogonal ring spectra and of Salgebras.
Theorem 1.3. For a cofibrant orthogonal ring spectrum R, the pair (N; N# ) *
*in
duces a Quillen equivalence between the categories of Rmodules and of NRmodul*
*es.
By [18, 12.1(iv)], the assumption that R is cofibrant results in no loss of *
*gen
erality. As in [18, x13], this result implies the following one.
Corollary 1.4. For an Salgebra R, the categories of Rmodules and of N# R
modules are Quillen equivalent.
3
4 I. ORTHOGONAL SPECTRA AND SMODULES
Theorem 1.5. The pair (N; N# ) induces a Quillen equivalence between the ca*
*t
egories of commutative orthogonal ring spectra and of commutative Salgebras.
Theorem 1.6. Let R be a cofibrant commutative orthogonal ring spectrum.
The categories of Rmodules, Ralgebras, and commutative Ralgebras are Quillen
equivalent to the categories of NRmodules, NRalgebras, and commutative NR
algebras.
By [18, 12.1(iv) and 15.2(ii)], the assumption that R is cofibrant results i*
*n no
loss of generality. Again, as in [18, x16], this result implies the following o*
*ne.
Corollary 1.7. Let R be a commutative Salgebra. The categories of R
modules, Ralgebras, and commutative Ralgebras are Quillen equivalent to the
categories of N# Rmodules, N# Ralgebras, and commutative N# Ralgebras.
These last results are the crucial comparison theorems since most of the dee*
*p
est applications of structured ring and module spectra concern E1 ring spectra*
* or,
equivalently by [5], commutative Salgebras. By [18, 22.4], commutative orthog
onal ring spectra are the same objects as commutative orthogonal FSP's. Under
the name "I*prefunctor", these were defined and shown to give rise to E1 ring
spectra in [21]. Theorem 1.5 shows that, up to equivalence, all E1 ring spectra
arise this way. The second author has wondered since 1973 whether or not that is
the case.
The analogues of the results above with orthogonal spectra and Smodules
replaced by symmetric spectra and orthogonal spectra are proven in [18]. This h*
*as
the following immediate consequence, which reproves all of the results of [29].
Theorem 1.8. The analogues of the results above with orthogonal spectra re
placed by symmetric spectra are also true.
The functor N that occurs in the results above has all of the formal and hom*
*o
topical properties that one might desire. However, a quite different and consid*
*erably
more intuitive functor M from orthogonal spectra to Smodules is implicit in [2*
*1].
The functor M gives the most natural way to construct Thom spectra as commu
tative Salgebras, and its equivariant version was used in an essential way in *
*the
proof of the localization and completion theorem for complex cobordism given in
[12]. We define M and compare it with N in x6.
Under the name of "I*prespectra", orthogonal spectra were first introduced *
*in
[22], where they were used as intermediaries in the passage from pairings of sp*
*aces
with operad actions and pairings of permutative categories to pairings of spect*
*ra.
As we explain in x7, symmetric spectra could have been used for the same purpos*
*e,
and the theory of this paper sharpens the conclusions of [22] by showing how to
obtain pointset level rather than homotopy category level pairings of spectra *
*from
the given input data.
It is a pleasure to thank our collaborators Brooke Shipley and Stefan Schwed*
*e.
Like Schwede's paper [29], which gives a blueprint for some of x3 here, this ch*
*apter
is an outgrowth of our joint work in [18].
2.Right exact functors on categories of diagram spaces
To clarify our arguments, we first give the formal structure of our construc*
*tion
of the adjoint pair (N; N# ) in a suitably general framework. We consider categ*
*ories
DT of Dshaped diagrams of based spaces for some domain category D, and we
2. RIGHT EXACT FUNCTORS ON CATEGORIES OF DIAGRAM SPACES 5
show that, to construct left adjoint functors from DT to suitable categories C *
*, we
need only construct contravariant functors D ! C . The proof is an exercise in
the use of representable functors and must be standard category theory, but we *
*do
not know a convenient reference.
Let T be the category of based spaces, where spaces are understood to be
compactly generated (= weak Hausdorff kspaces). Let D be any based topologi
cal category with a small skeleton skD. A Dspace is a continuous based functor
D ! T . Let DT be the category of Dspaces. As observed in [18, x1], the
evident levelwise constructions define limits, colimits, smash products with sp*
*aces,
and function Dspaces that give DT a structure of complete and cocomplete, ten
sored and cotensored, topological category. We call such a category topologica*
*lly
bicomplete. We fix a topologically bicomplete category C for the rest of this s*
*ection.
We write C ^A for the tensor of an object C of C and a based space A. All funct*
*ors
are assumed to be continuous.
Definition 2.1.A functor between topologically cocomplete categories is right
exact if it commutes with colimits and tensors. For example, any functor that i*
*s a
left adjoint is right exact.
For a contravariant functor E : D ! C and a Dspace X, we have the coend
Z d
(2.2) E D X = E(d) ^ X(d)
in C . Explicitly, E D X is the coequalizer in C of the diagram
W "^id//_W
d;eE(e) ^ D(d; e) ^ X(d)_//_dE(d) ^ X(d);
id^"
where the wedges run over pairs of objects and objects of skD and the parallel
arrows are wedges of smash products of identity and evaluation maps of E and X.
For an object d 2 D, we have a left adjoint Fd : T ! DT to the functor
given by evaluation at d. If d* is defined by d*(e) = D(d; e), then FdA = d* ^ *
*A.
In particular, FdS0 = d*.
Definition 2.3.Let D = DD : D ! DT be the evident contravariant func
tor that sends d to d*.
The following observation is [18, 1.6].
Lemma 2.4. The evaluation maps D(d; e) ^ X(d) ! X(e) of Dspaces X
induce a natural isomorphism of Dspaces D D X ! X.
Together with elementary categorical observations, this has the following im
mediate implication. It shows that (covariant) right exact functors F : DT  ! C
determine and are determined by contravariant functors E : D ! C .
Theorem 2.5. If F : DT  ! C is a right exact functor, then (F O D) D X ~=
FX. Conversely, if E : D  ! C is a contravariant functor, then the functor
F : D ! C specified by FX = E D X is right exact and F O D ~=E.
Notation 2.6. Write F $ F* for the correspondence between right exact func
tors F : DT ! C and contravariant functors F* : D ! C . Thus, given F,
F* = FOD, and, given F*, F = F*D (). In particular, on representable Dspaces,
Fd* ~=F*d.
6 I. ORTHOGONAL SPECTRA AND SMODULES
Corollary 2.7. Via * = Fj O D and = * D (), natural transformations
: F ! G between right exact functors DT  ! C determine and are determined
by natural transformations * : F* ! G* between the corresponding contravariant
functors D ! C .
Proposition 2.8. Any right exact functor F : DT  ! C has the right adjoint
F# specified by
(F# C)(d) = C (F*d; C)
for C 2 C and d 2 D. The evaluation maps
D(d; e) ^ C (F*d; C) ! C (F*e; C)
of the functor F# are the adjoints of the composites
F*e ^ D(d; e) ^ C (F*d; C) "^id!F*d ^ C (F*d; C) i!C;
where " is an evaluation map of the functor F* and i is an evaluation map of the
category C .
Proof. We must show that
(2.9) C (FX; C) ~=DT (X; F# C):
The description of FX as a coend implies a description of C (FX; C) as an end
constructed out of the spaces C (F*d^X(d); C). Under the adjunction isomorphisms
C (F*d ^ X(d); C) ~=T (X(d); C (F*d; C));
this end transforms to the end that specifies DT (X; F# C). _*
*__
As an illustration of the definitions, we show how the prolongation and forg*
*etful
functors studied in [18] fit into the present framework.
Example 2.10. A (covariant) functor : D ! D0 induces the forgetful func
tor U : D0T ! DT that sends Y to Y O . It also induces the contravariant
functor DD0O : D ! D0T . Let PX = (DD0O)D X. Then P is the prolongation
functor left adjoint to U.
Now let D be symmetric monoidal with product and unit uD . By [18, x21],
DT is symmetric monoidal with unit u*D. We denote the smash product of DT
by ^D . Actually, the construction of the smash product is another simple direct
application of the present framework.
Example 2.11. We have the external smash product Z : DT x DT !
(D x D)T specified by (X Z Y )(d; e) = X(d) ^ Y (e) [18, 21.1]. We also have t*
*he
contravariant functor DD O : D x D ! DT . The internal smash product is
given by
(2.12) X ^D Y = (DD O ) DxD (X Z Y ):
It is an exercise to rederive the universal property
(2.13) DT (X ^D Y; Z) ~=(D x D)T (X Z Y; Z O )
that characterizes ^D from this definition.
Proposition 2.14. Let F* : D ! C be a strong symmetric monoidal con
travariant functor. Then F : DT ! C is a strong symmetric monoidal functor
and F# : C ! DT is a lax symmetric monoidal functor.
3. THE PROOFS OF THE COMPARISON THEOREMS 7
Proof. We are given an isomorphism : F*uD ! uC and a natural isomor
phism
OE : F*d ^C F*e ! F*(d e):
Since F*uD ~=Fu*D, we may view as an isomorphism Fu*D! uC. By (2.9) and
(2.13), we have
C (F(X ^D Y ); C) ~=(DT x DT )(X Z Y; F# C O ):
Commuting coends past smash products and using isomorphisms
(F*d ^ X(d)) ^ (F*e ^ Y (e)) ~=F*(d e) ^ X(d) ^ Y (e)
induced by OE, we obtain the first of the following two isomorphisms. We obtain
the second by using the tensor adjunction of C and applying the defining univer*
*sal
property of coends.
Z (d;e)
C (FX ^C FY; C) ~= C ( F# (d e) ^ X(d) ^ Y (e); C)
~= (DT x DT )(X Z Y; F# C O ):
There results a natural isomorphism FX ^C FY ~= F(X ^D Y ), and coherence is
easily checked.
For F# , the adjoint u*D! F# uC of gives the unit map. Taking the smash
products of maps in C and applying isomorphisms OE, we obtain maps
C (F*(d); C) ^ C (F*(e); C0) ! C (F*(d e); C ^C C0)
that together define a map
F# C Z F# C0 ! F# (C ^C C0) O :
Using (2.13), there results a natural map
F# C ^D F# C0 ! F# (C ^C C0);
and coherence is again easily checked. ___
3. The proofs of the comparison theorems
We refer to [18] for details of the category I S of orthogonal spectra and to
[5] for details of the category M = MS of Smodules. Much of our work depends
only on basic formal properties. Both of these categories are closed symmetric
monoidal and topologically bicomplete. They are Quillen model categories, and
their model structures are compatible with their smash products. Actually, in [*
*18],
the category of orthogonal spectra is given two model structures with the same
(stable) weak equivalences. In one of them, the sphere spectrum is cofibrant, i*
*n the
other, the "positive stable model structure", it is not. In [18], use of the p*
*ositive
stable model structure was essential to obtain an induced model structure on the
category of commutative orthogonal ring spectra. It is also essential here, sin*
*ce the
sphere Smodule S is not cofibrant. We will review the model structures in x4.
We begin by giving a quick summary of definitions from [18], recalling how
orthogonal spectra fit into the framework of the previous section. Let I be the
symmetric monoidal category of finite dimensional real inner product spaces and
linear isometric isomorphisms. We call an I space an orthogonal space. The cat
egory I T of orthogonal spaces is closed symmetric monoidal under its smash
products X ^ Y and function objects F (X; Y ).
8 I. ORTHOGONAL SPECTRA AND SMODULES
The sphere orthogonal space SI has V th space the onepoint compactification
SV of V ; SI is a commutative monoid in I T . An orthogonal spectrum, or I 
spectrum, is a (right) SI module. The category I S of orthogonal spectra is cl*
*osed
symmetric monoidal. We denote its smash products and function spectra by X^I Y
and FI (X; Y ) (although this is not consistent with the previous section).
There is a symmetric monoidal category J with the same objects as I such
that the category of J spaces is isomorphic to the category of I spectra; J
contains I as a subcategory. The construction of J is given in [18, 2.1], where*
* it
is denoted IS. Its space of morphisms J (V; W ) is (V *^SI )(W ), where V *(W )*
* =
I (V; W )+ . In x5, we shall give a concrete alternative description of J in te*
*rms
of Thom spaces, and we shall construct a coherent family of cofibrant (V )sph*
*ere
Smodules N*(V ) that give us a contravariant "negative spheres" functor N* to
which we can apply the constructions of the previous section. Note that the unit
of I is 0, the unit of J is SI , and, as required for consistency, J (0; W ) = *
*SW .
Theorem 3.1. There is a strong symmetric monoidal contravariant functor
N* : J ! M . If V 6= 0, then N*(V ) is a cofibrant Smodule and the evaluation
map
" : N*(V ) ^ SV = N*(V ) ^ J (0; V ) ! N*(0) ~=S
of the functor is a weak equivalence.
Here N*(0) ~=S since N* is strong symmetric monoidal. Propositions 2.8 and
2.14 give the following immediate consequence.
Theorem 3.2. Define functors N : I S  ! M and N# : M  ! I S by
letting N(X) = N* J X and (N# M)(V ) = M (N*(V ); M). Then (N; N# ) is an
adjoint pair such that N is strong symmetric monoidal and N# is lax symmetric
monoidal.
This gives the formal properties of N and N# , and we turn to their homotopi*
*cal
properties. According to [18, A.2], to show that these functors give a Quillen
equivalence between I S and M , it suffices to prove the following three result*
*s.
Thus, since its last statement is formal [14, 4.3.3], these results will prove *
*Theorem
1.1. We give the proofs in the next section. A functor F : A ! B between model
categories is said to create the weak equivalences in A if the weak equivalence*
*s in
A are exactly the maps f such that F f is a weak equivalence in B, and similarly
for other classes of maps.
Lemma 3.3. The functor N# creates the weak equivalences in M .
Lemma 3.4. The functor N# preserves qfibrations.
Proposition 3.5. The unit j : X  ! N# NX of the adjunction is a weak
equivalence for all cofibrant orthogonal spectra X.
In Lemma 3.4, we are concerned with qfibrations of orthogonal spectra in the
positive stable model structure. To prove Theorem 1.1, we only need Proposition
3.5 for orthogonal spectra that are cofibrant in the positive stable model stru*
*cture,
but we shall prove it more generally for orthogonal spectra that are cofibrant *
*in
the stable model structure. We refer to positive cofibrant and cofibrant orthog*
*onal
spectra to distinguish these classes.
In the rest of this section, we show that these results imply their multipli*
*catively
enriched versions needed to prove Theorems 1.2, 1.3, 1.5, and 1.6. That is, in *
*all
3. THE PROOFS OF THE COMPARISON THEOREMS 9
cases, we have an adjoint pair (N; N# ) such that N# creates weak equivalences
and preserves qfibrations and the unit of the adjunction is a weak equivalence*
* on
cofibrant objects. The one subtlety is that, to apply Proposition 3.5, we must *
*relate
cofibrancy of multiplicatively structured orthogonal spectra with cofibrancy of*
* their
underlying orthogonal spectra.
The proof of Theorem 1.2. The category of orthogonal ring spectra has
two model structures. The respective weak equivalences and qfibrations are cre
ated in the category of orthogonal spectra with its stable model structure or i*
*ts
positive stable model structure. The category of Salgebras is a model category
with weak equivalences and qfibrations created in the category of Smodules. O*
*ur
claim is that (N; N# ) restricts to a Quillen equivalence relating the category*
* of or
thogonal ring spectra with its positive stable model structure to the category *
*of
Salgebras. It is clear from Lemmas 3.3 and 3.4 that N# creates weak equivalenc*
*es
and preserves qfibrations. We must show that j : R ! N# NR is a weak equiva
lence when R is a positive cofibrant orthogonal ring spectrum. More generally, *
*if R
is a cofibrant orthogonal ring spectrum, then the underlying orthogonal spectrum
of R is cofibrant (although not positive cofibrant) by [18, 12.1]. The conclus*
*ion_
follows from Proposition 3.5. __
The proof of Theorem 1.3. The category of Rmodules is a model cate
gory with weak equivalences and qfibrations created in the category of orthogo*
*nal
spectra with its positive stable model structure. The category of NRmodules is
a model category with weak equivalences and qfibrations created in the category
of Smodules. Again, it is clear that N# creates weak equivalences and preserves
qfibrations. We must show that j : Y ! N# NY is a weak equivalence when
Y is a positive cofibrant Rmodule. We are assuming that R is positive cofibrant
as an orthogonal ring spectrum, and it follows from [18, 12.1] that the underly*
*ing
orthogonal spectrum of a cofibrant Rmodule is cofibrant (although not necessar*
*ily_
positive cofibrant). The conclusion follows from Proposition 3.5. *
*__
The proof of Theorem 1.5. The category of commutative orthogonal ring
spectra has a model structure with weak equivalences and qfibrations created in
the category of orthogonal spectra with its positive stable model structure [18,
15.1]. The category of commutative Salgebras has a model structure with weak
equivalences and qfibrations created in the category of Smodules [5, VII.4.8].
Again, it is clear that N# creates weak equivalences and preserves qfibrations*
*, and
we must prove that j : R ! N# NR is a weak equivalence when R is a cofibrant
commutative orthogonal ring spectrum. Since the underlying orthogonal spectrum
of R is not cofibrant, we must work harder here. We use the notations and resul*
*ts of
[18, xx15, 16], where the structure of cofibrant commutative orthogonal ring sp*
*ectra
is analyzed and the precisely analogous proof comparing commutative symmetric
ring spectra and commutative orthogonal ring spectrum is given.
We may assume that R is a CF +Icell complex (see [18, 15.1]), where C is the
free commutative orthogonal ring spectrum functor, and we claim first that j is*
* a
weak equivalence when R = CX for a positive cofibrant orthogonal spectrum X.
It suffices to prove that j : X(i)=i ! N# N(X(i)=i) is a weak equivalence for
i 1. On the right, N(X(i)=i) ~=(NX)(i)=i, and NX is a cofibrant Smodule.
10 I. ORTHOGONAL SPECTRA AND SMODULES
Consider the commutative diagram
j
Ei+ ^i X(i) ____//_N# N(Ei+ ^i X(i)) ~= N# (Ei+ ^i (NX)(i))
q N#Nq N#q
fflffl fflffl fflffl
X(i)=i _____j____//_N# N(X(i)=i) ~= N# ((NX)(i)=i):
The q are the evident quotient maps, and the left and right arrows q are weak
equivalences by [18, 15.5] and [5, III.5.1]. The top map j is a weak equivalenc*
*e by
Proposition 3.5 since an induction up the cellular filtration of Ei, the succes*
*sive
subquotients of which are wedges of copies of i+^ Sn, shows that Ei+^i X(i)
is positive cofibrant since X(i)is positive cofibrant.
By passage to colimits, as in the analogous proof in [18, x16], the result f*
*or
general R follows from the result for a CF +Icell complex that is constructed *
*in
finitely many stages. We have proven the result when R requires only a single
stage, and we assume the result when R is constructed in n stages. Thus suppose
that R is constructed in n + 1 stages. Then R is a pushout (in the category
of commutative orthogonal ring spectra) of the form Rn ^CX CY , where Rn is
constructed in nstages and X ! Y is a wedge of maps in F +I. As in the proof *
*of
[18, 15.9], R ~=B(Rn; CX; CT ), where T is a suitable wedge of orthogonal spect*
*ra
FrS0. The bar construction here is the geometric realization of a proper simpli*
*cial
orthogonal spectrum and N commutes with geometric realization. Tracing through
the cofibration sequences used in the proof of the invariance of bar constructi*
*ons
in [5, X.4], we see that it suffices to show that j is a weak equivalence on the
commutative orthogonal ring spectrum
Rn ^ (CX)(q)^ CT ~=Rn ^ C(X _ . ._.X _ T )
of qsimplices for each q. By the definition of CF +Icell complexes, we see th*
*at this
smash product (= pushout) can be constructed in nstages, hence the conclusion_
follows from the induction hypothesis. __
The proof of Theorem 1.6. For a cofibrant commutative orthogonal ring
spectrum R, we must prove that the unit j : X ! N# NX of the adjunction is
a weak equivalence when X is a cofibrant Rmodule, Ralgebra, or commutative
Ralgebra. For Rmodules, this reduces as in [18, x16] to an application of [5,
III.3.8], which gives that the functor (NR) ^ () preserves weak equivalences. *
*The
case of Ralgebras follows since a cofibrant Ralgebra is cofibrant as an Rmod*
*ule
[18, 12.1]. The case of commutative Ralgebras follows from the previous proof
since a cofibrant commutative Ralgebra is cofibrant as a commutative orthogona*
*l_
ring spectrum. __
Remark 3.6. Consider the diagram
_P__//_ __N__//
I oUo__I S oo___ M ;
N#
where S is the category of symmetric spectra and U and P are the forgetful and
prolongation functors of [18] (see Example 2.10). By (5.8) below, we have
(U O N# )(M)(n) ~=M ((S1S)(n); M)
4. MODEL STRUCTURES AND HOMOTOPICAL PROOFS 11
as nspaces, where S1Sis the canonical cofibrant (1)sphere in the category
of Smodules. This is the right adjoint M  ! S used by Schwede [29], and
N O P is its left adjoint. Thus the adjunction studied in [29] is the composite*
* of the
adjunctions (P; U) and (N; N# ).
4. Model structures and homotopical proofs
To complete the proofs and to place our results in context, we recall the re*
*la
tionship of I S and M to various other model categories of prespectra and spect*
*ra.
We have two categories of prespectra, coordinatized and coordinatefree. In [18*
*],
the former was described as the category of N spectra, where N is the discrete
category with objects n, n 0, and it was given a stable model structure and a
positive stable model structure. We denote this category by N S . Then [18] giv*
*es
the following result.
Proposition 4.1. The forgetful functor U : I S ! N S has a left adjoint
prolongation functor P : N S ! I S , and the pair (P; U) is a Quillen equivale*
*nce
with respect to either the stable or the positive stable model structures.
We shall focus on prespectra in the coordinatefree sense of [17, 5]. Thus a
prespectrum X consists of based spaces X(V ) and based maps oe : WV X(V ) !
X(W ), where V ranges over the finite dimensional sub inner product spaces of a
countably infinite dimensional real inner product space U, which we may take to
be U = R1 . Let P denote the resulting category of prespectra. Exactly as in [1*
*8],
P has stable and positive stable model structures.
Remark 4.2. We obtain a forgetful functor U : P ! N S by restricting to
the subspaces Rn of U. We also have an underlying coordinatefree prespectrum
functor U : I S ! P. The composite of these two functors is the functor U of
Proposition 4.1. All three functors U have left adjoints P given by Example 2.1*
*0,
and Proposition 4.1 remains true with U replaced by either of our new functors *
*U.
A prespectrum X is said to be an spectrum if its adjoint structure maps
"oe: X(V ) ! WV X(W ) are weak equivalences; it is a positive spectrum if
these maps are weak equivalences for V 6= 0; it is a spectrum if these maps are
homeomorphisms. Let S denote the category of spectra. It is given a stable model
structure in [5, VIIxx4, 5]. The following result is implicit in [5, 18]. We in*
*dicate
the proof at the end of the section.
Proposition 4.3. The forgetful functor ` : S ! P has a left adjoint spec
trification functor L : P ! S , and the pair (L; `) is a Quillen equivalence w*
*ith
respect to the stable model structures.
Remark 4.4. This result applies to both the coordinatized and coordinate
free settings. The restriction of U : P ! N S to the respective subcategories
of spectra is an equivalence of categories [17, I.2.4]; both U and its restrict*
*ion to
spectra are the right adjoints of Quillen equivalences of model categories.
There is an evident underlying spectrum functor M  ! S ; that is, an S
module is a spectrum with additional structure. This functor is not the right
adjoint of a Quillen equivalence, but it is weakly equivalent to such a right a*
*djoint.
The following result is implicit in [5], as we explain at the end of the sectio*
*n.
12 I. ORTHOGONAL SPECTRA AND SMODULES
Proposition 4.5. There is a "free functor" F : S  ! M that has a right
adjoint V : M ! S , and there is a natural weak equivalence of spectra M !
VM for Smodules M. The pair (F; V) is a Quillen equivalence of stable model
categories.
Thus, even before constructing N* we have Quillen equivalences relating the
categories N S , I S , P, S , and M , so that we know that all of our homo
topy categories are equivalent. Of course, these equivalences are much less hig*
*hly
structured than the one we are after since N S , P, and S are not symmetric
monoidal under their smash products. To help orient the reader, we display our
Quillen equivalences in the following (noncommutative) diagram:
_____P______// _____L_____//_
N S Doo_________POOoo__________SOO
bDDbDDUD  ` 
DDDDDD  
DDDPDDDD  
U DDDDDU P VF
DDDDDD 
DD""fflfflDDN fflffl
I S ___________//Mo:o_
N#
To make our comparisons, we recall the definition of the model structures on*
* all
of our categories. The homotopy groups of a prespectrum, spectrum, orthogonal
spectrum, or Smodule are the homotopy groups of its underlying coordinatized
prespectrum. A map in any of these categories is a weak equivalence if it induc*
*es
an isomorphism of homotopy groups.
A map of spectra is a qfibration if each of its component maps of spaces is
a Serre fibration, and the functor V creates the qfibrations of Smodules. The
(positive) qfibrations of prespectra or of orthogonal spectra are the (positiv*
*e) level
Serre fibrations such that certain diagrams are homotopy pullbacks [18, 9.5]; a*
*ll
that we need to know about the latter condition is that it holds automatically *
*for
maps between (positive) spectra.
In all of our categories, the qcofibrations are the maps that satisfy the L*
*LP with
respect to the acyclic qfibrations. Equivalently, they are the retracts of rel*
*ative cell
complexes in the respective categories. These cell complexes are defined as usu*
*al in
terms of attaching maps whose domains are appropriate "spheres". We have nth
space or V th space evaluation functors from the categories N S , I S , P, and S
to the category T . These have left adjoint shift desuspension functors, denoted
Fn : T  ! N S ; FV : T  ! I S ; FV : T  ! P; and 1V: T  ! S :
We write Fn = FRn in P and I S and 1n= 1Rnin S . Obvious isomorphisms
between right adjoints imply isomorphisms between left adjoints
PFn ~=Fn and LFV ~=1V:
The domains of attaching maps are the FnSq in N S , I S , and P where, for the
positive stable model structures, we restrict to n > 0. The domains of attaching
maps are the 1nSq in S and the F1nSq in M .
Returning to N*, we will obtain the following description of its values on o*
*bjects.
Lemma 4.6. For an object V 6= 0 of I , the Smodule N*(V ) is noncanonical*
*ly
isomorphic to F1VS0.
4. MODEL STRUCTURES AND HOMOTOPICAL PROOFS 13
The subtlety in the construction of N* lies in its orthogonal functoriality.*
* We
cannot just define N*(V ) to be F1VS0, since that would not give a functor of V*
* .
We begin our proofs with the following observation.
Lemma 4.7. For Smodules M, N# M is a positive spectrum.
Proof. We have (N# M)(V ) = M (N*(V ); M). For V W ,
WV (N# M)(W ) ~=M (WV N*(W ); M)
and the adjoint structure map "oe: N# (V ) ! WV N# (W ) is induced from the
evaluation map " : WV N*(W ) ! N*(V ). Let V 6= 0. Then " is a weak *
* __
equivalence between cofibrant Smodules and "oeis thus a weak equivalence. *
*__
Proof of Lemma 3.3. By Lemma 4.6 and Proposition 4.5, for V U we
have
(N# M)(V ) ~=M (F1VS0; M) ~=S (1VS0; VM) ~=T (S0; (VM)(V )) = (VM)(V );
which is weakly equivalent to M(V ). Since a map of orthogonal positive spectra
or of Smodules is a weak equivalence if and only if its map on V th spaces is a
weak equivalence for V 6= 0 in U, this implies that a map f of Smodules is a w*
*eak__
equivalence if and only if N# f is a weak equivalence of orthogonal spectra. *
* __
Proof of Lemma 3.4. Let f : M ! N be a qfibration of Smodules. We
must show that N# f is a positive qfibration of orthogonal spectra. Since N# f*
* is
a map of positive spectra, we need only show that the V th space map of N# f
is a Serre fibration for V 6= 0, and it suffices to show this for V = Rn, n > 0*
*. By
[5, VII.4.6], f is a qfibration if and only if it satisfies the RLP with respe*
*ct to all
maps
i0 : F1nCSq ! F1nCSq ^ I+ :
An easy adjunction argument from the isomorphism N*(Rn) ~=F1nS0 and the
fact that F and the 1n are right exact shows that
f* : M (N*(Rn); M) ! M (N*(Rn); N)
satisfies the RLP with respect to the maps i0 : CSq ! CSq ^ I+ and is therefor*
*e_
a Serre fibration. __
Remark 4.8. In principle, the specified RLP property states that f* is a ba*
*sed
Serre fibration, whereas what we need is that f* is a classical Serre fibration*
*, that
is, a based map that satisfies the RLP in T with respect to the maps i0 : Dq+!
Dq+^ I. However, when n > 0, f* is isomorphic to the loop of a based Serre
fibration, and the loop of a based Serre fibration is a classical Serre fibrati*
*on.
Proof of Proposition 3.5. We first prove that j : FnA ! N# NFnA is a
weak equivalence for any based CW complex A; the only case we need is when A is
a sphere. Here Fn = PFn and it suffices to prove that the adjoint map of prespe*
*ctra
j : FnA ! UN# NFnA
is a weak equivalence. By a check of definitions and use of Lemma 4.6,
NFnA ~=NFnS0 ^ A ~=N*(Rn) ^ A ~=F1nS0 ^ A ~=F1nA:
Therefore, using Lemma 4.6 and Proposition 4.5, we have weak equivalences
(N# NFnA)(Rq) ~=M (F1qS0; F1nA) ~= S (1qS0; VF1nA)
' S (1qS0; 1nA) ~=(1nA)(Rq):
14 I. ORTHOGONAL SPECTRA AND SMODULES
Tracing through definitions, we find that, up to homotopy, the structural maps
coincide under these weak equivalences with those of 1nA ~=LFnA and the map
j induces the same map of homotopy groups as the unit FnA ! `LFnA of the
adjunction of Lemma 4.3. Therefore jis a weak equivalence. By standard results *
*on
the homotopy groups of prespectra [18, x7] and their analogues for spectra, we *
*see
that the class of orthogonal spectra for which j is a weak equivalence is close*
*d under
wedges, pushouts along hcofibrations, sequential colimits of hcofibrations, a*
*nd __
retracts. Therefore j is a weak equivalence for all cofibrant orthogonal spectr*
*a. __
The proofs of Propositions 4.1, 4.3, and 4.5.In all of these results, it
is immediate that the right adjoint creates weak equivalences and preserves q
fibrations. It remains to show that the units of the adjunctions are weak equiv*
*a
lences when evaluated on cofibrant objects. For Proposition 4.1, this is [18, 1*
*0.3].
For Proposition 4.3, Lemma 4.9 below gives the conclusion. For Proposition_4.5,_
Lemma 4.11 below gives the conclusion. __
A prespectrum X is an inclusion prespectrum if its adjoint structure maps
"oe: X(V ) ! WV X(W ) are inclusions, and this holds when X is cofibrant. The
following result is immediate from [17, I.2.2].
Lemma 4.9. Let X be an inclusion prespectrum. Then
LX(V ) = colimWV WV X(W ):
The V th map of the unit j : X ! `LX of the (L; `)adjunction is the map from
the initial term X(V ) into the colimit, and j is a weak equivalence of prespec*
*tra.
Remark 4.10. For later use, we note a variant. We call X a positive inclusi*
*on
prespectrum if "oeis an inclusion when V 6= 0. The description of LX(V ) is st*
*ill
valid and j is still a weak equivalence.
The notion of a tame spectrum is defined in [5, I.2.4]; cofibrant spectra are
tame.
Lemma 4.11. For tame spectra E, the unit j : E ! VFE is a weak equiva
lence. For Smodules M, there is a natural weak equivalence " : M ! VM.
Proof. With the notations of [5, I.4.1, I.5.1, I.7.1]
FE = S ^L LE and VM = FL (S; M):
The unit j is the composite of the homotopy equivalence j : E ! LE of [5, I.4.*
*6],
the weak equivalence " : LE ! FL (S; LE) of [5, I.8.7], and the isomorphism
FL (S; LE) ~=FL (S; S ^L LE) of [5, II.2.5]. The result [5, I.8.7] also gives *
*the
weak equivalence " : M ! VM. ___
5. The construction of the functor N*
We prove Theorem 3.1 here. Implicitly, we shall give two constructions of the
functor N*. The theory of Smodules is based on a functor called the twisted
halfsmash product, denoted n, the definitive construction of which is due to
Cole [5, App]. The theory of orthogonal spectra is the theory of diagram spaces
with domain category J . Both n and J are defined in terms of Thom spaces
associated to spaces of linear isometries. We first define N* in terms of twis*
*ted
halfsmash products. We then outline the definition of twisted halfsmash produ*
*cts
in terms of Thom spaces and redescribe N* in those terms. That will make the
5. THE CONSTRUCTION OF THE FUNCTOR N* 15
connection with the category J transparent, since the morphism spaces of J are
Thom spaces closely related to those used to define the relevant twisted halfs*
*mash
products.
Here we allow the universe U on which we index our coordinatefree prespectra
and spectra to vary. We write PU and S U for the categories of prespectra and
spectra indexed on U. We have a forgetful functor ` : S U ! PU with a left
adjoint spectrification functor L : PU  ! S U. We have a suspension spectrum
functor U that is left adjoint to the zeroth space functor U . Let SU = U (S0).
The functors U and U are usually denoted 1 and 1 , but we wish to emphasize
the choice of universe rather than its infinite dimensionality. We write 1 and
1 when U = R1 , and we then write SU = S. More generally, for a finite
dimensional sub inner product space V of U, we have a shift desuspension functor
UV: T  ! S U, denoted 1V when U = R1 . It is left adjoint to evaluation at V .
For inner product spaces U and U0, let I (U; U0) be the space of linear isom*
*etries
U  ! U0, not necessarily isomorphisms. It is contractible when U0 is infinite
dimensional [21, 1.3]. We have a twisted halfsmash functor
0
I (U; U0) n () : S U ! S U;
whose definition we shall recall shortly. It is a "change of universe functor" *
*that
converts spectra indexed on U to spectra indexed on U0 in a wellstructured way.
Now fix U = R1 and consider the universes V U for V 2 I . Identify V with
V R V U. In the language of [5], we define
(5.1) N*(V ) = S ^L (I (V U; U) n VVU (S0)):
To make sense of this, we must recall some of the definitional framework of
[5]. We have the linear isometries operad L with nth space L (n) = I (Un ; U).
The operad structure maps are given by compositions and direct sums of linear
isometries, and they specialize to give a monoid structure on L (1), a left act*
*ion
of L (1) on L (2), and a right action of L (1) x L (1) on L (2). For a spectrum
E 2 S , L (1) n E is denoted LE. The monoid structure on L (1) induces a monad
structure on the functor L : S ! S .
Definition 5.2.An Lspectrum is an algebra over the monad L. Let S [L]
denote the category of Lspectra. The functor L takes values in Lspectra and g*
*ives
the free Lspectrum functor L : S ! S [L].
By [5, Ix5], we have an "operadic smash product"
(5.3) E ^L E0= L (2) nL (1)xL (1)E Z E0
between Lspectra E and E0, where EZE0is the external smash product indexed on
U2 [5, Ix2]. The sphere S is an Lspectrum, and the action of L (1) by composit*
*ion
on I (V U; U) induces a structure of Lspectrum on I (V U; U) n VVU (S0).
An Lspectrum E has a unit map : S ^L E ! E that is always a weak
equivalence and sometimes an isomorphism [5, I.x8 and IIx1]; we redescribe it in
VIx6. In particular, is an isomorphism when E = S, when E = S ^L E0 for any
Lspectrum E0, and when E is the operadic smash product of two Smodules [5,
I.8.2, II.1.2].
Definition 5.4.An Smodule is an Lspectrum E such that is an isomor
phism. The smash product ^ in the category M of Smodules is the restriction to
16 I. ORTHOGONAL SPECTRA AND SMODULES
Smodules of ^L . The functor J : S [L] ! M specified by
JE = S ^L E
carries Lspectra to weakly equivalent Smodules. The functor F : S  ! M of
Proposition 4.5 is the composite J O L.
We can rewrite (5.1) as
(5.5) N*(V ) = J(I (V U; U) n VVU (S0)):
This makes sense of (5.1). It even makes sense when V = {0}. Here we interpret
spectra indexed on the universe {0}as based spaces. The space I ({0}; U) is a p*
*oint,
namely the inclusion iU : {0} ! U. The functor iU*= iU n () : T  ! S U is
left adjoint to the zeroth space functor, hence iU*= U . Thus (5.5) specializes*
* to
give N*(0) = JS and, as we have noted, : JS ! S is an isomorphism.
The evident homeomorphisms
0V W0W (V 0V )(W0W)
V A ^ B ~= (A ^ B)
for V V 0in V U and W W 0in W U, induce an isomorphism
(5.6) VVU (A) Z WUW (B) ~=(VVW)UW (A ^ B)
upon spectrification, where
Z : S V U x S WU ! S (V W)U
is the external smash product. Using the formal properties [5, A.6.2 and A.6.3]*
* of
twisted halfsmash products, the canonical homeomorphism
L (2) xL (1)xL (1)(I (V U; U) x I (W U; U)) ~=I ((V W ) U; U)
given by Hopkins' lemma [5, I.5.4], and the associative and unital properties o*
*f ^L
of [5, Ixx5,8], we see that the isomorphisms (5.6) induce isomorphisms
(5.7) OE : N*(V ) ^ N*(W ) ! N*(V W ):
We may identify Rn U with Un . With the notations of [5, II.1.7], the canon
ical cofibrant sphere Smodules are SnS= FSn, where Sn is the canonical sphere
spectrum. For n 0, Sn = 1nS0. Thus N*(R) = S1Sand, for n 1,
(5.8) N*(Rn) ~=(S1S)(n)~=SnS= F1nS0;
where the middle isomorphism is only canonical up to homotopy. The first isomor
phism is nequivariant, which is the essential point of Remark 3.6. If dimV = n,
n > 0, then N*(V ) is isomorphic to N*(Rn) and is thus cofibrant. Moreover, 1Vis
isomorphic to 1n, so that Lemma 4.6 holds.
Intuitively, (5.5) gives a coordinatefree generalization of the canonical c*
*ofi
brant negative sphere Smodules used in [5]. We must still prove the contravari*
*ant
functoriality in V of N*(V ), check the naturality of OE, and prove that the ev*
*aluation
maps " : N*(V ) ^ SV  ! N*(0) are weak equivalences. While this can be done
directly in terms of the definitions on hand, it is more illuminating to review*
* the
definition of the halfsmash product and relate it directly to the morphism spa*
*ces
of the category J . We introduce a category of Thom spaces for this purpose.
Its objects will be inclusions V U, which we secretly think of as symbols UVsi*
*nce
these objects are0closely related to the functors UVused in our definition of N*
**.
We think of TVU;U;Vi0n the following definition as a slightly abbreviated notat*
*ion for
the morphism space (UV; U0V)0.
5. THE CONSTRUCTION OF THE FUNCTOR N* 17
Definition 5.9.Let U and U0 be finite or countably infinite dimensional real
inner product spaces. Let0V and V 0be finite dimensional sub inner product spac*
*es
of U and U0. Let IVU;U;Vb0e the space of linear isometries f : U ! U0 such that
f(V ) V00. For V W , let W  V denote the orthogonal complement0of V in W .
Let EU;UV;Vb0e the subbundle of the product bundle IVU;U;Vx0V 0whose points are
0 U;*
*U0
the pairs (f; x) such that x 2 V 0 f(V ). Let TVU;U;Vb0e the Thom space of EV;*
*V 0; it
is obtained by applying fiberwise0onepoint compactification and identifying al*
*l of
the points at 1. The spaces TVU;U;Va0re the morphism spaces of a based topologi*
*cal
Thom category whose objects are the inclusions V U. Composition
0;U00 U;U0 U;U00
(5.10) O : TVU0;V^00TV;V 0! TV;V 00
is defined0by (g; y)O(f; x)0= (gOf; g(x)+y). Points (idU; 0) give identity morp*
*hisms.
If IVU;U;Vi0s empty, TVU;U;Vi0s a point. For any U and any object V 0 U0,
0 0 V 0
(5.11) T0U;U;V=0I (U; U )+ ^ S :
The category is symmetric monoidal with respect to direct sums of inner product
spaces. On morphism spaces, the map
0 U2;U0 U1U2;U0U0
(5.12) : TVU1;U11;V10^ TV2;V220! TV1V2;V1102V20
sends ((f1; x1); (f2; x2)) to (f1 f2; x1 + x2). Note that we have a trivializ*
*ation
isomorphism of bundles
0 U;U0 0
EU;UV;Vx0V ~=IV;V 0x V
and thus an "untwisting isomorphism"
0 V U;U0 V 0
(5.13) TVU;U;V^0S ~=IV;V 0+^ S :
The theory of orthogonal spectra is based on the full subcategory of whose
objects are the identity inclusions V V . If V V 0, then it is easily verifie*
*d that
0 0 V 0V
TVV;V;V~0=O(V )+ ^O(V 0VS) :
Comparing with the definitions in [18, 2.1, 4.4], we obtain the following resul*
*t.
Proposition 5.14. The full subcategory of whose objects are the identity
maps V V is isomorphic as a based symmetric monoidal category to the category
J such that an orthogonal spectrum is a continuous based functor J ! T .
We regard this isomorphism of categories as an identification.
In contrast, the twisted halfsmash product is defined in terms of the full *
*sub
category of whose objects are the inclusions V U in which U is infinite dimen
sional. The following definition and lemma are taken from [5, A.4.1A.4.3].
0
Definition 5.15.Fix V U and U0. Define a prespectrum TVU;U;indexed on
0
U0 by letting its V 0th space be TVU;U;Va0nd letting its structure map for V 0 *
*W 0be
induced by passage to Thom spaces from the evident bundle map
0 0 0 U;U0 U;U0
EU;UV;V 0(W  V ) ~=EV;W0I U;U0! EV;W0:
V;V 0
18 I. ORTHOGONAL SPECTRA AND SMODULES
0 U;U0
For V W , define a map o : WV TWU;U;! TV; of prespectra indexed on U0
by letting its V 0th map be induced by passage to Thom spaces from the evident
bundle map
0 U;U0 U;U0
EU;UW;V 0(W  V ) ~=EV;V 0I U;U0! EV;V 0:
W;V 0
0 U;U0 U;U0
Observe that TVU;U;is an inclusion prespectrum and define MV; = LTV; . (That
is, write M consistently for Thom spectra associated to Thom prespectra T .)
Lemma 5.16. The spectrified map
0 WV U;U0 U;U0 U;U0
Lo : WV MU;UW;~=L( TW; ) ! LTV; = MV;
is an isomorphism of spectra indexed on U0.
The following is a special case of the definition of the twisted half smash *
*product
given in [5, A.5.1].
Definition 5.17.Let E be a spectrum indexed on U. Define
0
I (U; U0) n E = colimVMU;UV;^ EV
where the colimit (in S U0) is taken over the maps
0 WV U;U0 U;U0 WV U;U0
MU;UV;^ EV ~= MW; ^ EV ~=MW; ^ EV ! MW; ^ EW
induced by the structure maps of E.
The following result of Cole [5, A.3.9] is pivotal.
Proposition 5.18. For based spaces A, there is a natural isomorphism
0
I (U; U0) n UVA ~=MU;UV;^ A
of spectra indexed on U0.
The proof is simply the observation that, in this case, the defining colimit
stabilizes at the V th stage. Returning to the fixed choice of U = R1 and taki*
*ng
A = S0, this gives the alternative description
(5.19) N*(V ) ~=JMVVU;U;:
We regard this isomorphism as an identification and use it to show the required
functoriality of the N*(V ).
Definition 5.20.Tensoring linear isometries V ! W with idU, we obtain a
map : TVV;W;W! TVV;U;WUW . The evaluation maps N*(W )^J (W; V ) ! N*(V )
of the contravariant functor N* are defined to be the maps
JMWU;UW;^ TVV;W;Wid^!JMWU;UW;^ TVV;U;WUW
~= JL(TWWU;U;^ TVV;U;WUW )
JL(O)!JL(T V U;U V U;U
V; ) = JMV;
induced by composition in the category .
The naturality of the maps OE of (5.7) is now checked by rewriting these maps
in terms of Thom complexes, using (5.12). Finally, we have the following lemma.
6. THE FUNCTOR M AND ITS COMPARISON WITH N 19
Lemma 5.21. The evaluation map " : N*(V )^SV ! N*(0) ~=S of the functor
N* is a weak equivalence. When V = R, " factors as the composite of the canonic*
*al
isomorphism N*(R)^S1 ~=SS and the canonical cofibrant approximation SS ! S.
Proof. Using the untwisting isomorphisms
0
TVV;U;UV^0SV ~=IVV;U;UV^0SV
and applying L, we obtain an isomorphism of Lspectra
MVVU;U;^ SV ~=I (V U; U)+ ^ S:
Applying J and using JS ~=S, we find by (5.19) that
(5.22) N*(V ) ^ SV ~=J((I (V U; U)+ ^ S) ~=I (V U; U)+ ^ S:
Under this isomorphism, the evaluation map corresponds to the homotopy equiva
lence induced by the evident homotopy equivalence I (V U; U)+ ! S0. When
V = R, LS ~=L (1)+ ^ S and the isomorphism just given is the cited canonical_
isomorphism N*(R) ^ S1 ~=SS. __
6. The functor M and its comparison with N
We begin with the underlying prespectrum and spectrification functors:
(6.1) I S _U___//P__L_//_S :
The functor M is the composite of the following three functors:
J
(6.2) I S _U__//_P[L]_L_//_S [L]___//M :
The categories P[L] and S [L] are the categories of Lprespectra and Lspectra.
We have already indicated what Lspectra are, and we shall define Lprespectra
shortly. The functors U and L in (6.2) are restrictions of those of (6.1), and *
*the
functor J is specified in Definition 5.4. Thus, to construct M, we must define *
*L
prespectra and show that the functors U and L induce functors from orthogonal
spectra to Lprespectra and from Lprespectra to Lspectra. The arguments are
already implicit in [21].
Definition 6.3.For a prespectrum X and a linear isometry f : U  ! U,
define a prespectrum f*X by (f*X)(V ) = X(fV ), with structure maps
f oe
X(fV ) ^ SWV id^S//_X(fV ) ^ Sf(WV_)__//X(fW ):
Observe that f*X is a spectrum if X is a spectrum.
Definition 6.4.An Lprespectrum is a prespectrum X together with maps
(f) : X ! f*X of prespectra for all linear isometries f : U ! U such that
(id) = id, (f0) O (f) = (f0O f), and the function
: TVU;U;W^ X(V ) ! X(W )
specified by
((f; w); x)) = oe((f)(x); w)
is a continuous.
20 I. ORTHOGONAL SPECTRA AND SMODULES
In Definition 5.2, we defined a Lspectrum to be an algebra over the monad
L. Inspection of the construction of twisted half smash products in x5 (compare
[25, XXII.5.3]) gives the following consistency statement. While this equivalen*
*ce
of definitions is not difficult, we emphasize that it is central to the mathema*
*tics: it
converts structures that are defined one isometry at a time into structures tha*
*t are
defined globally in terms of spaces of isometries.
Lemma 6.5. An Lspectrum is an Lprespectrum that is a spectrum.
Lemma 6.6. The functor L : P ! S induces a functor P[L] ! S [L].
Proof. For a linear isometry f : U ! U, the functor f* : P ! P and its
restriction f* : S ! S have left adjoints f*. The functor f* on spectra is def*
*ined
in terms of the functor f* on prespectra by f* = Lf*` [17, IIx1]. Let X be an L
prespectrum. The map (f) has an adjoint map f*X ! X; applying L, we obtain
a map f*LX ! LX, and its adjoint gives an induced map (f) : LX ! f*LX.
The properties (id) = id and (f0 O f) = (f0) O (f) are inherited from their
prespectrum level analogues. Since the functor L is continuous and commutes with
smash products with spaces, the continuity and equivariance condition on in__
Definition 6.4 are also inherited by LX. __
Lemma 6.7. The functor U : I S ! P takes values in P[L].
Proof. We obtain (f) : X ! f*X by applying the functoriality of X and
the naturality of oe to the restrictions of linear isometries f : U ! U to lin*
*ear
isometric isomorphisms f : V ! f(V ) for indexing spaces V . It is clear by
functoriality that (id) = id and (f0 O f) = (f0) O (f). The continuity and
equivariance condition on in Definition 6.4 follow from the continuity, natura*
*lity_
and equivariance of oe. __
Remark 6.8. For general Lprespectra, the map (f) : X(V ) ! X(fV ) de
pends on the linear isometry f : U ! U and not just on its restriction V ! f(*
*V ).
For those Lprespectra that come from orthogonal spectra, this map does de
pend solely on the restriction of f. For this reason, there is no obvious func*
*tor
P[L] ! I S .
The following lemmas give the basic formal properties of the functor M.
Lemma 6.9. The functor M is right exact.
Proof. The functors U, L, and J are each right exact. This is obvious for U
from the spacewise specification of colimits and smash products with based spac*
*es,_
and it holds for L and J since these functors are left adjoints. *
* __
Lemma 6.10. There is a canonical isomorphism : M(SI ) ! S.
Proof. Clearly U(SI ) is the usual sphere prespectrum and thus S = LU(SI_)._
As we have already used, JS ~=S by [5, I.8.2]. __
Lemma 6.11. The functor M is lax symmetric monoidal.
Proof. We have MSI ~=S, and we must construct a natural map
OE : M(X) ^ M(X0) ! M(X ^I X0)
for orthogonal spectra X and X0. The functor J is strong symmetric monoidal, so
(JE) ^ (JE0) ~=J(E ^L E0)
6. THE FUNCTOR M AND ITS COMPARISON WITH N 21
for Lspectra E and E0. Thus it suffices to construct a map of Lspectra
OE : LU(X) ^L LU(X0) ! LU(X ^I X0);
and OE is obtained by passage to coequalizers from a map
: L (2) n LU(X) Z LU(X0) ! LU(X ^I X0):
To construct , it suffices to construct maps
(f) : LU(X)(V ) ^ LU(X0)(V 0) ! LU(X ^I X0)(f(V V 0))
for linear isometries f 2 L (2) such that the (f) satisfy analogs of the condit*
*ions
in Definition 6.4 [25, XXII.5.3]. The functoriality of X and X0 gives maps
X(V ) ^ X0(V 0) ! X(f(V )) ^ X0(f(V 0)):
The universal property (2.13) that relates the external and internal smash prod*
*uct
of orthogonal spectra gives a map of (J x J )spaces
X Z X0 ! (X ^I X0) O ;
and this gives maps
X(f(V )) ^ X0(f(V 0)) ! (X ^I X0)(f(V V 0)):
We obtain the required maps (f) from the composites
X(V ) ^ X0(V 0) ! (X ^I X0)(f(V V 0))
by passing to prespectra and then to spectra, as in the proof of Lemma 6.6. The
coherence properties of the maps OE obtained from these maps are shown by form*
*al_
verifications from the properties of the various smash products. *
*__
Turning to homotopical properties, we have the following observation. Recall
Remark 4.10.
Lemma 6.12. If X is a positive inclusion orthogonal spectrum, then there are
natural isomorphisms
ss*(X) ~=ss*(M(X)):
Proof. We have a natural weak equivalence : M(X) = JLU(X) ! LU(X) __
for any X, and the unit map j : UX ! `LU(X) is also a weak equivalence. __
Now the following theorem compares M and N.
Theorem 6.13. There is a symmetric monoidal natural transformation
ff : N ! M
such that ff : NX ! MX is a weak equivalence if X is cofibrant.
Proof. Recall the definition M* = MODJ : J ! M (see Definition 2.3 and
Notation 2.6). By Corollary 2.7, to construct ff, it suffices to construct a na*
*tural
transformation ff* : N* ! M*. Thus consider the orthogonal spectra V *specified
by V *(W ) = J (V; W ). By definition, M*V = MV *= JLUV *. By Proposition
5.14, for W U,
UV *(W ) ~=TVV;W;W:
For V W Z, the structural map agrees under this isomorphism with
: TVV;W;W^ SZW ~=TVV;W;W^ T00;ZW;ZW! TVV;Z;Z:
22 I. ORTHOGONAL SPECTRA AND SMODULES
We obtain a map of Thom spaces TVV;U;UW! TVV;W;Wby restricting to V the linear
isometries f : V U ! U such that f(V ) W . These maps define a map of
prespectra TVV;U;U! UV *. Applying JL and using (5.19), there results a map of
Smodules
ff* : N*(V ) = JLTVV;U;U! JLUV *= M*(V ):
It is an exercise to verify from Proposition 5.14 and the definitions that thes*
*e maps
specify a natural transformation that is compatible with smash products. Using
Theorem 2.5, define
ff = ff* J id : NX = N* J X ! M* J X ~=MX:
Then ff is a symmetric monoidal natural transformation, and it remains to prove
that ff : NX ! MX is a weak equivalence if X is cofibrant. It suffices to assu*
*me
that X is an FIcell complex (see [18, x6]). Since M and N are right exact, it
follows by the usual induction up the cellular filtration of X, using commutati*
*ons
with suspension, wedges, pushouts, and colimits, that it suffices to prove that*
* ff
is a weak equivalence when X = V *. In this case, ff reduces to ff*. Again by
suspension, it suffices to prove that
V ff* : V N*(V ) ! V M*(V )
is a weak equivalence. We have an untwisting isomorphism (5.22) for the source *
*of
V ff* and an analogous isomorphism
M(V *) ^ SV ~=I (V; U)+ ^ S
for its target. Under these isomorphisms, V ff* is the smash product with S of *
*the
map I (V U; U) ! I (V; U) induced by restriction of linear isometries, and th*
*is__
map is a homotopy equivalence since its source and target are contractible. *
* __
Remark 6.14. By Proposition 2.8, the functor M has right adjoint M# . How
ever, M does not appear to preserve cofibrant objects and does not appear to be
part of a Quillen equivalence.
7. A revisionist view of infinite loop space theory
In 1971 [19], the second author gave an infinite loop space machine for the
passage from space level data to spectra. That machine gave coordinatized spect*
*ra
as its output. He improved the machine and showed how to feed category level da*
*ta
into it a little later [20]. In 1980 [22], he retooled the machine to give coor*
*dinate
free spectra as its output. The main motivation for the retooling was to show t*
*hat
space level and category level pairing data give rise to pairings X ^ Y  ! Z of
spectra. Of course, this long preceded the formal introduction of diagram spect*
*ra.
Nevertheless, their use was implicit in [19] and explicit in [22], as we now ex*
*plain.
We first show that the original machine of [19] takes values in symmetric sp*
*ec
tra. We retain most of the notations of [19] and refer to it for details. The m*
*achine
of [19] was based on the little ncubes operads Cn, which, in the earlier langu*
*age
of PROP's, were introduced by Boardman and Vogt [3]. The jth space Cn(j) con
sists of jtuples of little ncubes with disjoint interiors. A little ncube is*
* a map
f : In ! In that is the product of n linear maps fi: I ! I, fi(t) = (yixi)t+*
*xi
with 0 xi< yi 1. Obviously n acts on In by permuting coordinates and acts
on little ncubes by conjugation, (oe . f)(t) = oef(oe1t) for t 2 In. This me*
*ans
7. A REVISIONIST VIEW OF INFINITE LOOP SPACE THEORY 23
that the Cn give a symmetric sequence of operads. Therefore they give an associ
ated symmetric sequence of monads Cn on the category T of based spaces. Given
another operad C , in practice an E1 or at least spacewise constractible opera*
*d,
one can form the product operads Dn = C x Cn. Via the action of the symmetric
groups on the Cn, this is another symmetric sequence of operads, and it gives r*
*ise
to another symmetric sequence of monads Dn. Let X be a C space. By pullback
along the projections to C , X is a Dnspace for all n. There is a map of mon
ads ffn : Cn ! nn for each n, and an adjoint right action of the monad Cn
on the nfold suspension functor n. With the evident actions of the symmetric
groups, the ffn give a map of symmetric sequences of monads. Via the projections
Dn ! Cn, these statements remain true with the Cn replaced by the Dn. Define
Tn(X) = B(n; Dn; X):
This is a nspace. Taking the product of a little ncube with the identity map
on Im gives a little (n + m)cube. This gives a map of operads Cn ! Cn+m , and
thus a map of operads Dn ! Dn+m . The functor m is given by smashing with
Sm , and we obtain a canonical map
oe : m Tn(X) ~=B(n+m ; Dn; X) ! B(n+m ; Dn+m ; X):
This map is (n x m )equivariant. This proves the following result.
Theorem 7.1. For a C space X, the spaces Tn(X) and structure maps oe spec
ify a symmetric spectrum T (X).
The main theorem in this approach to infinite loop space theory can be stated
as follows [19, 20].
Theorem 7.2. If C is spacewise contractible, then the adjoint structure maps
"oe: Tn(X) ! Tn+1(X) are weak equivalences for n > 0, and there is a canonical
map j : X ! nTn(X) that is a group completion for n > 1.
From the point of view of symmetric spectra, this means that T (X) is a posi*
*tive
spectrum (a fibrant object in the positive stable model structure), and the ze*
*roth
space of its associated spectrum (a fibrant approximation in the stable model
structure) is a group completion of X. Taking the original point of view of [1*
*9,
20], we note that the "oeare inclusions, so that we can pass to colimits to obt*
*ain
a spectrum E(X) with nth space En(X) = colimm m Tm+n (X) together with
a group completion j : X  ! E0(X). Implicitly, E(X) is obtained from the
symmetric spectrum T (X) by applying the forgetful functor to prespectra and
then the spectrification functor. We can instead prolong T (X) to an orthogonal
spectrum and apply the functor N (or M) from orthogonal spectra to Smodules,
a process which retains more precise information.
In [26], this machine is generalized to take ^Cspaces as input, where ^Cis *
*the
"category of operators" associated to C . The discussion above applies just as *
*well
to the generalized machine, which again gives symmetric spectra as output. The
generalized machine accepts Segal's spaces as special cases of its input. Roug*
*hly
speaking, the uniqueness theorem of [26] says that, up to equivalence, the func*
*tor
E and natural group completion j : X ! E(X) from the category of ^Cspaces to
the category of spectra is unique.
There is a (m x n)equivariant pairing of operads (Cm ; Cn) ! Cm+n [19,
8.3]. These pairings fit naturally and easily into the theory of external smash
24 I. ORTHOGONAL SPECTRA AND SMODULES
products of symmetric spectra. Using the internalization of the smash product
obtained by Kan extension [15, 18], this gives the starting point for an elabor*
*ation
of infinite loop space that shows how to pass from pairings of spaces with oper*
*ad
actions (or category of operator actions) to pairings X ^ Y  ! Z of symmetric
spectra.
While Theorem 7.1 is a new observation, its coordinatefree analogue was ex
plained in detail in 1980 [22, xx5, 6], where orthogonal spectra were introduced
under the name of I*prespectra. Moreover, the analogue was used there to give
the elaboration of infinite loop space theory that shows how to pass from pairi*
*ngs of
spaces with operad actions (or category of operator actions) to pairings of ort*
*hog
onal spectra defined in terms of external smash products. Now that we understand
the internalization of the smash product, the arguments given there have strong*
*er
conclusions. Implicitly the passage from orthogonal spectra to spectra in [22] *
*was
obtained by applying the forgetful functor to prespectra and then the spectrifi*
*cation
functor. This does not preserve pointset leve smash products, and we can inste*
*ad
use the functor N (or just M). We conclude that all statements in [22] about the
construction of pairings X ^ Y ! Z of spectra in the homotopy category actually
give pairings of Smodules that are welldefined and enjoy good algebraic prope*
*rties
on the pointset level.
We briefly recall how the theory of [22] goes. For a finite dimensional real
inner product space V , there is a Steiner operad KV [31]. The group I (V; V )
acts on it in a similar fashion to the action of n on Cn. In fact, the KV give *
*the
object function of a functor K from I to the category of operads [22, 6.7]. We
can mimic the discussion above, but replacing the little cubes operads Cn with *
*the
Steiner operads KV , setting DV = C x KV . For a ^Cspace X, we construct spaces
TV (X) = B(V ; ^D; X)
and maps
oe : W TV (X) ! TW (X):
(Technically, we have suppressed use of a forgetful functor in writing down the*
* bar
construction [22, p. 325]). We obtain the following conclusion, which is [22, 6*
*.1].
Theorem 7.3. For a ^Cspace X, the spaces TV (X) and structure maps oe spec
ify an orthogonal spectrum T (X).
Pairings of operads, categories of operators, C spaces, ^Cspaces, and perm*
*u
tative categories are studied and interrelated in [22, xx14]. There are pairi*
*ngs
(KV ; KW ) ! KV W analogous to the pairings (Cm ; Cn) ! Cm+n . This mate
rial provides input for the infinite loop space theory of pairings and is uncha*
*nged by
the present revisionist attitude towards the output of that theory. By internal*
*izing
the output external pairings, we obtain the following reinterpretation of [22, *
*6.2].
Theorem 7.4. Let ^ : (C ; D) ! E be a pairing of operads. Then pairings
f : (X; Y ) ! Z of a ^Cspace X and a ^Dspace Y to an ^Espace Z functorially
determine maps T f : T X ^ T Y ! T Z of orthogonal spectra.
In particular, by [22, 2.2], this applies to pairings of permutative categor*
*ies.
There is an analogous result [22, 6.3] for ring spectra. In [22], ring spectra *
*were
thought of in the classical, up to homotopy, sense. While [22, 6.3] can now be *
*rein
terpreted on the point set level, the result then seems to be without applicati*
*on since
the resulting input data are too stringent to arise in nature; see [22, p. 310]*
*. Thus
7. A REVISIONIST VIEW OF INFINITE LOOP SPACE THEORY 25
the present reinterpretation of output data does not obviate the need for the m*
*uch
more elaborate multiplicative infinite loop space theory of [23]. That theory s*
*hows
how to pass from bipermutative categories to E1 ring spectra, alias commutative
Salgebras. By the comparisons here and in [18], it follows that bipermutative
categories give rise to commutative symmetric ring spectra and commutative or
thogonal ring spectra. It is plausible that a different line of argument might *
*give a
direct construction of this sort.
The equivariant generalization of infinite loop space theory shows how to co*
*n
struct Gspectra from equivariant space and category level input. The natural
output of the equivariant version of the machine that we have been discussing is
given by orthogonal Gspectra, which are the objects of study of the rest of th*
*is
monograph. We intend to return to equivariant infinite loop space theory elsewh*
*ere.
CHAPTER II
Equivariant orthogonal spectra
This chapter parallels [18, Part I], and we focus on points of equivariance.*
* It
turns out that we need to distinguish carefully between topological Gcategories
CG , which are enriched over Gspaces, and their Gfixed topological categories*
* GC ,
which are enriched over spaces. After explaining this in x1, we define orthogon*
*al
Gspectra in x2, discuss their smash product in x3, and reinterpret the definit*
*ion in
terms of diagram spaces in x4.
1. Preliminaries on equivariant categories
Recall that T denotes the category of based spaces and let GT denote the
category of based Gspaces and based Gmaps. Then GT is complete and cocom
plete, and it is a closed symmetric monoidal category under its smash product a*
*nd
function Gspace functors. For based Gspaces A and B, we write F (A; B) for the
function Gspace of all maps A ! B, with G acting by conjugation. Thus
GT (A; B) = F (A; B)G :
That is, a Gmap A ! B is a fixed point of F (A; B).
It is useful to think of GT in a different fashion. Let TG be the category *
*of
based Gspaces (with specified action of G) and nonequivariant maps, so that
TG (A; B) = F (A; B):
Then TG is enriched over GT : its morphism spaces are Gspaces, and composition
is given by Gmaps. The objects of GT and TG are the same. If we think of G as
acting trivially on the collection of objects (after all, gA = A for all g 2 G)*
*, then
we may think of GT as the Gfixed point category (TG )G .
Observe that TG is also closed symmetric monoidal under the smash product
and function Gspace functors, with S0 as unit. If we ignore the fact that TG is
enriched over GT , we obtain inverse equivalences of categories TG  ! T and
T ! TG by forgetting the action of G on Gspaces and by giving spaces the
trivial action by G. Of course, limits and colimits of diagrams of Gspaces (ta*
*ken
in T ) only inherit sensible Gactions when the maps in the diagrams are Gmaps,
so that we are working in GT .
Many of our equivariant categories will come in pairs like this: we will have
a category CG consisting of Gobjects and nonequivariant maps, and a category
GC with the same objects and the Gmaps between them. We can think of GC as
(CG )G , although the notation would be inconvenient. Formally, CG will be enri*
*ched
over GT , so that its hom sets CG (C; D) are based Gspaces and composition is *
*given
by continuous Gmaps. We call such a category a topological Gcategory. As in [*
*18],
when the morphism spaces of CG are given without basepoints, we implicitly give
them disjoint Gfixed basepoints.
26
1. PRELIMINARIES ON EQUIVARIANT CATEGORIES 27
In all cases that we will encounter, we will have a faithful underlying based
Gspace functor S : CG ! TG , so that S embeds CG (C; D) as a sub Gspace of
F (SC; SD). We can define a Gmap C ! D to be a fixed point of CG (C; D), and
we will have
GC (C; D) = CG (C; D)G ~=SCG (C; D) \ F (SC; SD)G :
We emphasize that it is essential to think in terms of such topological Gcateg*
*ories
CG even when the categories of ultimate interest are the associated categories *
*GC
of Gobjects and Gmaps between them. Note that, when constructing model
structures, we must work in GC in order to have limits and colimits.
A continuous Gfunctor X : CG ! DG between topological Gcategories is a
functor X such that
X : CG (C; D) ! DG (X(C); X(D))
is a map of Gspaces for all pairs of objects of CG . In terms of elementwise a*
*ctions,
this means that gX(f)g1 = X(gfg1). It follows that X takes Gmaps to G
maps. From now on, all functors defined on topological categories are assumed to
be continuous.
A natural Gmap ff : X ! Y between Gfunctors CG  ! DG consists of
Gmaps ff : X(C) ! Y (D) such that the evident naturality diagrams
X(C) _____//X(D)
ff ff
fflffl fflffl
Y (C)_____//Y (D)
commute in DG for all maps (and not just all Gmaps) C ! D.
Since the present point of view has not appeared explicitly in previous stud*
*ies
of equivariant stable homotopy theory, we give the definitions of the categorie*
*s PG
and GP of Gprespectra and their full subcategories SG and GS of Gspectra.
See [17] or [25] for more details. In fact, we have such categories for any Gu*
*niverse
U, and we write PUG, etc, when necessary for clarity.
Definition 1.1.A Guniverse U is a sum of countably many copies of each
real Ginner product space in some set of irreducible representations of G that
includes the trivial representation; U is complete if it contains all irreduci*
*ble rep
resentations; U is trivial if it contains only trivial representations. An in*
*dexing
Gspace in U is a finite dimensional sub Ginner product space of U. When V W ,
we write W  V for the orthogonal complement of V in W . Define V (U) to be
the collection of all real Ginner product spaces that are isomorphic to indexi*
*ng
Gspaces in U.
Write SV for the onepoint compactification of V , and write V A = A ^ SV
and V A = F (SV ; A) for the resulting generalized loop and suspension functors.
Definition 1.2.A Gprespectrum X consists of based Gspaces X(V ) for in
dexing Gspaces V U and based Gmaps oe : WV X(V ) ! X(W ) for V W ;
here oe is the identity if V = W , and the evident transitivity diagram must co*
*m
mute when V W Z. A map f : X ! Y of prespectra consists of based maps
f(V ) : X(V ) ! Y (V ) that commute with the structure maps oe; f is a Gmap if
the f(V ) are Gmaps. A Gspectrum is a Gprespectrum whose adjoint structure
maps "oe: X(V ) ! WV X(W ) are homeomorphisms of Gspaces.
28 II. EQUIVARIANT ORTHOGONAL SPECTRA
Q To fit this into the general framework above, define SX to be the based Gsp*
*ace
V U X(V ). Then S : PG ! TG is a faithful functor, and a map f : X ! Y of
prespectra is a Gmap if and only if Sf is a fixed point of F (SX; SY ). In pre*
*vious
work in this area, the focus is solely on the Gfixed categories GP and GS .
When U is the trivial universe, GS is the category of naive Gspectra, or
spectra with Gactions. When U is a complete universe, GS is the category of
genuine Gspectra (and the adjective is omitted): these Gspectra are the objec*
*ts
of the equivariant stable homotopy category of [17].
2. The definition of orthogonal Gspectra
Just as for Gspectra, we have several kinds of orthogonal Gspectra, depend*
*ing
on an initial choice of a set of irreducible representations of G. The reader i*
*s warned
that, as explained in [18, 7.1], nontrivial orthogonal Gspectra are never Gs*
*pectra
in the sense of the previous definition.
Definition 2.1.Let V = V (U) for some universe U. Define IGV to be the
(unbased) topological Gcategory whose objects are those of V and whose mor
phisms are the linear isometric isomorphisms, with G acting by conjugation on t*
*he
space IGV(V; W ) of morphisms V ! W . Let GI Vbe the category with the same
objects and the Glinear isometric isomorphisms between them, so that
GI V(V; W ) = IGV(V; W )G :
Define a canonical Gfunctor SVG: IGV ! TG by sending V to SV . Clearly IGV is
a symmetric monoidal category under direct sums of Ginner product spaces, and
the functor SVGis strong symmetric monoidal.
Variant 2.2. We could relax the conditions on V by allowing any cofinal sub
collection W of V that is closed under finite direct sums. Here "cofinal" me*
*ans
that, up to Gisomorphism, every V in V is contained in some W in W . We shall
need the extra generality when we consider change of groups.
We usually abbreviate IG = IGV and SG = SVG. The case of central interest
is V = A ``, the collection of all finite dimensional real Ginner product spa*
*ces,
but we shall work with the general case until we specify otherwise. From here, *
*the
basic categorical definitions and constructions of [18] go through without esse*
*ntial
change. The only new point to keep track of is which maps are Gmaps and which
are not. We give a quick summary. We shall not spell out diagrams, referring to
[18] instead. We choose and fix a skeleton skIG of IG .
Definition 2.3.An IG space is a (continuous) Gfunctor X : IG  ! TG .
Let IG T be the category whose objects are the IG spaces X and whose morphisms
are the natural maps X ! Y . Topologize the set IG T (X; Y ) of maps X ! Y
as a subspace of the product over V 2 skIG of the spaces of maps X(V ) ! Y (V )
and let G act on it by conjugation; this implicitly specifies an underlying bas*
*ed
Gspace functor S : IG T  ! TG . Let GI T be the category of IG spaces and
natural Gmaps, so that
GI T (X; Y ) = IG T (X; Y )G :
It is essential to keep in mind the distinction between maps and Gmaps of
IG spaces. We are interested primarily in the Gmaps.
2. THE DEFINITION OF ORTHOGONAL GSPECTRA 29
Definition 2.4.For IG spaces X and Y , define the "external" smash product
X Z Y by
X Z Y = ^ O (X x Y ) : IG x IG ! TG ;
thus (X Z Y )(V; W ) = X(V ) ^ Y (W ). For an IG space Y and an (IG x IG )spa*
*ce
Z, define the external function IG space F(Y; Z) by
F(Y; Z)(V ) = IG T (Y; Z);
where Z(W ) = Z(V; W ).
Remark 2.5. The definition generalizes to give the external smash product
functor 0 0
IGVT x IGVT  ! (IGVx IGV)T :
Definition 2.6.An IG spectrum, or orthogonal Gspectrum, is an IG space
X : IG ! TG together with a natural structure Gmap oe : X Z SG ! X O
such that the evident unit and associativity diagrams commute [18, xx1,8]. Let
IG S denote the topological Gcategory of IG spectra and maps f : X ! Y
that commute with the structure maps. Explicitly, the following diagrams must
commute, the maps oe being Gmaps and the maps f being nonequivariant maps
in general:
X(V ) ^ SW oe__//_X(W )
f^id f
fflffl fflffl
Y (V ) ^ SWoe__//Y (W )
If these diagrams commute, then so do the diagrams obtained by replacing f by gf
for g 2 G, so that IG S (X; Y ) is indeed a sub Gspace of IG T (X; Y ). Let GI*
* S
denote the category of IG spectra and the Gmaps between them, so that
GI S (X; Y ) = IG S (X; Y )G :
IG spectra are Gprespectra by neglect of structure.
Definition 2.7.Let V = V (U). Define a discrete subcategory (identity mor
phisms only) of IG whose objects are the indexing Gspaces in U. By restricting
functors IG ! TG to this subcategory and using structure maps for V W , so
that W = V (W  V ), we obtain forgetful functors
U : IG S ! PG and U : GI S ! GP:
Working with orthogonal Gspectra, we have an equivariant notion of a functor
with smash product (FSP). It was used in [12] and, implicitly, [13].
Definition 2.8.An IG FSP is an IG space X with a unit Gmap j : S ! X
and a natural product Gmap : X Z X ! X O of functors IG x IG ! TG
such that the evident unit, associativity, and centrality of unit diagrams comm*
*ute
[18, 22.3]. An IG FSP is commutative if the evident commutativity diagram also
commutes.
We have the topological Gcategory of IG FSP's and its Gfixed point catego*
*ry
of Gmaps of IG FSP's. An IG FSP is an IG spectrum with additional structure.
30 II. EQUIVARIANT ORTHOGONAL SPECTRA
Lemma 2.9. An IG FSP has an underlying IG spectrum with structure G
map
oe = O (idZj) : X Z S ! X O :
We emphasize that all structure maps (oe, j, ) in the definitions above must
be Gmaps. All maps must commute with these structure maps; the dichotomy
between structurepreserving maps and structurepreserving Gmaps persists.
3. The smash product of orthogonal Gspectra
Just as nonequivariantly, we can reinterpret FSP's in terms of a pointset l*
*evel
internal smash product on the category of orthogonal Gspectra that is associat*
*ive,
commutative, and unital up to coherent natural isomorphism.
Theorem 3.1. The category IG S of orthogonal Gspectra has a smash prod
uct ^SG and function spectrum functor FSG under which it is a closed symmetric
monoidal category with unit SG .
Passing to Gfixed points on morphism spaces, we obtain the following coroll*
*ary.
Corollary 3.2. The category GI S is also closed symmetric monoidal under
^SG and FSG.
After this section, we will abbreviate ^SG to ^ and FSG to F , but the more
cumbersome notations clarify the definitions.
Definition 3.3.A Gmonoid X in IG S is a monoid in GI S ; that is, the
unit SG ! X and product X ^SG X ! X must be Gmaps. Allowing morphisms
of such monoids that are not necessarily Gmaps, we obtain the Gcategory of G
monoids in IG S ; its associated fixed point category is the category of monoid*
*s in
GI S . Similarly, we obtain the Gcategory of commutative Gmonoids in IG S ;
its fixed point category is the category of commutative monoids in GI S .
The external notion of an IG FSP translates to the internal notion of a G
monoid in IG S .
Theorem 3.4. The categories of IG FSP's and of commutative IG FSP's are
isomorphic to the categories of Gmonoids in IG S and of commutative Gmonoids
in IG S .
We adopt a more familiar language for these objects.
Definition 3.5.A (commutative) orthogonal ring Gspectrum is a (commu
tative) monoid in GI S .
Theorem 3.4 asserts that (commutative) orthogonal ring Gspectra are the same
as (commutative) IG FSP's. That is, they are the same structures, but specified
in terms of the internal rather than the external smash product.
We outline the proof of Theorem 3.1, which is the same as in [18]. We first
construct a smash product ^ on the category of IG spaces [18, 21.4]. This in
ternalization of the external smash product is given by left Kan extension and *
*is
characterized by the adjunction homeomorphism of based Gspaces
(3.6) IG T (X ^ Y; Z) ~=(IG x IG )T (X Z Y; Z O ):
4. A DESCRIPTION OF ORTHOGONAL GSPECTRA AS DIAGRAM GSPACES 31
An explicit description of ^ is given in [18, 21.4]. There is one key subtle po*
*int.
The Kan extension is a kind of colimit, and our Gcategories of diagrams do not
admit colimits in general. However, the assumption that the maps
X : IG (V; W ) ! TG (X(V ); X(W ))
given by an IG space X must be Gmaps ensures that the equivalence relation
that defines the Kan extension is Ginvariant, producing a welldefined IG spa*
*ce
X ^ Y : IG ! TG from IG spaces X and Y .
There is a concomitant internal function IG space functor F constructed from
F [18, 21.6].
Proposition 3.7. The category of IG spaces is closed symmetric monoidal
under ^ and F . Its unit object is the functor IG ! TG that sends 0 to S0 and
sends V 6= 0 to a point.
We can reinterpret orthogonal Gspectra in terms of the internal smash produ*
*ct.
Proposition 3.8. The IG space SG is a commutative Gmonoid in IG T ,
and the category of orthogonal Gspectra is isomorphic to the category of SG 
modules.
From here, we imitate algebra, thinking of ^ and F as analogues of and Hom .
Definition 3.9.For orthogonal Gspectra X and Y , thought of as right and
left SG modules, define X ^SG Y to be the coequalizer in the category of IG s*
*paces
(constructed spacewise) displayed in the diagram
__^id_____//_
X ^ SG ^ Y __________//_X ^ Y___//_X ^SG Y;
id^0
where and 0are the given actions of SG on X and Y . Then X ^SG Y inherits
an IG spectrum structure from the IG spectrum structure on X or, equivalently,
Y . The function orthogonal Gspectrum FSG(X; Y ) is defined dually in terms of*
* a
suitable equalizer [18, x22]
FSG(Y; Z)_____//F (Y; Z)__//_//_F (Y ^ SG ; Z):
Theorem 3.1 follows easily from the definitions and the universal property (*
*3.6).
4.A description of orthogonal Gspectra as diagram Gspaces
As in [18, 2.1], there is a category JG = JGV constructed from IG and SG
such that if we define JG spaces exactly as in Definition 2.3, then a JG space
is the same structure as an IG spectrum. This reduces the study of orthogonal
Gspectra to a special case of the conceptually simpler study of diagram Gspac*
*es.
Rather than repeat the cited formal definition, we give a more concrete alterna*
*tive
description of JG in terms of Thom complexes, implicitly generalizing I.5.14 to
the equivariant context.
Definition 4.1.We define the topological Gcategory JGV. The objects of
JGVare the same as the objects of IGV. For objects V and V 0, let L (V; V 0) be*
* the
(possibly empty) Gspace of linear isometries from V to V 0; G acts by conjugat*
*ion.
Of course, a linear isometry is necessarily a monomorphism, but, in contrast to*
* our
definition of I , we no longer restrict attention to linear isometric isomorphi*
*sms.
Let E(V; V 0) be the subbundle of the product Gbundle L (V; V 0)xV 0consisting*
* of
32 II. EQUIVARIANT ORTHOGONAL SPECTRA
the points (f; x) such that x 2 V 0 f(V ). The Gspace JGV(V; V 0) of morphisms
V ! V 0in JGVis the Thom Gspace of E(V; V 0); it is obtained from the fiberwi*
*se
onepoint compactification of E(V; V 0) by identifying the points at infinity, *
*and it
is interpreted to be a point if L (V; V 0) is empty. Define composition
(4.2) O : JGV(V 0; V 00) ^ JGV(V; V 0) ! JGV(V; V 00)
by (g; y) O (f; x) = (g O f; g(x) + y). The points (idV; 0) give identity morph*
*isms.
Observe that JGVis symmetric monoidal under the operation specified by V V 0
on objects and
(f; x) (f0; x0) = (f f0; x + x0)
on morphisms. Let GJ V be the Gfixed category with the same objects, so that
GJ V(V; W ) = JGV(V; W )G :
We usually abbreviate JG = JGV. If dimV = dimV 0, then a linear isometry
V ! V 0is an isomorphism and JG (V; V 0) = IG (V; V 0)+ . This embeds IG as a
sub symmetric monoidal category of JG . If V V 0, then
0V
JG (V; V 0) ~=O(V 0)+ ^O(V 0VS)V :
In particular, the functor JG (0; ) : IG ! TG coincides with SG . The category
of JG spaces is symmetric monoidal, as in Proposition 3.7 but with unit SG , a*
*nd
we have the following result.
Theorem 4.3. The symmetric monoidal category of IG spectra is isomorphic
to the symmetric monoidal category of JG spaces.
Using this reinterpretation, we see immediately that the category GI S is
complete and cocomplete, with limits and colimits constructed levelwise. The ca*
*te
gory IG S is tensored and cotensored over the category TG of based Gspaces. For
an orthogonal Gspectrum X and a based Gspace A, the tensor X ^ A is given by
the levelwise smash product, (X ^ A)(V ) = X(V ) ^ A, and the cotensor F (A; X)
is given similarly by the levelwise function space. We have both
(4.4) IG S (X ^ A; Y ) ~=TG (A; IG S (X; Y )) ~=IG S (X; F (A; Y ))
and, by passage to fixed points,
(4.5) GI S (X ^ A; Y ) ~=GT (A; IG S (X; Y )) ~=GI S (X; F (A; Y )):
When G acts trivially on A, we can replace IG S (X; Y ) by GI S (X; Y ) in the *
*last
display. Therefore the category GI S is tensored and cotensored over T , althou*
*gh
not over GT . We define homotopies between maps of orthogonal Gspectra by use
of the cylinders X ^ I+ , and similarly for Ghomotopies between Gmaps.
We also use JG to define represented orthogonal Gspectra that give rise to
left adjoints to evaluation functors, as in [18, x3].
Definition 4.6.For an object V of IG , define the orthogonal Gspectrum V *
represented by V by V *(W ) = JG (V; W ). In particular, 0* = SG . Define the s*
*hift
desuspension spectrum functors FV : TG  ! IG S and the evaluation functors
EvV : IG S ! TG by FV A = V *^ A and EvV X = X(V ). Then FV and EvV
are left and right adjoint:
IG S (FV A; X) ~=TG (A; EvV X):
4. A DESCRIPTION OF ORTHOGONAL GSPECTRA AS DIAGRAM GSPACES 33
In order to mesh with notations in [17, 25], we introduce alternative names
for these functors.
Notations 4.7. Let 1 = F0 and 1 = Ev0. These are the suspension
orthogonal Gspectrum and zeroth space functors. Note that 1 A = SG ^ A.
Similarly, let 1V = FV and 1V = EvV ; SV = 1VS0 is the canonical (V )
sphere.
As in [18, 1.8], we have the following commutation with smash products.
Lemma 4.8. There is a natural isomorphism
FV A ^ FW B ~=FV W (A ^ B):
As in [18, 1.6], but with the tensor product of functors notation of Ix2, we*
* have
the following description of general orthogonal Gspectra in terms of represent*
*ed
ones. Observe that V *varies contravariantly in V , so that we have a contravar*
*iant
functor D : JG ! IG S specified by DV = V *.
Lemma 4.9. The evaluation maps V *^ X(V ) ! X of IG spectra X, thought
of as JG spaces, induce a natural isomorphism
Z V 2skIG
D JG X = V *^ X(V ) ! X:
The definitions and results of this section have analogues for prespectra. R*
*ecall
Definition 2.7.
Definition 4.10.Let V = V (U). We have a subcategory KG = KGV of JG
such that a KG space is the same thing as a Gprespectrum. The objects0of KG
are the indexing Gspaces in U; the morphism space KG (V; V 0) is SV V if V V*
* 0
and a point otherwise. The forgetful Gfunctor U : IG S ! PG has a left adjoint
prolongation functor P. With the notation of I.2.10, PX = D O KG X, where
: KG ! JG is the inclusion. See also [18, x3].
CHAPTER III
Model categories of orthogonal Gspectra
We explain the model structures on the category of orthogonal Gspectra and
on its various categories of rings and modules. The material here is parallel t*
*o the
material of [18, xx512]. However, since we are focusing on orthogonal spectra,*
* some
features that were made more complicated by the inclusion of symmetric spectra
in the theory there become simpler here. We focus on points of equivariance. One
new equivariant feature is the notion of a Gtopological model category, which *
*is
an equivariant analogue of the classical notion of a topological (or simplicial*
*) model
category. To make sense of this, we must take into account the dichotomy between
CG and GC : only GC can have a model structure, but use of CG is essential to
encode the Gtopological structure, which is used to prove the model axioms.
1. The model structure on Gspaces
We take for granted the generalities on nonequivariant topological model cat
egories explained in [18, x5]. In particular we have the notion of a compactly
generated model category, for which the small objects argument for verifying t*
*he
factorization axioms requires only sequential colimits. All of our examples of *
*model
categories will be of this form. However there are a few places where equivaria*
*nce
plays a role. We discuss these and then describe the appropriate model structure
on the category GT of based Gspaces. This must be known, although we have
not seen details in the literature. In any case, we find our treatment amusing.
We begin with a topological Gcategory CG and its Gfixed category GC of
Gmaps. We assume that GC is complete and cocomplete and that CG is tensored
and cotensored over GT , so that (II.4.4) and (II.4.5) hold with IG S and GI
replaced by CG and GC . The discussion in [18, x5] applies to GC . One place
where equivariance is relevant is in the Cofibration Hypothesis, [18, 5.3]. Tha*
*t uses
the concept of an hcofibration in GC , namely a map that satisfies the homotopy
extension property (HEP) in GC . Since the maps in GC are Gmaps, the HEP is
automatically equivariant. That is, hcofibrations in GC satisfy the GHEP. As *
*in
[18], we write qcofibration and qfibration for model cofibrations and cofibra*
*tions,
but we write cofibrant and fibrant rather than qcofibrant and qfibrant.
A more substantial point of equivariance concerns the notion of a topological
model category. As defined in [18, 5.12], that notion remembers only that GC is
tensored and cotensored over T , which is insufficient for our applications. We*
* shall
return to this point and define the notion of a "Gtopological model category" *
*after
giving the model structure on GT .
Definition 1.1.Let I be the set of cell hcofibrations
i : (G=H x Sn1)+ ! (G=H x Dn)+
34
1. THE MODEL STRUCTURE ON GSPACES 35
in GT , where n 0 (S1 being empty) and H runs through the (closed) subgroups
of G. Let J be the set of hcofibrations
i0 : (G=H x Dn)+ ! (G=H x Dn x I)+
and observe that each such map is the inclusion of a Gdeformation retract.
Recall that, for unbased spaces A and B, (A x B)+ ~= A+ ^ B+ . Recall too
that, for a based Hspace A and a based Gspace B,
(1.2) GT (G+ ^H A; B) ~=HT (A; B):
If A is a Gspace, then we have a natural homeomorphism of Gspaces
(1.3) G+ ^H A ~=(G=H)+ ^ A;
where G acts diagonally on the right. Also, for a based space A regarded as a
Gtrivial Gspace,
(1.4) GT (A; B) ~=T (A; BG )
and therefore
(1.5) GT ((G=H)+ ^ A; B) ~=T (A; BH ):
As a right adjoint, the Gfixed point functor preserves limits. It also pres*
*erves
some, but not all, colimits.
Lemma 1.6. The functor ()G on based Gspaces preserves pushouts of dia
grams one leg of which is a closed inclusion and colimits of sequences of inclu*
*sions.
For a based space A and a based Gspace B, F (A; B)G ~= F (A; BG ). For based
Gspaces A and B, (A ^ B)G ~=AG ^ BG .
Definition 1.7.A map f : A ! B of Gspaces is a weak equivalence or Serre
fibration if each fH : AH ! BH is a weak equivalence or Serre fibration; by (1*
*.5),
f is a Serre fibration if and only if it satisfies the RLP (right lifting prope*
*rty) with
respect to the maps in J. Note that a relative Gcell complex is a relative Ic*
*ell
complex as defined in [18, 5.4].
Theorem 1.8. GT is a compactly generated proper Gtopological model cate
gory with respect to the weak equivalences, Serre fibrations, and retracts of r*
*elative
Gcell complexes. The sets I and J are the generating qcofibrations and the ge*
*n
erating acyclic qcofibrations.
We have not yet defined what it means for GT to be "Gtopological", and we
shall turn to that concept after proving the rest of the theorem. For the proof*
*, we
compare GT to an appropriate model category of diagram spaces. Thus let GO be
the (unbased) topological category of orbit Gspaces G=H and Gmaps. We have
the category GOopT of GOopspaces, namely contravariant functors GO ! T .
This is an example of a category of diagram spaces, so the theory of [18, x6] a*
*pplies
to it; see also Piacenza [27]. We have functors
: GT  ! GOopT and : GOopT  ! GT
specified by (A)(G=H) = AH and (D) = D(G=e). It is the contravariance of
(A) as a functor on GO that motivates the use of GOop. Clearly O = Id. In
fact, we have the following elementary observation.
Lemma 1.9. The functor is full and faithful.
36 III. MODEL CATEGORIES OF ORTHOGONAL GSPECTRA
Lemma 1.6 implies the following further properties of .
Lemma 1.10. The functor preserves limits. It also preserves pushouts of dia
grams one leg of which is a closed inclusion and colimits of sequences of inclu*
*sions.
For a based space A and a based Gspace B, F (A; B) ~=F (A; B). For based G
spaces A and B, (A^B) ~=(A)^(B), where ((A)^(B))(G=H) = AH ^BH .
Note that (G=H+ )(G=K) = (G=H)K+~=GO(G=K; G=H)+ , so that (G=H+ )
is a represented diagram.
Definition 1.11.A map f : D ! E of GOopspaces is a level equivalence or
level fibration if each f(G=H) is a weak equivalence or Serre fibration. Let GOI
and GOJ be the sets of maps of the form (G=H+ ) ^ i and (G=H+ ) ^ j, where
i 2 I and j 2 J. A relative GOcell complex is a relative GOIcell complex (see
[18, 5.4]).
The following result is a special case of [18, 6.5]; most of it is in Piacen*
*za [27].
Theorem 1.12. The category GOopT is a compactly generated proper topo
logical model category with respect to the level equivalences, level fibrations*
*, and
retracts of relative GOcell complexes. The sets GOI and GOJ are the generating
qcofibrations and generating acyclic qcofibrations.
Proof of Theorem 1.8. By definition, a map f of Gspaces is a weak equiva
lence or Serre fibration if and only if f is a level equivalence or level Serre*
* fibration.
The amusing thing is that we have an analogue for cofibrations. By the preserva*
*tion
properties of already specified, it is clear that carries Gcell complexes to*
* GO
cell complexes. Moreover, because is full and faithful, it is elementary to ch*
*eck
inductively that if g : (X) ! D is a relative GOcell complex, then g = (f)
for a unique relative Gcell complex f; compare [25, VI.6.2]. We now see that b*
*oth
lifting axioms, both factorization axioms, and the left and right properness fo*
*r_GT
follow directly from the corresponding results for GOopT . _*
*_
We must still explain what it means for GT to be a "Gtopological" model
category. We revert to our general categories CG and GC , and we suppose that
GC has a given model structure. For maps i : A ! X and p : E ! B in GC , let
(1.13) CG (i*; p*) : CG (X; E) ! CG (A; E) xCG(A;B)CG (X; B)
be the map of Gspaces induced by CG (i; id) and CG (id; p) by passage to pullb*
*acks.
Definition 1.14.A model category GC is Gtopological if the map CG (i*; p*)
is a Serre fibration (of Gspaces) when i is a qcofibration and p is a qfibra*
*tion and
is a weak equivalence (as a map of Gspaces) when, in addition, either i or p i*
*s a
weak equivalence.
The point is that we must go beyond the category GC to the category CG to
formulate this equivariant notion. It follows on passage to Gfixed point spac*
*es
that GC is nonequivariantly topological, in the sense of [18, 5.12], but we nee*
*d the
equivariant version. The nonequivariant version has the following significance.
Lemma 1.15. The pair (i; p) has the lifting property if and only if GC (i*; *
*p*)
is surjective.
2. THE LEVEL MODEL STRUCTURE ON ORTHOGONAL GSPECTRA 37
As in [18, x5], we will need two pairs of analogues of the maps CG (i*; p*).*
* First,
for a map i : A ! B of based Gspaces and a map j : X ! Y in GC , passage to
pushouts gives a map
(1.16) ij : (A ^ Y ) [A^X (B ^ X) ! B ^ Y
and passage to pullbacks gives a map
(1.17) F (i; j) : F (B; X) ! F (A; X) xF(A;Y )F (B; Y );
where ^ and F denote the tensor and cotensor in CG .
Second, assume that CG is a closed symmetric monoidal category with product
^C and internal function object functor FC. Then, for maps i : X ! Y and
j : W ! Z in GC , passage to pushouts gives a map
(1.18) ij : (Y ^C W ) [X^CW (X ^C Z) ! Y ^C Z;
and passage to pullbacks gives a map
(1.19) F (i; j) : FC(Y; W ) ! FC(X; W ) xFC(X;Z)FC(Y; Z):
Inspection of definitions gives adjunctions relating these maps.
Lemma 1.20. Let i : A ! B be a map of based Gspaces and let j : X ! Y
and p : E ! F be maps in GC . Then there are natural isomorphisms of maps
CG ((ij)*; p*) ~=TG (i*; CG (j*; p*)*) ~=CG (j*; F (i; p)*):
Therefore, passing to Gfixed points, (ij; p) has the lifting property in GC if*
* and
only if (i; CG (j*; p*)) has the lifting property in GT .
Lemma 1.21. Let i, j, and p be maps in CG , where CG is closed symmetric
monoidal. Then there is a natural isomorphism of maps
CG ((ij)*; p*) ~=CG (i*; F (j; p)*):
Returning to TG and using Lemma 1.20, it is formal that the following lemma
is equivalent to the assertion that GT is Gtopological.
Lemma 1.22. Let i : A ! X and j : B ! Y be qcofibrations of Gspaces.
Then ij is a qcofibration and is acyclic if i or j is acyclic.
Proof. By passage to wedges, pushouts, colimits, and retracts, it suffices *
*to
prove the first part for a pair of generating qcofibrations. Here the conclusi*
*on holds
because products G=H x G=K with the diagonal action are triangulable as (finite)
GCW complexes. This is trivial when G is finite and holds for general compact
Lie groups by [16]. It suffices to prove the second part when i is a generating
acyclic qcofibration, but then i is the inclusion of a Gdeformation retract_a*
*nd the
conclusion is clear. __
2. The level model structure on orthogonal Gspectra
We here give the category GI S of orthogonal Gspectra and Gmaps a level
model structure, following [18, x3]; maps will mean Gmaps throughout. We need
three definitions, the first of which concerns nondegenerate basepoints. A Gsp*
*ace
is said to be nondegenerately based if the inclusion of its basepoint is an unb*
*ased
hcofibration (satisfies the GHEP in the category of unbased Gspaces). As in
[32, Prop. 9], a based hcofibration between nondegenerately based Gspaces is *
*an
unbased hcofibration. Each morphism space JG (V; W ) is nondegenerately based.
38 III. MODEL CATEGORIES OF ORTHOGONAL GSPECTRA
Definition 2.1.An orthogonal Gspectrum X is nondegenerately based if
each X(V ) is a nondegenerately based Gspace.
Definition 2.2.Define F I to be the set of all maps FV i with V 2 skIG and
i 2 I. Define F J to be the set of all maps FV j with V 2 skIG and j 2 J, and
observe that each map in F J is the inclusion of a Gdeformation retract.
Definition 2.3.We define five properties of maps f : X ! Y of orthogonal
Gspectra.
(i)f is a level equivalence if each map f(V ) : X(V ) ! Y (V ) of Gspaces *
*is a
weak equivalence.
(ii)f is a level fibration if each map f(V ) : X(V ) ! Y (V ) of Gspaces is*
* a
Serre fibration.
(iii)f is a level acyclic fibration if it is both a level equivalence and a l*
*evel
fibration.
(iv)f is a qcofibration if it satisfies the LLP with respect to the level ac*
*yclic
fibrations.
(v) f is a level acyclic qcofibration if it is both a level equivalence and *
*a q
cofibration.
Theorem 2.4. The category GI S of orthogonal Gspectra is a compactly gen
erated proper Gtopological model category with respect to the level equivalenc*
*es,
level fibrations, and qcofibrations. The sets F I and F J are the generating *
*q
cofibrations and the generating acyclic qcofibrations, and the following ident*
*ifica
tions hold.
(i)The level fibrations are the maps that satisfy the RLP with respect to F J
or, equivalently, with respect to retracts of relative F Jcell complexes*
*, and
all orthogonal Gspectra are level fibrant.
(ii)The level acyclic fibrations are the maps that satisfy the RLP with respe*
*ct
to F I or, equivalently, with respect to retracts of relative F Icell co*
*mplexes.
(iii)The qcofibrations are the retracts of relative F Icell complexes.
(iv)The level acyclic qcofibrations are the retracts of relative F Jcell co*
*mplexes.
Moreover, every cofibrant orthogonal Gspectrum X is nondegenerately based.
The proof is the same as that of [18, 6.5]. As there, the following analogue*
* of
[18, 5.5] plays a role.
Lemma 2.5. Every qcofibration is an hcofibration.
The following analogue of [18, 3.7] also holds. The proof depends on II.4.8 *
*and
Lemma 1.22 and thus on the fact that products of orbit spaces are triangulable *
*as
GCW complexes.
Lemma 2.6. If i : X ! Y and j : W ! Z are qcofibrations, then
ij : (Y ^ W ) [X^W (X ^ Z) ! Y ^ Z
is a qcofibration which is level acyclic if either i or j is level acyclic. In*
* particular,
if Z is cofibrant, then i ^ id: X ^ Z ! Y ^ Z is a qcofibration, and the smash
product of cofibrant orthogonal Gspectra is cofibrant.
Let [X; Y ]`Gdenote the set of maps X ! Y in the level homotopy category
Ho`GI S and let ss(X; Y )G denote the set of homotopy classes of maps X ! Y .
Then [X; Y ]`G~=ss(X; Y )G , where X ! X is a cofibrant approximation of X.
2. THE LEVEL MODEL STRUCTURE ON ORTHOGONAL GSPECTRA 39
Fiber and cofiber sequences of orthogonal Gspaces behave the same way as for
based Gspaces, starting from the usual definitions of homotopy cofibers and fi*
*bers
[18, 6.8]. We record the analogue of [18, 6.9]. Most of the proof is the same*
* as
there. Some statements, such as the last clause of (i), are most easily proven *
*by
using (1.5) and Lemma 1.6 to reduce them to their nonequivariant counterparts by
levelwise passage to fixed points.
Theorem 2.7. (i)If A is a based GCW complex and X is a nondegen
erately based orthogonal Gspectrum, then X ^ A is nondegenerately based
and
[X ^ A; Y ]`G~=[X; F (A; Y )]`G
for any Y . If f : X ! Y is a level equivalence of nondegenerately based
orthogonal Gspectra, then f ^Wid: X ^ A ! Y ^ A is a level equivalence.
(ii)For nondegenerately based Xi, iXi is nondegenerately based and
_ Y
[ Xi; Y ]`G~= [Xi; Y ]`G
i i
for any Y . A wedge of level equivalences of nondegenerately based orthog*
*onal
Gspectra is a level equivalence.
(iii)If i : A ! X is an hcofibration and f : A ! Y is any map of orthogonal
Gspectra, where A, X, and Y are nondegenerately based, then X [A Y
is nondegenerately based and the cobase change j : Y ! X [A Y is an
hcofibration. If i is a level equivalence, then j is a level equivalence.
(iv)If i and i0 are hcofibrations and the vertical arrows are level equivale*
*nces
in the following commutative diagram of nondegenerately based orthogonal
Gspectra, then the induced map of pushouts is a level equivalence.
X ooi___A _____//_Y
  
  
fflffl fflfflfflffl
X0ooi0_ A0_____//Y 0
(v) If X is the colimit of a sequence of hcofibrations in : Xn ! Xn+1 of
nondegenerately based orthogonal Gspectra, then X is nondegenerately bas*
*ed
and there is a lim1exact sequence of pointed sets
* ! lim1[Xn; Y ]`G! [X; Y ]`G! lim[Xn; Y ]`G! *
for any Y . If each in is a level equivalence, then the map from the init*
*ial
term X0 into X is a level equivalence.
(vi)If f : X ! Y is a map of nondegenerately based orthogonal Gspectra, then
Cf is nondegenerately based and there is a natural long exact sequence
. .!.[n+1X; Z]`G! [nCf; Z]`G! [nY; Z]`G! [nX; Z]`G! . .!.[X; Z]`G:
We shall also need a variant of the level model structure, called the positi*
*ve
level model structure, as in [18, x14]. It is obtained by ignoring representati*
*ons V
that do not contain a positive dimensional trivial representation. We can obtai*
*n a
similar model structure by ignoring only V = 0, but that would not give the rig*
*ht
model structure for some of our applications.
Definition 2.8.Define the positive analogues of the classes of maps specified
in Definition 2.3 by restricting attention to those levels V with V G 6= 0.
40 III. MODEL CATEGORIES OF ORTHOGONAL GSPECTRA
Definition 2.9.Let F +I and F +J be the sets of maps in F I and F J that
are specified in terms of the functors FV with V G 6= 0.
Theorem 2.10. The category GI S is a compactly generated proper Gtopo
logical model category with respect to the positive level equivalences, positiv*
*e level
fibrations, and positive level qcofibrations. The sets F +I and F +J are the *
*gen
erating sets of positive qcofibrations and positive level acyclic qcofibratio*
*ns. The
positive qcofibrations are the qcofibrations that are homeomorphisms at all l*
*evels
V such that V G = 0.
Proof. As in [18, x14], this is a special case of a general relative versio*
*n of
Theorem 2.4. For the last statement, the positive qcofibrations are the retrac*
*ts of
the relative F +Icell complexes, and a relative F Icell complex is a homeomor*
*phism
at levels V with V G 6= 0 if and only if no standard cells FV i with V G 6=_0 o*
*ccur in
its construction. __
Variants 2.11. There are other variants. Rather than using varying categories
IGV, we could work with orthogonal Gspectra defined with respect to V = A ``
and define a "V level model structure" by restricting to those levels V that *
*are
isomorphic to representations in V when defining level equivalences, level fibr*
*ations,
and the generating sets of qcofibrations and acyclic qcofibrations. This allo*
*ws us to
change V by changing the model structure on a single fixed category of orthogon*
*al
Gspectra; see Remark 1.9.
Remark 2.12. Everything in this section applies verbatim to the category GP
of Gprespectra. Recall II.2.7 and II.4.10. Because KG contains all objects of *
*JG ,
the forgetful functor U : GI ! GP creates the level equivalences and level
fibrations of orthogonal Gspectra. That is, a map f of orthogonal Gspectra is*
* a
level equivalence or level fibration if and only if Uf is a level equivalence o*
*r level
fibration of prespectra. In particular (P; U) is a Quillen adjoint pair [18, A.*
*1].
3.The homotopy groups of Gprespectra
By II.2.7, an orthogonal Gspectrum has an underlying Gprespectrum indexed
on a universe U such that V (U) = V . The homotopy groups of orthogonal G
spectra are defined to be the homotopy groups of their underlying Gprespectra,
and we discuss the homotopy groups of Gprespectra here. We first define G
spectra (more logically, prespectra).
Definition 3.1.A Gprespectrum X is an Gspectrum if each of its adjoint
structure maps "oe: X(V ) ! WV X(W ) is a weak equivalence of Gspaces. An
orthogonal Gspectrum is an orthogonal Gspectrum if each of its adjoint struc
ture maps is a weak equivalence or, equivalently, if its underlying Gprespectr*
*um
is an Gspectrum.
It is convenient to write
ssHq(A) = ssq(AH )
for based Gspaces A.
Definition 3.2.For subgroups H of G and integers q, define the homotopy
groups ssHq(X) of a Gprespectrum X by
ssHq(X) = colimVssHq(V X(V )) ifq 0;
3. THE HOMOTOPY GROUPS OF GPRESPECTRA 41
where V runs over the indexing Gspaces in U, and
q
ssHq(X) = colimV RqssH0(V R X(V )) ifq > 0:
A map f : X ! Y of Gprespectra is a ss*isomorphism if it induces isomorphisms
on all homotopy groups. A map of orthogonal Gspectra is a ss*isomorphism if i*
*ts
underlying map of Gprespectra is a ss*isomorphism.
As H varies, the ssHq(X) define a contravariant functor from the homotopy
category hGO of orbits to the category of Abelian groups, but the functoriality*
* is
not important to us at the moment.
We state the results of this section for Gprespectra but, since the forgetf*
*ul
functor U preserves all relevant constructions, they apply equally well to orth*
*ogonal
Gspectra. The previous section gives GP a level model structure.
Lemma 3.3. A level equivalence of Gprespectra is a ss*isomorphism.
Proof. Since each SV is triangulable as a finite GCW complex [16], this
follows from the fact that if A is a GCW complex and f : B ! C is a weak
equivalence of Gspaces, then f* : F (A; B) ! F (A; C) is a weak equivalence_of
Gspaces. __
The nonequivariant version [18, 7.3] of the following partial converse is tr*
*ivial.
The equivariant version is the key result, [17, I.7.12], in the classical devel*
*opment of
the equivariant stable homotopy category, and it is also the key result here. W*
*hile
the result there is stated for Gspectra, the argument is entirely homotopical *
*and
applies verbatim to Gspectra. To make this paper selfcontained, we will rewo*
*rk
the proof in x9.
Theorem 3.4. A ss*isomorphism between Gspectra is a level equivalence.
Using that spacelevel constructions commute with passage to fixed points, as
in Lemma 1.6, all parts of the following equivariant analogue of [18, 7.4] eith*
*er
follow from or are proven in the same way as the corresponding part of that res*
*ult.
As there, the nondegenerate basepoint hypotheses in Theorem 2.7 are not needed
here.
Theorem 3.5. (i)A map of Gprespectra is a ss*isomorphism if and only
if its suspension is a ss*isomorphism.
(ii)The homotopy groups of a wedge of Gprespectra are the direct sums of the
homotopy groups of the wedge summands, hence a wedge of ss*isomorphisms
of Gprespectra is a ss*isomorphism.
(iii)If i : A ! X is an hcofibration and a ss*isomorphism of Gprespectra *
*and
f : A ! Y is any map of Gprespectra, then the cobase change j : Y !
X [A Y is a ss*isomorphism.
(iv)If i and i0are hcofibrations and the vertical arrows are ss*isomorphism*
*s in
the comparison of pushouts diagram of Theorem 2.7(iv), then the induced
map of pushouts is a ss*isomorphism.
(v) If X is the colimit of a sequence of hcofibrations Xn ! Xn+1, each of
which is a ss*isomorphism, then the map from the initial term X0 into X
is a ss*isomorphism.
(vi)For any map f : X ! Y of Gprespectra and any H G, there are natural
long exact sequences
. ..! ssHq(F f) ! ssHq(X) ! ssHq(Y ) ! ssHq1(F f) ! . .;.
42 III. MODEL CATEGORIES OF ORTHOGONAL GSPECTRA
. ..! ssHq(X) ! ssHq(Y ) ! ssHq(Cf) ! ssHq1(X) ! . .;.
and the natural map j : F f ! Cf is a ss*isomorphism.
Equivariant stability requires consideration of general representations V 2 *
*V ,
rather than just the trivial representation as in (i).
Theorem 3.6. Let V 2 V . A map f : X  ! Y of Gprespectra is a ss*
isomorphism if and only if V f : V X ! V Y is a ss*isomorphism
We prove half of the theorem in the following lemma, which will be used in o*
*ur
development of the stable model structure.
Lemma 3.7. Let V 2 V . If f : X ! Y is a map of Gprespectra such that
V f : V X ! V Y is a ss*isomorphism, then f is a ss*isomorphism.
Proof. By Proposition 3.9 below, V V f is a ss*isomorphism. The conclu __
sion follows by naturality from the following lemma. __
Lemma 3.8. For Gprespectra X and V 2 V , the unit j : X ! V V X of
the (V ; V ) adjunction is a ss*isomorphism.
Proof. Up to isomorphism, we may write the universe U as U0 V 1. We
choose an expanding sequence of indexing Gspaces U0iin U0 whose union is U0.
For q 0, j
ssHq(X) = colimi;jssHq(UiV X(Ui V j))
and j
ssHq(V V X) = colimi;jssHq(UiV V V X(Ui V j)):
The unit j induces a map from the first colimit to the second, and the structure
maps V X(UiV j) ! X(UiV j+1) induce a map from the second colimit to the *
* __
first. These are inverse isomorphisms. A similar argument applies when q < 0. *
*__
Proposition 3.9. If f : X ! Y is a ss*isomorphism of Gprespectra and
A is a finite based GCW complex, then F (id; f) : F (A; X) ! F (A; Y ) is a s*
*s*
isomorphism.
Proof. By inspection of colimits, using the standard adjunctions, we see th*
*at
(3.10) ssH*(F (A; X) ~=ssG*(F (G=H+ ^ A; X)):
Thus, since G=H+ ^ A is a finite GCW complex [16], we may focus on ssG*. Since
the functor F (; X) converts cofiber sequences of Gspaces to fiber sequences *
*of
Gprespectra, we see by the first long exact sequence in Theorem 3.5(vi) and co*
*m
mutation relations with suspension that the result holds in general if it holds*
* when
A = G=K+ for any K. Here (3.10) gives that ssG*(F (G=K+ ; X)) ~=ssK*(X), and_the
conclusion follows. __
The analogue for smash products is a little more difficult and gives the con*
*verse
of Lemma 3.7 that is needed to complete the proof of Theorem 3.6.
Theorem 3.11. If f : X ! Y is a ss*isomorphism of Gprespectra and A is
a based GCW complex, then f ^ id: X ^ A ! Y ^ A is a ss*isomorphism.
We can reduce this to the case A = G=H+ by use of Theorem 3.5, but that
case seems hard to handle directly. We prove a partial result here. The rest wi*
*ll
drop out model theoretically in the next section.
4. THE STABLE MODEL STRUCTURE ON ORTHOGONAL GSPECTRA 43
Lemma 3.12. If f : X ! Y is a level equivalence of Gprespectra and A is a
based GCW complex, then f ^ id: X ^ A ! Y ^ A is a ss*isomorphism.
Proof. We consider ssHqfor q 0. The case q < 0 is similar. Let U = R1 U0,
where (U0)G = 0. We may write
ssHq(X ^ A) = colimV U0colimWR1 ssHq(V W (X(V W ) ^ A)):
For fixed V and K G, the X(V W )K for varying W specify a nonequivariant
prespectrum X[V ]K indexed on R1 , and we have
(W (X(V W ) ^ A))K ~=W (X(V W )K ^ AK ):
Since f is a level equivalence, it induces a nonequivariant ss*isomorphism f[V*
* ]K :
X[V ]K ! Y [V ]K . By the nonequivariant version [18, 7.4(i)] of Theorem 3.11,
f[V ]K ^ id: X[V ]K ^ AK ! Y [V ]K ^ AK
is a ss*isomorphism. Therefore, for each V , the induced map of Gspaces
hocolimW W (X(V W ) ^ A) ! hocolimWW (Y (V W ) ^ A)
is a weak Gequivalence. Applying V to this map still gives a weak Gequivalenc*
*e._
Passage to homotopy groups ssHqand then to colimits over V gives the result. *
*__
4. The stable model structure on orthogonal Gspectra
We give the categories of orthogonal Gspectra and Gprespectra stable model
structures and prove that they are Quillen equivalent. The arguments are like t*
*hose
in the nonequivariant context of [18], except that we work with ss*isomorphisms
rather than the equivariant analogue of the stable equivalences used there. As *
*we
explain in x6, that analogue gives a formally equivalent condition for a map to*
* be
a ss*isomorphism. All of the statements and most of the proofs are identical *
*in
GI S and GP. Definition 2.3 specifies the level equivalences, level fibrations,*
* level
acyclic fibrations, qcofibrations, and level acyclic qcofibrations in these c*
*ategories.
Definition 4.1.Let f : X ! Y be a map of orthogonal Gspectra or G
prespectra.
(i)f is an acyclic qcofibration if it is a ss*isomorphism and a qcofibrat*
*ion.
(ii)f is a qfibration if it satisfies the RLP with respect to the acyclic q*
*cofibra
tions.
(iii)f is an acyclic qfibration if it is a ss*isomorphism and a qfibration.
Theorem 4.2. The categories GI S and GP are compactly generated proper
Gtopological model categories with respect to the ss*isomorphisms, qfibratio*
*ns,
and qcofibrations. The fibrant objects are the Gspectra.
The set of generating qcofibrations is the set F I specified in Definition *
*1.1.
The set K of generating acyclic qcofibrations properly contains the set F J sp*
*ecified
there. As nonequivariantly [18, xx8, 9], it is defined in terms of the followin*
*g maps
V;W, which turn out to be ss*isomorphisms.
Definition 4.3.For V; W 2 V , define V;W : FV W SW ! FV S0 to be the
adjoint of the evident canonical map
SW ! (FV S0)(V W ) ~=O(V W )+ ^O(W) SW :
44 III. MODEL CATEGORIES OF ORTHOGONAL GSPECTRA
Lemma 4.4. For any orthogonal Gspectrum or Gprespectrum X,
*V;W: IG S (FV S0; X) ! IG S (FV W SW ; X)
coincides with "oe: X(V ) ! W X(V W ) under the canonical homeomorphisms
X(V ) = TG (S0; X(V )) ~=IG S (FV S0; X)
and
W X(V W ) = TG (SW ; X(V W )) ~=IG S (FV W SW ; X):
Proof. With X = FV S0, "oemay be identified with a map
"oe: IG S (FV S0; FV S0) ! IG S (FV W SW ; FV S0);
and V;W is the image of the identity map under "oe. __*
*_
The following result is the equivariant version of [18, 8.6].
Lemma 4.5. For all based GCW complexes A, the maps
V;W ^ id: FV W W A ~=FV W SW ^ A ! FV S0 ^ A ~=FV A
are ss*isomorphisms.
Proof. We prove this separately in the two cases.
Gprespectra. As in [18, 4.1], we have (FV A)(Z) = SZV ^ A, where
SZV = * if V is not contained in Z. Thus FV A is essentially a reindexing of
the suspension Gprespectrum of A. The map V;W(Z) is the identity unless Z
contains V but does not contain W , when it is the inclusion * ! SZV . Passing
to colimits, we see that V;W ^ idis a ss*isomorphism.
Orthogonal Gspectra. As in [18, 4.4], for Z V we have
(FV A)(Z) = O(Z)+ ^O(ZV )SZV ^ A:
By Lemma 3.7, it suffices to prove that V (V;W ^ id) is a ss*isomorphism. When
Z contains V W , V V;W(Z) can be identified with the quotient map
O(Z)+ ^O(Z(V W)) SZ ! O(Z)+ ^O(ZV )SZ :
Under Ghomeomorphisms (1.3), V V;W(Z) corresponds to the map
ss ^ id: O(Z)=O(Z  (V W )) +^ SZ ! O(Z)=O(Z  V )+^ SZ ;
where ss is the evident quotient map. Smashing with A, restricting to indexing *
*G
spaces in a universe U, taking homotopy groups, and passing to colimits, we obt*
*ain_
an isomorphism between copies of equivariant stable homotopy groups of A. _*
*_
Recall the operation from (1.16).
Definition 4.6.Let MV;W be the mapping cylinder of V;W. Then V;W
factors as the composite of a qcofibration kV;W : FV W SW ! MV;W and a
deformation retraction rV;W : MV;W ! FV S0. Let jV;W : FV S0 ! MV;W be
the evident homotopy inverse of rV;W. Restricting to V and W in skIG , let K be
the union of F J and the set of all maps of the form ikV;W, i 2 I.
We need a characterization of the maps that satisfy the RLP with respect to
K. It is the equivariant analogue of [18, 9.5], but we give the proof since thi*
*s is
the place where we need the new notion of a Gtopological model category.
4. THE STABLE MODEL STRUCTURE ON ORTHOGONAL GSPECTRA 45
Definition 4.7.A commutative diagram of based Gspaces
g
D _____//E
p  q
fflfflfflffl
A __f__//B
in which p and q are Serre fibrations is a homotopy pullback if the induced map
D ! A xB E is a weak equivalence of Gspaces.
Proposition 4.8. A map p : E ! B satisfies the RLP with respect to K if
and only if p is a level fibration and the diagram
EV _o"e//_W E(V W )

(4.9) pV Wp(V W)
fflffl fflffl
BV _o"e//_W B(V W )
is a homotopy pullback for all V and W .
Proof. The map p has the RLP with respect to F J if and only if it is a lev*
*el
fibration. By Lemma 1.20, p has the RLP with respect to ikV;W if and only if
IG S (k*V;W; p*) has the RLP with respect to I, which means that IG S (k*V;W; p*
**)
is an acyclic Serre fibration of Gspaces. Since kV;W is a qcofibration and p *
*is a
level fibration, IG S (k*V;W; p*) is a Serre fibration because the level model *
*structure
is Gtopological. We conclude that p satisfies the RLP with respect to K if and
only if p is a level fibration and each IG S (k*V;W; p*) is a weak equivalence.*
* Since
kV;W ' jV;WV;W and jV;W is a homotopy equivalence, this holds if and only if
IG S (*V;W; p*) is a weak equivalence. Using the fact that V;W corresponds to "*
*oe
under adjunction, we see that the map IG S (*V;W; p*) is isomorphic to the map
EV ! BV xW B(V W) E(V W )
and is thus a weak equivalence if and only if (4.9) is a homotopy pullback. *
* ___
From here, the proof of Theorem 4.2 is virtually identical to that of its no*
*nequiv
ariant version in [18, x9]. We record the main steps of the argument since they
give the order of proof and encode useful information about the qfibrations and
qcofibrations. Rather than repeat the proofs, we point out the main input. The
following corollary is immediate.
Corollary 4.10. The trivial map F ! * satisfies the RLP with respect to
K if and only if F is an Gspectrum.
It is at this point that the key result, Theorem 3.4, comes into play.
Corollary 4.11. If p : E ! B is a ss*isomorphism that satisfies the RLP
with respect to K, then p is a level acyclic fibration.
Proof. If F = p1(*), then F is an Gspectrum and F  ! * is a ss*
isomorphism. Thus, by Theorem 3.4, F is level acyclic. The rest is as_in_[18,
9.8]. __
Proposition 4.12. Let f : X ! Y be a map of orthogonal Gspectra.
46 III. MODEL CATEGORIES OF ORTHOGONAL GSPECTRA
(i)f is an acyclic qcofibration if and only if it is a retract of a relativ*
*e Kcell
complex.
(iii)f is a qfibration if and only if it satisfies the RLP with respect to K*
*, and
X is fibrant if and only if it is an orthogonal Gspectrum.
(iii)f is an acyclic qfibration if and only if it is a level acyclic fibrati*
*on.
Proof. Lemma 4.5 implies that the maps ikV;W in K are ss*isomorphisms.
Thus all maps in K are ss*isomorphisms. Now Theorem 3.5 implies that all retra*
*cts __
of relative Kcell complexes are ss*isomorphisms. The rest is as in [18, 9.9].*
* __
The proof of the model axioms is completed as in [18, x9]. The properness of
the model structure is implied by the following more general statements.
Lemma 4.13. Consider the following commutative diagram:
f
A _____//B
i j
fflfflfflffl
X __g__//Y:
(i)If the diagram is a pushout in which i is an hcofibration and f is a ss*
isomorphism, then g is a ss*isomorphism.
(ii)If the diagram is a pullback in which j is a level fibration and g is a s*
*s*
isomorphism, then f is a ss*isomorphism.
The following model theoretical observation leads to the proof of Theorem 3.*
*11.
Lemma 4.14. If A is a based GCW complex A, then ( ^ A; F (A; )) is a
Quillen adjoint pair on GI S or GP with its stable model structure.
Proof. Since the functor F (A; ) preserves fibrations, level equivalences,*
* and
homotopy pullbacks, it preserves qfibrations and acyclic qfibrations by their*
*_char
acterizations in Propositions 4.8 and 4.12. __
The proof of Theorem 3.11. Let f : X ! Y be a ss*isomorphism and A
be a based GCW complex. We must show that f ^ idAis a ss*isomorphism. By
cofibrant approximation in the level model structure and use of Lemma 3.12, we
may assume that X and Y are cofibrant. However, as a Quillen left adjoint, the *
*__
functor () ^ A preserves weak equivalences between cofibrant objects. *
*__
The following result, which is immediate from Lemmas 4.14 and 3.8, implies
that the homotopy category with respect to the stable model structure really is
an "equivariant stable homotopy category", in the sense that the functors V and
V on it are inverse equivalences of categories for V 2 V . Recall the notion of*
* a
Quillen equivalence (e.g. from [18, A.1]).
Theorem 4.15. For V 2 V , the pair (V ; V ) is a Quillen equivalence.
Finally, we have the following promised comparison theorem.
Theorem 4.16. The pair (P; U) is a Quillen equivalence between the categori*
*es
GP and GI S with their stable model structures.
5. THE POSITIVE STABLE MODEL STRUCTURE 47
Proof. In view of II.2.12 and [18, A.2], we need only show that the unit
j : X ! UPX of the adjunction is a ss*isomorphism when X is a cofibrant G
prespectrum. As in [18, 10.3], Theorems 3.5 and 3.6 imply that it suffices to p*
*rove
this when X = V FV A for an indexing Gspace V and a based GCW complex
A. The functors FV for PG and IG S are related by FV ~=PFV , by inspection of
their right adjoints, and these functors commute with smash products with based
Gspaces. We have the commutative diagram
0;V
FV V A ______//_F0A
j  j
fflffl fflffl
UFV V A U0;V//_UF0A:
The right vertical arrow is an isomorphism and the maps 0;V and U0;V are_ss*
isomorphisms by Lemma 4.5 __
5. The positive stable model structure
In x6, we explained the positive level model structure, and we need the con
comitant positive stable model structure, as in [18, x14].
Definition 5.1.A Gprespectrum or orthogonal Gspectrum X is a positive
Gspectrum if "oe: X(V ) ! WV X(W ) is a weak equivalence for V G 6= 0.
Definition 5.2.Define acyclic positive qcofibrations, positive qfibrations*
*, and
acyclic positive qfibrations as in Definition 4.1, but starting with the posit*
*ive level
classes of maps specified in Definition 2.8.
Theorem 5.3. The categories GI S and GP are compactly generated proper
Gtopological model categories with respect to the ss*isomorphisms, qfibratio*
*ns,
and qcofibrations. The positive fibrant objects are the positive Gspectra.
The set of generating positive qfibrations is the set F +I specified in Def*
*inition
2.9. The set of generating acyclic positive qcofibrations is the union, K+ , o*
*f the
set F +J specified there and the set of maps of the form ikV;W with i 2 I and
V G 6= 0 from Definition 4.6.
To prove the theorem, we need a comparison between positive Gspectra and
Gspectra. For V 2 V and an orthogonal Gspectrum or Gprespectrum X, the
map = 0;V: FV SV ! F0S0 = S induces a map
(5.4) * : X ~=F (S; X) ! F (FV SV ; X):
of function orthogonal Gspectra or Gprespectra. Standard adjunctions imply th*
*at
F (FV SV ; X)(W ) ~=V X(V W );
and this leads to the following observation. Write F1S1 = FV SV when V = R.
Lemma 5.5. If E is a positive Gspectrum, then F (F1S1; E) is an G
spectrum and * is a positive level equivalence.
Now Theorem 3.4, applied to F (F1S1; ), implies its positive level analogue.
Theorem 5.6. A ss*isomorphism between positive Gspectra is a positive
level equivalence.
48 III. MODEL CATEGORIES OF ORTHOGONAL GSPECTRA
From here, Theorem 5.3 is proven by the same arguments as for the stable
model structure, with everything restricted to positive levels. The proof of t*
*he
following comparison result is the same as the proof of Theorem 4.16.
Theorem 5.7. The pair (P; U) is a Quillen equivalence between the categories
GP and GI S with their positive stable model structures.
The relationship between the stable model structure and the positive stable
model structure is given by the following equivariant analogue of [18, 14.6].
Proposition 5.8. The identity functor from GI S with its positive stable
model structure to GI S with its stable model structure is the left adjoint of*
* a
Quillen equivalence, and similarly for GP.
6. Stable equivalences of orthogonal Gspectra
Let [X; Y ]`Gand [X; Y ]+`Gdenote the set of maps X ! Y in the homotopy
category of GI S or GP with respect to the level model structure or the positive
level model structure and let [X; Y ]G denote the set of maps X  ! Y in the
homotopy category with respect to the stable (or, equivalently, positive stable)
model structure. The following observation applies to both GI S and GP.
Theorem 6.1. The following conditions on a map f : X ! Y are equivalent.
(i)f is a ss*isomorphism.
(ii)f* : [Y; E]`G! [X; E]`Gis an isomorphism for all Gspectra E.
(iii)f* : [Y; E]+`G! [X; E]+`Gis an isomorphism for all positive Gspectra *
*E.
Proof. We prove that (i) and (ii) are equivalent. The same proof shows that
(i) and (iii) are equivalent. Since the qcofibrations are the same in the leve*
*l and
stable model structures and level equivalences are ss*isomorphisms, the identi*
*ty
functor is the left adjoint of a Quillen adjoint pair from the level model stru*
*cture to
the stable model structure. Since Gspectra are stably fibrant, this implies t*
*hat
(6.2) [X; E]G ~=[X; E]`G
when E is an Gspectrum. Since every object of the stable homotopy category is
isomorphic to an Gspectrum, the Yoneda lemma gives that f is an isomorphism
in that category if and only if f* : [Y; E]G ! [X; E]G is an isomorphism for a*
*ll __
Gspectra E. In view of (6.2), this says that (i) and (ii) are equivalent. *
* __
The maps that satisfy (ii) or (iii) are called stable equivalences or positi*
*ve stable
equivalences. The nonequivariant analogues of the stable equivalences play a ce*
*n
tral role in the theory of symmetric spectra [15, 18], where the previous theor*
*em
fails, but they need not be introduced explicitly in the study of orthogonal sp*
*ectra.
An easy formal argument shows that a stable equivalence between Gspectra
is a level equivalence. Nonequivariantly, there is a direct proof [18, 8.8] th*
*at a
ss*isomorphism of symmetric spectra or orthogonal spectra is a stable equivale*
*nce.
That argument precedes the development of the stable model structure there. How
ever, the argument depends on the trivial nonequivariant version of the key res*
*ult
Theorem 3.4 and is of no help in the present approach to the equivariant stable
homotopy category.
7. MODEL CATEGORIES OF RING AND MODULE GSPECTRA 49
7. Model categories of ring and module Gspectra
In this section and the next, we study model structures induced by the stabl*
*e or
positive stable model structure on GI S . We prove here that the categories of *
*or
thogonal ring spectra and of modules over an orthogonal ring spectrum are Quill*
*en
model categories. The proofs are essentially the same as those in the nonequiva*
*ri
ant case given in [18, xx12, 14], but some of the cases covered there dictated *
*a more
complicated line of argument than is necessary here. We give an outline.
In the language of [30], we show that the monoid and pushoutproduct axioms
hold for orthogonal Gspectra. As in [17, 11.2], the following elementary compl*
*e
ment to Lemmas 2.5 and 2.6 is used repeatedly.
Lemma 7.1. If i : X ! Y is an hcofibration of orthogonal Gspectra and Z
is any orthogonal Gspectrum, then i ^ id: X ^ Z ! Y ^ Z is an hcofibration.
The following lemma is the key step in the proof of the cited axioms. Its
nonequivariant analogue is part of the proof of [18, 12.3].
Lemma 7.2. Let Y be an orthogonal Gspectrum such that ss*(Y ) = 0. Then
ss*(FV SV ^ Y ) = 0 for any V 2 V .
Proof. Let flV = 0;V: FV SV ! F0S0 = S be the canonical ss*isomorphism
of Lemma 4.5. Let ff 2 ssHq(FV SV ^ Y ). Taking q 0 for definiteness, the proo*
*f for
q < 0 being similar, choose a map Sq ! (W (FV SV ^Y )(W ))H that represents ff.
By standard adjunctions, this map is determined by a map of orthogonal Gspectra
f : FW (G=H+ ^ Sq ^ SW ) ! FV SV ^ Y:
Since ssH*(Y ) = 0, we can choose W large enough that the composite
(flV ^ id) O f : FW (G=H+ ^ Sq ^ SW ) ! FV SV ^ Y ! S ^ Y ~=Y
is null homotopic. Let g = (flV ^ id) O f and let g0 be the map
id^g :FV W (SV ^G=H+ ^Sq^SW ) ~=FV SV ^FW (G=H+ ^Sq^SW ) ! FV SV ^Y
obtained from g by smashing with FV SV . Then g0is also null homotopic. Now let
f0 : FV W (SV ^G=H+ ^Sq^SW ) ~=FV SV ^FW (G=H+ ^Sq^SW ) ! FV SV ^Y
be the composite f O (flV ^ id). Then f0 also represents ff. We show that ff = *
*0 by
showing that the maps f0 and g0 are homotopic. We can rewrite f0 and g0 as the
composites of the map
id^f : FV SV ^ FW (G=H+ ^ Sq ^ SW ) ! FV SV ^ FV SV ^ Y
and the maps FV SV ^ FV SV ^ Y ! FV SV ^ Y obtained by applying flV to the
first or second factor FV SV . Thus, it suffices to show that the maps flV ^ id*
*and
id^flV from FV V SV V ~=FV SV ^ FV SV to FV SV are homotopic. The adjoints
SV V ! (FV SV )(V V ) = O(V V )+ ^exO(V )SV V
of these two maps are the Gmaps that send (v; v0) 2 SV V to o ^ (v0; v) and to
e ^ (v; v0), where o 2 O(V V ) is the evident transposition on V V . Writing *
*out
the homeomorphism of O(V V )spaces
O(V V )+ ^exO(V )SV V ~=O(V V )=O(V )+ ^ SV V
given by (1.3), we see that it is a Gmap. Under this homeomorphism, our two G
maps send s 2 SV V to o^s and to e^s. The elements e and o are in O(V V )G . By
50 III. MODEL CATEGORIES OF ORTHOGONAL GSPECTRA
writing V as a sum of irreducible representations Viniand considering Ginvaria*
*nt
isometries, we find that O(V V )=O(V )G is the product of groups O(2ni)=O(ni),
U(2ni)=U(ni) or Sp(2ni)=Sp(ni), depending on the type of Vi (compare e.g. [7,
3.6]) and is thus connected. A choice of path connecting e and o in O(V V )=O(V*
*_)G
determines a choice of homotopy between our two Gmaps. __
Proposition 7.3. For a cofibrant orthogonal Gspectrum X, the functor X ^
() preserves ss*isomorphisms.
Proof. When X = FV SV , this is implied by Lemma 7.2, as we see by using
the usual mapping cylinder construction to factor a given ss*isomorphism as a
composite of an hcofibration and a Ghomotopy equivalence and comparing long
exact sequences given by Lemma 7.1 and Theorem 3.5(vi). As in the proof of [18,*
* __
12.3], the general case follows by use of Theorems 3.5, 3.6, and 3.11. *
* __
As in [18, 12.5 and 12.6], this together with other results already proven i*
*mplies
the monoid and pushoutproduct axioms. These apply to GI S with both its stable
and its positive stable model structures.
Proposition 7.4 (Monoid axiom).For any acyclic (positive) qcofibration
i : X ! Y of orthogonal Gspectra and any orthogonal Gspectrum Z, the map
i ^ id: X ^ Z ! Y ^ Z is a ss*isomorphism and an hcofibration. Moreover,
cobase changes and sequential colimits of such maps are also ss*isomorphisms a*
*nd
hcofibrations.
Proposition 7.5 (Pushoutproduct axiom).If i : X ! Y and j : W ! Z
are (positive) qcofibrations of orthogonal Gspectra and i is a ss*isomorphis*
*m,
then the (positive) qcofibration ij : (Y ^ W ) [X^W (X ^ Z) ! Y ^ Z is a
ss*isomorphism.
As in [18, xx12, 14], the methods and results of [30], together with Proposi*
*tion
5.8, entitle us to the following conclusions. More explicitly, [18, 5.13] spec*
*ifies
conditions for the category of algebras over a monad in a compactly generated
topological model category C to inherit a structure of topological model catego*
*ry,
and that result generalizes to Gtopological model categories. The pushoutprod*
*uct
and monoid axioms allow the verification of the conditions in the cases on hand.
Theorem 7.6. Let R be an orthogonal ring Gspectrum, and consider the stable
model structure on GI S .
(i)The category of left Rmodules is a compactly generated proper Gtopologi*
*cal
model category with weak equivalences and qfibrations created in GI S .
(ii)If R is cofibrant as an orthogonal Gspectrum, then the forgetful functor
from Rmodules to orthogonal Gspectra preserves qcofibrations, hence ev*
*ery
cofibrant Rmodule is cofibrant as an orthogonal Gspectrum.
(iii)If R is commutative, the symmetric monoidal category GI SR of Rmodules
also satisfies the pushoutproduct and monoid axioms.
(iv)If R is commutative, the category of Ralgebras is a compactly generated *
*right
proper Gtopological model category with weak equivalences and qfibratio*
*ns
created in GI S .
(v) If R is commutative, every qcofibration of Ralgebras whose source is co*
*fi
brant as an Rmodule is a qcofibration of Rmodules, hence every cofibra*
*nt
Ralgebra is cofibrant as an Rmodule.
8. THE MODEL CATEGORY OF COMMUTATIVE RING GSPECTRA 51
(vi)If f : Q ! R is a weak equivalence of orthogonal ring Gspectra, then
restriction and extension of scalars define a Quillen equivalence between*
* the
categories of Qmodules and of Rmodules.
(vii)If f : Q ! R is a weak equivalence of commutative orthogonal ring G
spectra, then restriction and extension of scalars define a Quillen equiv*
*alence
between the categories of Qalgebras and of Ralgebras.
Parts (i), (iii), (iv), (vi), and (vii) also hold for the positive stable model*
* structure.
Parts (ii) and (v) do not hold for the positive stable model structure, in w*
*hich
SG is not cofibrant. As in [18, 12.7], we have the following generalization of *
*Propo
sition 7.3, which is needed in the proofs of parts (vi) and (vii) of the theore*
*m.
Proposition 7.7. For a cofibrant right Rmodule M, the functor M ^R N of
N preserves ss*isomorphisms.
8. The model category of commutative ring Gspectra
Let A be the monad on orthogonalWGspectra that defines commutative orthog
onal ring Gspectra. Thus AX = i0 X(i)=i, where X(i)denotes the ith smash
power, with X(0)= SG .
Theorem 8.1. The category of commutative orthogonal ring Gspectra is a
compactly generated proper Gtopological model category with qfibrations and w*
*eak
equivalences created in the positive stable model category of orthogonal Gspec*
*tra.
The sets AF +I and AK+ are the generating sets of qcofibrations and acyclic q
cofibrations.
This is a consequence of the following two results, which (together with two
general results on colimits [5, I.7.2, VII.2.10]) verify the criteria for inher*
*itance of
a model structure specified in [18, 5.13].
Lemma 8.2. The sets AF +I and AK+ satisfy the Cofibration Hypothesis.
Proof. The Cofibration Hypothesis is specified in [18, 5.3]. Its verificat*
*ion
here amounts to the following results:
(i)The functor A preserves hcofibrations.
(ii)The cobase change R ! R^AX AY associated to a wedge X ! Y of maps
in F I+ and a map AX ! R of commutative orthogonal ring Gspectra is
an hcofibration.
(iii)Sequential colimits of maps of commutative orthogonal ring Gspectra that
are hcofibrations are computed as the colimits of their underlying ortho*
*go
nal Gspectra.
Here (i) and (iii) are easy, but (ii) requires the methods of [5, VIIx3], as_ex*
*plained
in the proof of [18, 15.9]. __
Lemma 8.3. Every relative AK+ cell complex is a ss*isomorphism.
Proof. First, one needs that each map in AK+ is a ss*isomorphism or, more
generally, that the functor A preserves ss*isomorphisms between positive cofib*
*rant
orthogonal Gspectra. That is a consequence of the second statement of the foll*
*ow
ing lemma. From there, as explained in [18, x15], the proof reduces to showing *
*that
if R ! R0is a relative AF +Icell complex, then the functor () ^R R0preserves
ss*isomorphisms. In turn, using the methods of [5, VIIx4], that reduces to sho*
*wing
52 III. MODEL CATEGORIES OF ORTHOGONAL GSPECTRA
that the functor AFV A ^ () on orthogonal Gspectra preserves ss*isomorphisms
when A is a based GCW complex and V G 6= 0. That is a consequence of the first_
statement of the following lemma. Compare [18, 15.6, 15.7]. __
Lemma 8.4. Let A be a based GCW complex, X be an orthogonal Gspectrum,
and V G 6= 0. Then the quotient map
q : (Ei+ ^i (FV A)(i)) ^ X ! ((FV A)(i)=i) ^ X
is a ss*isomorphism. If X is a positive cofibrant orthogonal Gspectrum, then
q : Ei+ ^i X(i)! X(i)=i
is a ss*isomorphism.
Proof. For the first statement, we show that q is an eventual level Ghomot*
*opy
equivalence. Precisely, we prove that the W th map of q is a Ghomotopy equival*
*ence
for all W that, up to Gisomorphism, contain V i. We may as well assume that
W V i. Then, by [18, 4.4] and inspection of coequalizers,
((FV A)(i)^ X)(W ) ~=O(W )+ ^O(WV i)(A(i)^ X(W  V i)):
The action of oe 2 iis to permute the factors in A(i)and to act through oeidWV*
* i
on O(W ), where oe 2 O(V i) permutes the i summands of V in V i. Since iacts on
O(W ) as a subgroup of O(V i), the action commutes with the action of O(W  V i*
*).
Therefore, the Gmap at level W in the first statement is obtained by passing to
orbits over ix O(W  V i) from the projection
(Eix O(W ))+ ^ (A(i)^ X(W  V i)) ! O(W )+ ^ (A(i)^ X(W  V i)):
This map is equivariant with respect to the evident actions of the semidirect *
*prod
uct = Gn(ixO(W V i)), where G acts on ixO(W V i) through its (right)
action on O(W  V i). Therefore, to show that the level Gmap is a Ghomotopy
equivalence, it suffices to show that the projection
Ei+ ^ O(W )+ ! O(W )+
is a homotopy equivalence. Since both sides are CW complexes, it suffices
to observe that this map becomes a homotopy equivalence on passage to fixed
points for each . The fixed points of the source and target are empty unless
G n O(W  V i), in which case acts trivially on the contractible space Ei
and the projection is a homotopy equivalence. This proves the first statement.
For the second statement, we may assume that X is an F +Icell spectrum, and the
proof then is an induction on i and on the cellular filtration of X that is ess*
*entially_
the same as the proof of [5, III.5.1]. __
9. Level equivalences and ss*isomorphisms of Gspectra
We must prove Theorem 3.4. It asserts that a ss*isomorphism f : X ! Y
between Gspectra indexed on any given universe U is a level equivalence. The
proof is a somewhat streamlined version of the proof of [17, 7.12]. Consider t*
*he
fiber F f. By the long exact sequence of homotopy groups, ssH*(F f) = 0 for all
H. Since F f is constructed by taking levelwise fibers, it suffices by the leve*
*l exact
sequences of homotopy groups to prove that F f is level acyclic. For the case o*
*f ssH0,
this verification uses that, up to homotopy, the map f(V ) : X(V ) ! Y (V ) is*
* the
loop of the map f(V + R) : X(V + R) ! Y (V + R), where R U and R \ V = 0.
Since F f is again an Gspectrum, it suffices to prove the following result.
9. LEVEL EQUIVALENCES AND ss*ISOMORPHISMS OF GSPECTRA 53
Lemma 9.1. If X is an Gspectrum such that ssHn(X) = 0 for all integers n
and all H G, then ssHnX(V ) = 0 for all n 0, H G, and V U.
We need an observation about Gspectra. It is adapted from [21, p. 30].
Lemma 9.2. Let X be an Gspectrum and let V and V 0be indexing Gspaces
in U that are isomorphic as Hinner product spaces. Then the Gspaces X(V ) and
X(V 0) are weakly equivalent as Hspaces.
Proof. Choose an indexing Gspace Z that contains V and V 0and let W and
W 0be the orthogonal complements of V and V 0in Z. Then W and W 0are indexing
Gspaces that are isomorphic as Hinner product spaces. Together with structural
Gequivalences of X, any choice of Hisomorphism gives a diagram
X(V )_____//W X(Z) ~=W0 X(Z) oo___X(V 0)
that displays the claimed weak Hequivalence. ___
The main tool in the proof of Lemma 9.1 is a familiar fiber sequence. For any
V , let S(V ) be the unit sphere in the unit disk D(V ). We may identify SV wit*
*h the
Gspace D(V )=S(V ) = D(V )+ =S(V )+ , and the quotient map D(V )+ ! S0 is a
Ghomotopy equivalence. For any Gspace A, application of the functor F (; A) *
*to
the cofiber sequence S(V )+ ! D(V )+ ! SV gives a fiber sequence of Gspaces,
and passage to Gfixed points gives a fiber sequence of spaces.
Proof of Lemma 9.1. By cofibrant approximation in the level model struc
ture, we may as well assume that X is a cell Gprespectrum, so that each X(V )
has the homotopy type of a GCW complex. We may identify R1 with UG and
so fix Rd U for all d 0. We observe first that ssH*X(Rd) = 0 for all d and H.
Indeed, by the definition of the homotopy groups of an Gspectrum, we have
ssHn(X) ~=ssHndX(Rd) = ssHn+dX(Rd) = 0
if n 0 and, if 0 < n d,
ssHn(X) = ssH0dnX(Rd) = ssHdnX(Rd) = 0:
If V = V G, then V is Gisomorphic to some Rd. Thus, by Lemma 9.2, ssH*X(V ) = 0
for all H in this case as well. Similarly, any V is eisomorphic to some Rd, an*
*d thus
ss*X(V ) = sse*X(V ) = 0 for all V .
Since G is a compact Lie group, it contains no infinite descending chain of
(closed) subgroups and we can argue by induction over subgroups. Assume induc
tively that, for all V and all proper subgroups K of H, ssK*X(V ) = 0. The indu*
*ctive
step is to prove that ssH*X(V ) = 0, and we have already proven this when H = e.
Fix V , let W be the orthogonal complement of V G in V , and let Z be the
orthogonal complement of W H in W . Then W is a GspaceHand Z is an Hspace.
Let d = dim(W H). We begin by proving that ssHnW X(V ) = ssHn+dX(V ) = 0 for
n > 0. We have the fiber sequence
H H WH H WH H
F (SZ ; W X(V )) ! F (D(Z)+ ; X(V )) ! F (S(Z)+ ; X(V )) :
The middle term is equivalent to WH X(V )H . In the left term, the Hspace
F (SZ ; WH X(V )) is Hhomeomorphic to the Gspace W X(V ). We have a struc
tural Gequivalence X(V G) ! W X(V ) since V = V G + W is a direct sum
splitting of Gspaces. Thus, up to homotopy, our fiber sequence may be written
H H WH H
X(V G)H ! W X(V ) ! F (S(Z)+ ; X(V )) :
54 III. MODEL CATEGORIES OF ORTHOGONAL GSPECTRA
We have shown that ssH*X(V G) = 0. Thus to show that ssHnWH X(V ) = 0 for
n > 0, it suffices to show that ssHnF (S(Z)+ ; WH X(V )) = 0 for n > 0. We
may triangulate S(Z) as a finite (unbased) HCW complex [16]. By construction,
S(Z)H is the empty set, so the triangulation only has cells of the form Dm x H=K
with K a proper subgroup of H. By the induction hypothesis, ssKn(WH X(V )) =
ssKn+dX(V ) = 0 for n 0. It follows by induction on the number of cells in the
chosen triangulation that ssHnF (S(Z)+ ; WH X(V )) = 0 for n > 0.
We have proven that ssHnX(V ) = 0 for n > d. Choose a copy of Rd+1 in R1
such that V \Rd+1 = 0. Then V +Rd+1 = (V G+Rd+1)+W . Applied to V +Rd+1,
the argument just given shows that ssHnX(V + Rd+1) = 0 for n > d. Since X(V ) is
Gequivalent to d+1X(V + Rd+1), we conclude that ssHnX(V ) = 0 for all n._This
completes the proof. __
Let F be a family of subgroups of G, that is a set of subgroups closed under
passage to subgroups and conjugates. The inductive nature of the argument makes
it clear that the following generalization holds. We shall need it later.
Theorem 9.3. Let f : X ! Y be a map of Gprespectra. Assume that
f* : ssH*(X) ! ssH*(Y ) is an isomorphism for all H 2 F . Then, for V U,
f(V )* : ssH*(X(V )) ! ssH*(Y (V )) is an isomorphism for all H 2 F .
CHAPTER IV
Orthogonal Gspectra and SG modules
1. Introduction and statements of results
Taking Gspectra and orthogonal Gspectra to be indexed on a complete uni
verse, we shall prove the following precise analogues of the results stated in *
*Ix1.
Theorem 1.1. There is a strong symmetric monoidal functor N : GI S !
GM and a lax symmetric monoidal functor N# : GM ! GI S such that (N; N# )
is a Quillen equivalence between GI S and GM . The induced equivalence of ho
motopy categories preserves smash products.
Theorem 1.2. The pair (N; N# ) induces a Quillen equivalence between the ca*
*t
egories of orthogonal ring Gspectra and SG algebras.
Theorem 1.3. For a cofibrant orthogonal ring Gspectrum R, the pair (N; N# )
induces a Quillen equivalence between the categories of Rmodules and of NR
modules.
Corollary 1.4. For an SG algebra R, the categories of Rmodules and of
N# Rmodules are Quillen equivalent.
Theorem 1.5. The pair (N; N# ) induces a Quillen equivalence between the ca*
*t
egories of commutative orthogonal ring Gspectra and of commutative SG algebra*
*s.
Theorem 1.6. Let R be a cofibrant commutative orthogonal ring spectrum.
The categories of Rmodules, Ralgebras, and commutative Ralgebras are Quillen
equivalent to the categories of NRmodules, NRalgebras, and commutative NR
algebras.
Corollary 1.7. Let R be a commutative SG algebra. The categories of R
modules, Ralgebras, and commutative Ralgebras are Quillen equivalent to the c*
*at
egories of N# Rmodules, N# Ralgebras, and commutative N# Ralgebras.
These comparisons sheds new light on the original equivariant theory of G
spectra [17]. A subtle and somewhat mysterious aspect of the theory concerns wh*
*en
to use sequential indexing and when to use coordinatefree indexing. The object*
*s,
Gspectra or, in the modern version, SG modules, are intrinsically coordinate*
*free.
However, the homotopy groups are Zgraded. The theory of CW objects is at
the heart of the matter. These are special kinds of cell objects. There is no
explicit model structure in the existing literature but, in the model structure*
* that
is implicit in [5, 17], the cofibrations are the retracts of the cell Gcomplex*
*es, which
are coordinatized kinds of objects defined in terms of sphere Gspectra G=H+ ^ *
*Sn
for integers n, just as in the nonequivariant theory. We call this the cellular*
* model
structure. It is not the model structure relevant to the results above.
It has often been wondered why integers, which implicitly encode trivial (vi*
*r
tual) representations, appear here, rather than general (virtual) representatio*
*ns.
55
56 IV. ORTHOGONAL GSPECTRA AND SGMODULES
As we shall see in x2, the answer turns out to be that there is a choice. There
are two model structures with the same (stable) weak equivalences. The cofibra
tions in the (implicit) classical model structure are the retracts of the relat*
*ive cell
Gcomplexes. We shall present a second model structure, called the generalized
cellular model structure, in which the cofibrations are retracts of generalized*
* cell
Gcomplexes defined in terms of general representations of G. These two model
structures are Quillen equivalent; more precisely the identity functor is the l*
*eft
adjoint of a Quillen equivalence from GM with its cellular model structure to
GM with its generalized cellular model structure. Of course, both model struc
tures determine the same homotopy category since that depends only on the weak
equivalences. It is the generalized cellular model structure that is relevant t*
*o the
theorems above.
The relevant model structure on the orthogonal Gspectrum side is the positi*
*ve
stable model structure of IIIx5. In fact, there is no model structure on the ca*
*tegory
of orthogonal Gspectra that corresponds to the cellular model structure on the
category of SG modules, even though we have Quillen equivalences relating all *
*of
our model categories. The point is that the Quillen equivalences above cannot be
composed, since their left adjoints have the same target. However, all homotopy
category level information can be transported back and forth along the induced
equivalence of homotopy categories.
Many arguments in the classical theory are based on use of the classical kin*
*d of
cell structure. Some of these arguments can also be carried out using the gener*
*alized
kind of cell structure, but others cannot. In view of the prevalence of argumen*
*ts
based on inductive verifications on cell complexes, it is in principle preferab*
*le to
have a model structure with as few cofibrations as possible. More fundamentally,
very many arguments, both equivariant and nonequivariant, depend on the use
of CW complexes rather than just cell complexes, the advantage being the avail
ability of the cellular approximation theorem. It is worth emphasizing that th*
*is
fundamentally important refinement of cell theory is invisible to the model cat*
*e
gory formalities. The cellular approximation theorem applies to CW SG modules,
where it reads exactly the same way as it does nonequivariantly. However, this
result fails both for generalized cell SG modules and for cell orthogonal Gsp*
*ectra,
even when one requires cells to be attached only to cells of lower dimension.
For this reason, we have made no attempt to rederive the deeper results of
equivariant stable homotopy theory in terms of orthogonal Gspectra; just as wi*
*th
the nonequivariant theory of diagram spectra, the new category should be viewed
as complementary to the old one, rather than as a replacement for it.
We construct the functors N and N# in x3 and prove the comparison theorems
stated above in x4. We show that the functor N appearing in them is equivalent *
*to a
more intuitive comparison functor M in the brief x5. In x6 we consider families*
* and
cofamilies of subgroups of G. We define and compare new model structures on the
categories of Gspectra, SG modules, and orthogonal Gspectra whose associated
homotopy categories localize information at or away from a chosen collection of
subgroups. Some of these model structures are given by Bousfield localizations,
which admit a simple model theoretical construction in all of our categories.
2. MODEL STRUCTURES ON THE CATEGORY OF SGMODULES 57
2.Model structures on the category of SG modules
We shall not repeat the basic definitions given in [5, 25] and recalled none*
*quiv
ariantly in Ix4. The papers [4, 9, 10] give an introductory overview, and the b*
*asic
equivariant reference is [25, XXIV]. However, we must modify its perspective on
equivariance, in line with IIx1.
As observed in II.1.2, we have categories PG and SG of Gprespectra and G
spectra. Their maps are nonequivariant, but their spaces of maps are Gspaces; *
*that
is, they are enriched over GT . They have associated Gfixed categories GP and
GS , which are enriched over T . There is a spectrification Gfunctor L : PG !
SG that is left adjoint to the evident forgetful functor `.
There is a sphere Gspectrum SG , and a closed symmetric monoidal category
of SG modules. Its objects are Gspectra with additional structure, as specifi*
*ed in
[25, XXIV.1.2, 1.5]. We give a summary. There is a monad L on SG (and not
just on GS ). The unit and product maps j : E ! LE and : LLE ! LE
are Gmaps, and the action map : LE ! E of an Lspectrum is required to be
a Gmap. There is a smash product of Lspectra, denoted ^L , that is associative
and commutative; it has a natural unit Gmap : SG ^L E ! E that is always a
weak equivalence and sometimes an isomorphism. (We redescribe it in VIx6). An
SG module is an Lspectrum E for which is an isomorphism, and ^L restricts
to a smash product ^ between SG modules. We have a Gspace of nonequivariant
maps f : E ! E0 of Lspectra, where f must satisfy f O = 0O Lf. Its Gfixed
point space is the space of Gmaps E ! E0. Maps and Gmaps of SG modules
are just maps and Gmaps of the underlying Lspectra.
We let MG denote the category of SG modules and nonequivariant maps be
tween them. This category is enriched over GT . We let GM denote its Gfixed
category; its objects are the SG modules and its maps are the Gmaps.
All of these categories depend on a choice of a universe U: Gspectra are in*
*dexed
on the indexing spaces in U, as in II.1.2. When suppressed from the notation, as
above, U is generally assumed to be complete. However, everything in this secti*
*on
applies verbatim to an arbitrary universe U.
We have defined the stable and positive stable model structures on GP in
Chapter III. We now consider model structures on the categories GS and GM of
Gspectra and SG modules. We write 1V: TG ! SG for the shift desuspension
Gspectrum functor LFV , which again is left adjoint to evaluation at V . As in*
* I.4.5
and I.4.11, we have an adjunction (F; V) relating the categories SG and MG . It
induces an adjunction between the Gfixed categories GS and GM .
Proposition 2.1. Define F : SG ! MG by FE = SG ^L LE and V : MG !
SG by VM = FL (SG ; M). Then F and V are left and right adjoint, and there is a
natural weak equivalence of Gspectra " : M ! VM.
Definition 2.2.We define spheres and cells in GS and GM .
(i)A sphere Gspectrum is a Gspectrum of the form 1q(G=H+ ^ Sn), where
q 0, n 0, and H G. A generalized sphere Gspectrum is a Gspectrum
of the form 1V(G=H+ ^ Sn), where V is a representation in the universe U,
n 0, and H G. Write SnG= 1 Sn if n 0 and write SVG= 1VS0 for
any Grepresentation V .
58 IV. ORTHOGONAL GSPECTRA AND SGMODULES
(ii)A sphere SG module or a generalized sphere SG module is an SG module
of the form FE, where E is a sphere Gspectrum or a generalized sphere
Gspectrum.
(iii)A generating qcofibration or generalized generating qcofibration in GS
or GM is a map of the form E ! CE, where E is a sphere object or
generalized sphere object and CE is the cone on E.
(iv)A generating acyclic qcofibration or generalized generating acyclic qco*
*fi
bration is a map of the form i0 : CE ! CE ^ I+ , where E is a sphere
object or generalized sphere object.
Remark 2.3. It would serve no purpose to consider sphere Gspectra of the
more general form FV (G=H+ ^ SW ) for Grepresentations W since SW is trian
gulable as a finite GCW complex [16]. The use of functors FV rather than just
functors Fn is the fundamental distinction.
The definition of homotopy groups for Gspectra takes a simpler form than for
Gprespectra, as in [5, I.4.4]. Write ss(E; E0)G for the set of homotopy classe*
*s of
maps E ! E0 in GS . Then, for H G, n 2 Z, and E 2 GS ,
(2.4) ssHn(E) = ss(G=H+ ^ Sn; E)G :
Equivalently, for n 0,
(2.5) ssHn(E) = ssHn(E(0)) and ssHn(E) = ssH0(E(Rn)):
The homotopy groups of an SG module are the homotopy groups of its underlying
Gspectrum.
Remark 2.6. The homotopy groups of a Gspectrum E are the same as those of
the Gprespectrum `E. A Gprespectrum T is a (positive) inclusion Gprespectrum
if each adjoint structure map T (V ) ! WV T (W ) (with V G 6= 0) is an inclus*
*ion,
and the unit T ! `LT of the adjunction is then a weak equivalence [17, I.2.2].
This applies to cofibrant Gprespectra, for example.
Definition 2.7.Consider the categories GS and GM of Gspectra and SG 
modules.
(i)A map in either category is a weak equivalence if it induces an isomorphi*
*sm
on all homotopy groups ssHn.
(iia)A map is a qcofibration if it is a retract of a relative cell Gcomplex *
*defined
in terms of generating qcofibrations.
(iib)A map is a generalized qcofibration if it is a retract of a relative cel*
*l G
complex defined in terms of generalized generating qcofibrations.
(iiia)A map is a qfibration if it satisfies the RLP with respect to the gener*
*ating
acyclic qcofibrations.
(iiib)A map is a restricted qfibration if it satisfies the RLP with respect t*
*o the
generalized generating acyclic qcofibrations.
Theorem 2.8. Consider the categories GS and GM of Gspectra and SG 
modules.
(i)These categories are Gtopological model categories with respect to the w*
*eak
equivalences, qcofibrations, and qfibrations; we call this the cellular*
* model
structure.
2. MODEL STRUCTURES ON THE CATEGORY OF SGMODULES 59
(ii)These categories are also Gtopological model categories with respect to *
*the
weak equivalences, generalized qcofibrations, and restricted qfibration*
*s; we
call this the generalized cellular model structure.
(iii)The identity functors of GS and GM are the left adjoints of Quillen equi*
*v
alences from the cellular model structure to the generalized cellular mod*
*el
structure.
(iv)With either the cellular or the generalized cellular model structures on *
*both
categories, the pair (F; V) is a Quillen equivalence between GS and GM .
(v) With the generalized cellular model structure on GS and the stable model
structure on GP, (L; `) is a Quillen equivalence between GP and GS .
With either model structure, GS is right proper and GM is proper.
Proof. One can mimic the proofs of the model axioms in Chapter III or in [5,
Vx5]. With the second strategy, one starts with different definitions. One rede*
*fines
the qfibrations of Gspectra to be the levelwise Serre Gfibrations for levels*
* n
or levels V , and one redefines the qfibrations of SG modules to be the maps f
such that Vf is a qfibration of Gspectra; for both Gspectra and SG modules,
one redefines the qcofibrations to be the maps that satisfy the LLP with respe*
*ct
to the acyclic qfibrations. With the first proof, this characterization of th*
*e q
fibrations and qcofibrations follows. The second proof capitalizes on the fact*
*s that
the generating acyclic qcofibrations are inclusions of deformation retracts an*
*d that
homotopic maps induce isomorphisms of homotopy groups. These facts make the
proofs of the model axioms almost completely formal. The only point that requir*
*es
comment in the equivariant setting is the proof of [5, VII.5.8], where one need*
*s to
know that a map is an acyclic (restricted) qfibration if and only if it satisf*
*ies the
RLP with respect to the (generalized) generating qcofibrations. The interesti*
*ng
point is that the same weak equivalences work for both statements, and this is a
direct consequence of III.3.4.
To prove that our adjoint pairs are Quillen equivalences, it suffices to pro*
*ve that
their right adjoints create weak equivalences and preserve qfibrations and tha*
*t the
unit of the adjunction is a weak equivalence on cofibrant objects [18, A.2]. T*
*he
statements about right adjoints are immediate; for (iv), the definitions imply *
*that
V creates the qfibrations in GM . The statement about the unit of the adjuncti*
*on
is trivial in part (iii), follows as in I.4.5 from [5, I.4.6, I.8.7, II.2.5] in*
* part (iv), and
follows from Remark 2.6 in part (v).
The last statement is proven using long exact sequences of homotopy groups o*
*f __
fiber and cofiber sequences, as in [18, 9.10] or [5, I.6.6]. *
* __
Remark 2.9. It seems unlikely that GS is left proper. The problem is that a
cofiber sequence of Gspectra is only known to give rise to a long exact sequen*
*ce of
homotopy groups under a mild hypothesis [5, I.3.4], whereas any cofiber sequence
of SG modules gives rise to a long exact sequence of homotopy groups [5, I.6.4*
*].
Note that SG itself is cofibrant as an object of GS but is not cofibrant as *
*an
object of GM , where F(SG ) is a cofibrant approximation of SG .
Of course, the reason for the introduction of GM is that, unlike GS , it is
a closed symmetric monoidal category under the smash product, so that we can
define rings, called SG algebras, and modules over them; see also [6, 24]. Exa*
*ctly
as in [5, VIIxx4,5], we have model categories of such highly structured equivar*
*iant
ring and module spectra.
60 IV. ORTHOGONAL GSPECTRA AND SGMODULES
Theorem 2.10. The following categories admit cellular and generalized cellu*
*lar
Gtopological model structures whose weak equivalences and qfibrations or rest*
*ricted
qfibrations are created in GM .
(i)The category of SG algebras.
(ii)The category of commutative SG algebras.
(iii)The category of modules over an SG algebra R.
(iv)The category of algebras over a commutative SG algebra R.
(v) The category of commutative algebras over a commutative SG algebra R.
In all cases, the qcofibrations are the retracts of relative cellular or gener*
*alized
cellular objects in the specified category. All of these categories are right *
*proper,
and the categories in (iii) are also left proper.
A general notion of cellular object that applies in all cases is given in [5,
VII.4.11]. The fact that all of our categories are right proper is inherited f*
*rom
GM , as is the fact that the categories in (iii) are left proper.
Remark 2.11. The analogues for orthogonal Gspectra of the categories in (i*
*i),
(iii), and (v) are left proper [18, 12.1, 15.2], but not the analogues of the c*
*ategories
in (i) and (iv). The reason for the better behavior of commutative algebras of
orthogonal Gspectra can be seen by comparing [18, 15.5, 15.6] with [5, III.5.1*
*].
3. The construction of the functors N and N#
The following equivariant analogue of I.3.1 is the essential step in the con*
*struc
tion of the adjoint pair (N; N# ). We assume that our given fixed universe U is
complete, but the result holds more generally; see Remark 3.8. Recall that the
Gcategory IG S of orthogonal Gspectra is isomorphic to the Gcategory JG T
of JG spaces, where JG is the category constructed from the category IG and
the IG space SG in II.4.1.
Theorem 3.1. There is a strong symmetric monoidal contravariant Gfunctor
N* : JG  ! MG . If V G 6= 0, then N*(V ) is (noncanonically) isomorphic to
FSVG, and the evaluation Gmap
" : N*(V ) ^ SV = N*(V ) ^ JG (0; V ) ! N*(0) ~=SG
of the functor is a weak equivalence.
Proof. The proof is similar to the nonequivariant argument in Ix5, and we
shall not give full details. Two equivalent constructions of N* were given in *
*Ix5,
and both apply equivariantly. The first is in terms of twisted halfsmash produ*
*cts.
The required equivariant theory of twisted halfsmash products is given by Cole*
* in
[25, XXII], except that he writes entirely in terms of Gmaps. Reinterpretation*
* in
terms of the equivariant context of IIx1 is straightforward.
As in Ix5, we use superscripts to denote relevant universes. Consider the un*
*i
verses V U for indexing spaces V U, together with their subspaces V ~=V R.
We set
(3.2) N*(V ) = SG ^L (I (V U; U) n VVU (S0)):
Here, for inner product Gspaces U and U0, I (U; U0) is the Gspace of linear
isometries U ! U0. It is Gcontractible when there is a Glinear isometry U !*
* U0
[21, 1.3]. Since U is complete, there is such a Glinear isometry V U ! U for
3. THE CONSTRUCTION OF THE FUNCTORS N AND N# 61
any V . Note however that V U does not contain trivial representations (as we
usually require of a universe) unless V G 6= 0. The functor J given by
(3.3) JE = SG ^L E
converts Lspectra to weakly equivalent SG modules; we rewrite
(3.4) N*(V ) = J(I (V U; U) n VVU (S0)):
The functoriality in V can be proven directly or by comparison with the alte*
*r
native second construction, which is given in terms of Thom Gspectra. Replacing
inner product spaces in I.5.85.18 by Ginner product spaces, still using gener*
*al
linear isometries but taking the action of G by conjugation into account, we ob*
*tain
the alternative description
(3.5) N*(V ) ~=JMVVU;U;:
With this form of the definition, the functoriality in V is given by I.5.20.
The definition (3.4) makes sense even when V = 0, with Gspectra indexed
on 0 interpreted as Gspaces. Inspection of definitions shows that N*(0) ~=SG ,*
* as
required for N* to be strong symmetric monoidal. The natural isomorphism
(3.6) OE : N*(V ) ^ N*(W ) ! N*(V W )
required of a strong symmetric monoidal functor is constructed as in I.5.7.
The argument for the identification of the N(Rn) for n > 0 given in Ix5 does*
* not
generalize to an equivariant identification of the N(V ) for V G 6= 0. However,*
* since
U is complete, V U is complete when V G 6= 0, and we can choose an isomorphism
of Guniverses fV : V U ! U that restricts to the identification V R ~=V .
Then fV induces an isomorphism
I (V U; U) n VVU (S0) ~=I (U; U) n UVS0 = LSVG:
Applying J and using the definition in (3.4), we obtain the required isomorphism
(3.7) N*(V ) ! JLSVG= FSVG:
Comparing with I.5.21, we see that " : N*(V ) ^ SV ! SG corresponds under
this isomorphism to the composite of the identification FSVG^SV ~=FSG (see [17,
I.4.2]) and the cofibrant approximation FSG ! SG . Observe that, although fV
and the isomorphism (3.7) depend noncanonically on V , when we suspend (3.7)
by SV and map to SG , the resulting weak equivalence, namely ", is natural in V*
* .
Concretely, this holds since we are projecting the contractible Gspaces I (V U*
*; U)
and I (U; U) to a point, which makes the choice of isomorphism between them_
invisible. __
Remark 3.8. Let U be a universe and let V = V (U) be the collection of all
Ginner product spaces that are isomorphic to indexing spaces in U. Taking JG
and MG to be defined with respect to V and U, the construction of the Gfunctor
N* still applies. However, the identification (3.7) need not hold. We say tha*
*t U
is closed under tensor products if the tensor product of representations in U is
isomorphic to a representation in U; clearly this holds if and only if V is cl*
*osed
under tensor products. In this case, V U is isomorphic to U for any V in U such
that V G 6= 0, and Theorem 3.1 holds as stated. In fact, all of the results in *
*the
introduction hold in this generality. For example, these results hold when U is*
* the
trivial universe.
62 IV. ORTHOGONAL GSPECTRA AND SGMODULES
The categorical discussion of Ix2 applies verbatim in the equivariant contex*
*t,
provided that all functors are required to be (continuous) Gfunctors between "*
*bi
complete" (topological) Gcategories. Here a Gcategory CG is said to be bicom
plete if it is tensored and cotensored over the Gcategory TG of based Gspaces*
* and
if its associated Gfixed category GC is complete and cocomplete. A Gfunctor
F : CG ! CG0is defined to be "right exact" if it preserves tensors and if its *
*restric
tion F : GC ! GC 0to Gmaps is right exact. The cited discussion then gives the
following formal consequence of Theorem 3.1.
Theorem 3.9. Define Gfunctors N : IG S ! MG and N# : MG ! IG S
by letting N(X) = N* JG X and (N# M)(V ) = MG (N*(V ); M). Then (N; N# ) is
an adjoint pair of Gfunctors such that N is strong symmetric monoidal and N# is
lax symmetric monoidal.
4. The proofs of the comparison theorems
As in Ix3, to prove Theorem 1.1 it suffices to prove the following three res*
*ults.
Lemma 4.1. The functor N# creates weak equivalences.
Lemma 4.2. The functor N# preserves qfibrations.
Proposition 4.3. The unit j : X  ! N# NX of the adjunction is a weak
equivalence for all cofibrant orthogonal Gspectra X.
As in the proof of its nonequivariant analogue in Ix4, Lemma 4.1 is implied *
*by
the following result.
Lemma 4.4. For SG modules M, N# M is a positive Gspectrum.
Proof. The statement means that if V W , V G 6= 0, then the structure map
"oe: (N# M)(V ) = M (N*(V ); M) ! M (WV N*(W ); M) ~=WV (N# M)(W )
is a weak equivalence. This holds because "oeis induced by the evaluation map
" : N*(W ) ^ SWV ! N*(V ), which is a weak equivalence between cofibrant_
SG modules by Theorem 3.1. __
In Lemma 4.2, we are saying that N# carries restricted qfibrations of SG 
modules to qfibrations of orthogonal Gspectra in the positive stable model st*
*ruc
ture, and the proof is the same formal argument as in Ix4.
To prove Theorem 1.1, we only need Proposition 4.3 for orthogonal Gspectra
that are cofibrant in the positive stable model structure, but it holds more ge*
*nerally
for orthogonal Gspectra that are cofibrant in the stable model structure. The *
*proof
is the same as in the nonequivariant case Ix4. As there, we first prove the res*
*ult
when X = FV A for a GCW complex A, and we then deduce it for cell orthogonal
Gspectra. For the first step, it is convenient to work on the prespectrum leve*
*l, using
the Quillen equivalence (P; U) between GI S and GP of III.4.16 and III.5.7.
5. THE FUNCTOR M AND ITS COMPARISON WITH N 63
We display our Quillen equivalences in the following (noncommutative) dia
gram:
_____L______//
GPOOoo__________GSOO
 ` 
 
UP VF
 
 
fflfflN fflffl
GI S __________//_GMo:o_
N#
As in Ix3, to prove Theorems 1.2, 1.3, 1.5, and 1.6, we observe that Lemmas *
*4.1
and 4.2 and Proposition 4.3 imply their analogues for the adjoint pairs (N; N# *
*) in
duced on the categories of multiplicatively enriched objects considered in the *
*cited
theorems. Since the weak equivalences and qfibrations in the multiplicatively *
*en
riched categories are created in the underlying categories (of orthogonal Gspe*
*ctra
or of SG modules), this is obvious for the lemmas. For the proposition, we must
relate cofibrancy of multiplicatively enriched objects with cofibrancy of their*
* un
derlying orthogonal Gspectra. For Theorems 1.2 and 1.3, III.7.6 gives what is
needed. For Theorem 1.5, we argue as in Ix3. The essential point is comparison *
*of
the second statement of III.8.4 with the following analogue for SG modules, wh*
*ose
proof is precisely parallel to the proof of its nonequivariant analogue in [5, *
*III.5.1].
Lemma 4.5. For a cofibrant SG module M (in the generalized cellular model
structure), the quotient map
q : Ei+ ^i M(i)! M(i)=i
is a weak equivalence.
5. The functor M and its comparison with N
Exactly as in Ix6, there is another Gfunctor M : IG S  ! MG which, al
though less convenient for the comparison theorems, is considerably more natural
intuitively. We have the forgetful Gfunctor U from orthogonal Gspectra to G
prespectra and the spectrification Gfunctor L from Gprespectra to Gspectra. *
*As
in I.6.4 and I.6.5, we can define equivariant notions of Lprespectra and Lspe*
*ctra.
The latter notion is the one that forms the basis for the definition of SG mod*
*ules.
Writing PG [L] and SG [L] for the resulting Gcategories, we verify as in I.6.6*
* and
I.6.7 that the forgetful Gfunctor U takes values in PG [L] and the spectrifica*
*tion
Gfunctor L induces a Gfunctor L : PG [L] ! SG [L]. Moreover, we have the
Gfunctor J : SG [L] ! MG specified by JE = SG ^L E.
Definition 5.1.Define the Gfunctor M : IG S ! MG to be the composite
IG S __P__//PG [L]_L_//_SG [L]J__//MG ;
where P denotes the underlying prespectrum functor regarded as taking values in
the category of Lprespectra.
As in I.6.96.11, the functor M has formal properties much like those of N.
Lemma 5.2. The Gfunctor M is right exact and lax symmetric monoidal, with
MSG ~=SG (where SG on the left is the sphere orthogonal Gspectrum).
64 IV. ORTHOGONAL GSPECTRA AND SGMODULES
As in I.6.12, although M does not appear to preserve cofibrant objects, it h*
*as
the following basic homotopical property. Recall Remark 2.6.
Lemma 5.3. For positive inclusion orthogonal Gspectra X, there is a natural
isomorphism
ssH*(X) ~=ssH*(M(X)):
Arguing as in I.6.13, we obtain the following comparison theorem.
Theorem 5.4. There is a symmetric monoidal natural Gmap
ff : NX ! MX
such that ff : NX ! MX is a weak equivalence if X is cofibrant.
One advantage of M over N is that it is quite convenient for the study of
change of groups, as was exploited implicitly by Greenlees and May in [12]. We
turn to considerations of change of group and universe in our new model theoret*
*ic
framework after generalizing the theory to families.
6. Families, cofamilies, and Bousfield localization
We discuss in model theoretical terms the familiar idea of concentrating G
spaces or Gspectra at or away from a family of subgroups. We also relate this
idea to Bousfield localization. The theory works the same way for SG modules a*
*nd
for orthogonal Gspectra; we often use the neutral term "object", and we let GC
stand for either GM or GI S . Similar arguments lead to weaker conclusions in
the category of Gspectra, due to Remark 2.9. We write [X; Y ]G for maps X ! Y
in HoGC .
We call weak equivalences Gequivalences in this section. For H G, we say
that a Gmap is an Hequivalence if it is a weak equivalence when regarded as a*
*n H
map; we will treat restriction to subgroups systematically in the next section.*
* Let
F be a family of subgroups of G, namely a set of subgroups closed under passage
to conjugates and subgroups. There is a universal F space EF . It is a GCW
complex such that (EF )H is contractible for H 2 F and empty for H =2F . Of
course, its cells must be of orbit type G=H with H 2 F . The following definiti*
*ons
make sense for based Gspaces as well as for objects in GC .
Definition 6.1. (i)A map f : X  ! Y is a F equivalence if it is an
Hequivalence for all H 2 F .
(ii)An object X is an F object if the map ss : EF+ ^ X ! X induced by the
projection EF+ ! S0 is a Gequivalence.
Definition 6.2.Let E be a cofibrant object of GC or a cofibrant based G
space.
(i)A map f : X ! Y is an Eequivalence if id^f : E ^ X ! E ^ Y is a
Gequivalence.
(iii)Z is Elocal if f* : [Y; Z]G ! [X; Z]G is an isomorphism for all Eequi*
*va
lences f : X ! Y .
(iv)An Elocalization of X is an Eequivalence : X  ! Y from X to an
Elocal object Y .
Consider GM with either the cellular or the generalized cellular model struc*
*ture
and consider GI S with either the stable or the positive stable model structure.
6. FAMILIES, COFAMILIES, AND BOUSFIELD LOCALIZATION 65
Indexing can be on any universe. We have the following slightly digressive, but
important, omnibus theorem. Nonequivariantly, it is proven for Smodules in [5,
VIIIx1]. With only minor variations, the argument there applies equivariantly t*
*o all
of the model structures we are considering. It does not apply to Gspectra beca*
*use
it uses that GC is left proper.
Theorem 6.3. Let E be a cofibrant object of GC or a cofibrant based Gspace.
Then GC has an Emodel structure whose equivalences are the Eequivalences
and whose Ecofibrations are the qcofibrations of the given model structure. T*
*he
Efibrant objects are the Elocal objects, and Efibrant approximation construc*
*ts a
Bousfield localization : X ! LE X of X at E.
Taking E = EF+ , we call the resulting model structures Bousfield F model
structures. Here Bousfield localization takes the followng elementary form.
Proposition 6.4. The map : X ! F (EF+ ; X) induced by the projection
EF+ ! S0 is an EF+ localization of X.
Proof. The map is an EF+ equivalence by [11, 17.2], and it is immediate_
by adjunction that F (EF+ ; X) is EF+ local. __
Completion theorems in equivariant stable homotopy theory are concerned with
the comparison of this Bousfield localization with another, more algebraically *
*com
putable, Bousfield localization. See, for example, [8, 4.1], [9]. However, the *
*Bous
field F model structures are not the most natural ones to consider in the cont*
*ext
of families. In the model structures on GC , the generating qcofibrations and
generating acyclic qcofibrations are obtained by applying functors FV or 1V to
certain maps G=H+ ^ A ! G=H+ ^ B of Gspaces. We can restrict attention to
those H 2 F in all of these definitions. We refer to F cofibrations rather th*
*an
qcofibrations for the retracts of the resulting relative F cell complexes. In*
* contrast
with Theorem 6.3, the following theorem, with left properness deleted, applies *
*just
as well to Gspectra as to SG modules and orthogonal Gspectra.
Theorem 6.5. The category GC is a compactly generated proper Gtopological
model category with weak equivalences the F equivalences and with generating F*
* 
cofibrations and generating acyclic F cofibrations obtained from the original *
*gen
erating qcofibrations and generating acyclic qcofibrations by restricting to *
*orbits
G=H with H 2 F .
We refer to these as F model structures. The proofs of the model axioms
are the same as in the case F = A ``. Following the approach of Chapter III, we
first give GT an F model structure using its F equivalences and F cofibratio*
*ns.
The F fibrations are the maps that give Serre fibrations on passage to Hfixed
points for H 2 F . We then give Gprespectra and orthogonal Gspectra level
F model structures using level F equivalences and level F fibrations; the re*
*sult
ing F cofibrations are as described above. Finally, we give Gprespectra and o*
*r
thogonal Gspectra stable F model structures using the F equivalences and F 
cofibrations. Similarly, the approach of [5] applies verbatim to give Gspectra*
* and
SG modules cellular and generalized cellular F model structures. All of our c*
*om
parison theorems have F versions that admit the same proofs.
To relate the F model structure to the Bousfield F model structure, we must
show that the two structures have the same weak equivalences. This is not at all
obvious. We need the following lemma in the proof.
66 IV. ORTHOGONAL GSPECTRA AND SGMODULES
Lemma 6.6. If A is a based F CW complex and X is a cell object, then A ^ X
is an F cell object.
Proof. This is shown by inspection of the combinatorial structure of cell o*
*b
jects. The essential point is that, for H 2 F and any K, we can triangulate the
product G=H x G=K as a finite GCW complex by [16], and in any such trian
gulation the only orbit types that can occur are G=L with L 2 F . While the
details from here are straightforward, the reader should be aware that the unde*
*r
lying spaces of F cell objects in any of our categories are generally not F s*
*paces,_
so that the conclusion is much less obvious than its space level analogue. *
* __
Proposition 6.7. The following conditions on a map f : X ! Y are equiv
alent.
(i)f is an F equivalence.
(ii)f* : ssH*(X) ! ssH*(Y ) is an isomorphism for H 2 F .
(iii)f is an EF+ equivalence.
Proof. Parts (i) and (ii) are equivalent by definition. For H 2 F , EF+ !
S0 is an Hhomotopy equivalence and therefore (iii) implies (i). We must prove
that (i) implies (iii). Thus let f : X  ! Y be an F equivalence. We must
show that id^f : EF+ ^ X ! EF+ ^ Y is a Gequivalence, and it is certainly
an F equivalence. Since EF+ is a GCW complex, smashing with it preserves
Gequivalences. Using functorial cofibrant approximation in the original model
structure, we may assume without loss of generality that X and Y are cell objec*
*ts.
By Lemma 6.6, EF+ ^ X and EF+ ^ Y are then F cell objects. In the case of
SG modules, where all objects are fibrant, we conclude that id^f is a Ghomoto*
*py
equivalence because it is an F equivalence between F cofibrant objects. (In e*
*arlier
terminology, we are invoking the F Whitehead theorem [17, II.2.2]). In the case
of GI S , using functorial fibrant approximation we may assume further that X
and Y are orthogonal Gspectra. Using III.9.3, we conclude that f is a level
F equivalence. On the space level, it is clear that (i) implies (iii), and we_*
*conclude_
that id^f is a level Gequivalence and hence a ss*isomorphism. _*
*_
Remark 6.8. For a fibrant object X of GC , we have ssH*(X) = ss*(XH ), just
as for Gspaces. For orthogonal Gspectra, this is V.3.2 below. For Gspectra (*
*all
of which are fibrant), it is [17, I.4.5], and the analogue for SG modules foll*
*ows
(see VI.3.4 below). Thus, when X and Y are fibrant, a map f : X ! Y is an
F equivalence if and only fH : XH  ! Y H is a weak equivalence for all H 2 F .
The Bousfield F model structures have more cofibrations and the same weak
equivalences as the F model structures. This implies the following result.
Theorem 6.9. The identity functor GC ! GC is the left adjoint of a Quillen
equivalence from the F model structure to the Bousfield F model structure.
Now return to the notion of an F object in Definition 6.1. Observe that this
is an intrinsic notion, independent of any model structure.
Theorem 6.10. An object X is an F object if and only if its F cofibrant
approximation fl : X ! X is a Gequivalence.
Proof. Since EF+ ^G=H+ ! G=H+ is a Ghomotopy equivalence if H 2 F ,
an F cell complex is an F object. The standard functorial construction gives
6. FAMILIES, COFAMILIES, AND BOUSFIELD LOCALIZATION 67
X as an F cell complex. The conclusion follows from the evident commutative
diagram
EF+ ^ X _ss_//_X
id^fl fl
fflffl fflffl
EF+ ^ X __ss__//X;
in which id^fl and the top map ss are Gequivalences for any X by Lemma_6.6 and
Theorem 6.7. __
Observe that, in GM , this holds for both the cellular and the generalized
cellular F model structures. Let HoF C denote the homotopy category associated
to the F model structure on GC , or, equivalently, the Bousfield F model stru*
*cture,
and write [X; Y ]F for the set of maps X ! Y in this category. The results abo*
*ve
imply the following description of HoF C .
Theorem 6.11. Smashing with EF+ defines an isomorphism
[X; Y ]F ~=[EF+ ^ X; EF+ ^ Y ]G
and thus gives an equivalence of categories from HoF C to the full subcategory *
*of
objects EF+ ^ X in HoGC .
Proof. If X and Y are F cofibrant approximations of X and Y , then
[X; Y ]F ~=[X; Y ]G ~=[EF+ ^ X; EF+ ^ Y ]G ~=[EF+ ^ X; EF+ ^ Y ]G
since the definition of an F cell object, Lemma 6.6, and Theorem 6.10 imply th*
*at
X is an F object, X and EF+ ^X are cofibrant in our original model structure,
and EF+ ^ X is a cofibrant approximation of EF+ ^ X. Formally, we are using
that the identity functor is the right adjoint of a Quillen adjoint pair relati*
*ng_the
original model structure to the F model structure. __
There is an analogous theory for cofamilies, namely complements F 0of famili*
*es.
Thus F 0is the set of subgroups of G not in F . We define "EF to be the cofiber
of EF+ ! S0. Then (E"F )H is contractible if H 2 F and is S0 if H =2F . In
contrast to the situation for Gspaces, the evident analogue of Proposition 6.7*
* is
false for Gspectra. This motivates the following variant of Definition 6.1.
Definition 6.12. (i)A map f : X ! Y is an F 0equivalence if it is an
"EF equivalence.
(ii)An object X is an F 0object if the map : X ! "EF ^ X induced by the
inclusion S0 ! "EF is a Gequivalence.
Again, take GC to be GM or GI S with one of our usual model structures.
We do not obtain model structures by restricting attention to orbits G=H with
H 2 F 0, but we do still have the Bousfield F 0model structure obtained by tak*
*ing
E = "EF in Theorem 6.3. We have the following analogue of Theorem 6.10.
Theorem 6.13. The following conditions on an object X are equivalent.
(i)X is an F 0object.
(ii)X is an "EF local object.
(iii)The "EF fibrant approximation : X ! LE"FX is a Gequivalence.
(iv)ssH*(X) = 0 for H 2 F .
68 IV. ORTHOGONAL GSPECTRA AND SGMODULES
For such an X and any Y , * : [E"F ^ Y; X]G ! [Y; X]G is an isomorphism.
Proof. This is a strengthened version of [17, II.9.2]. Assume (i). The com
posite
[Y; X]G ! [E"F ^ Y; "EF ^ X]G ! [Y; "EF ^ X]G ~=[Y; X]G
is the identity, where the first map is given by smashing with E"F , the second
is *, and the isomorphism is given by *. Therefore, if f : Y ! Y 0is an
"EF equivalence, then f* : [Y 0; X]G  ! [Y; X]G is a retract of the isomorphi*
*sm
f* : [E"F ^ Y 0; "EF ^ X]G ! [E"F ^ Y; "EF ^ X]G and is thus an isomorphism.
This shows that (i) implies (ii), and (ii) and (iii) are equivalent by Theorem *
*6.3
and the uniqueness of localizations. If H 2 F , then G=H+ ^ "EF is Gcontractib*
*le
by a check of fixed point spaces, hence G=H+ ^ Y ! * is an "EF equivalence for
any Y . Letting Y run through the spheres Sn, this shows that (ii) implies (iv*
*).
Finally, assume (iv). We must prove (i). Smashing with the GCW complex "EF
preserves Gequivalences, so by cofibrant approximation we may assume that X is*
* a
cell complex. Clearly (i) holds if and only if EF+ ^X is trivial and, by Lemma *
*6.6,
EF+ ^ X is an F cell complex. If H 2 F , then EF+ ! S0 is an Hhomotopy
equivalence, hence ssH*(EF+ ^ X) = 0 by hypothesis. Thus EF+ ^ X is F 
equivalent to the trivial object and is therefore trivial. Since : "EF ! "EF *
*^E"F
is a Ghomotopy equivalence by a check on fixed points, : Y ! "EF ^ Y is an *
* __
F 0equivalence for any Y . Therefore the last statement follows from (ii). *
* __
Again, in GM this holds for both the cellular and the generalized cellular B*
*ous
field F 0model structures. Let HoF 0C denote the homotopy category associated
to the Bousfield F 0model structure on GC and write [X; Y ]F0 for the set of m*
*aps
X ! Y in this category. The previous theorem implies the following one.
Theorem 6.14. Smashing with "EF defines an isomorphism
[X; Y ]F0 ~=[E"F ^ X; "EF ^ Y ]G
and thus gives an equivalence of categories from HoF 0C to the full subcategory*
* of
objects "EF ^ X in HoGC .
CHAPTER V
"Change" functors for orthogonal Gspectra
We develop the analogues for orthogonal Gspectra of the central structural
features of equivariant stable homotopy theory: change of universe, change of g*
*roup,
fixed point and orbit spectra, and geometric fixed point spectra. The last noti*
*on
has turned out to be very important in many applications, and its treatment in
[17, IIx9] is decidedly ad hoc and conceptually unsatisfactory. The geometric f*
*ixed
point functor on orthogonal spectra turns out to be far more satisfactory.
1. Change of universe
Change of universe plays a fundamental role in the homotopical theory of [17*
*],
and, as explained in [6, 24], it takes a precise pointset level form in the th*
*eory of
SG modules. The theory for orthogonal Gspectra takes a similarly precise poin*
*t
set level form. The key fact is the following implication of the definition of*
* an
orthogonal Gspectrum.
Lemma 1.1. Let V and W be Grepresentations in V of the same dimension.
Then, for orthogonal Gspectra X, the evaluation Gmap JG (V; W ) ^ X(V ) !
X(W ) of the Gfunctor X induces a Ghomeomorphism
ff : JG (V; W ) ^O(V )X(V ) ! X(W ):
Its domain is homeomorphic, but not necessarily Ghomeomorphic, to X(V ).
Proof. Since V and W have the same dimension, JG (V; W ) = IG (V; W )+ ,
and IG (V; W ) is a free right O(V )space generated by any chosen linear isome*
*tric
isomorphism f : V ! W . We see that ff is a homeomorphism, hence a G
homeomorphism, by noting that the map that sends y in X(W ) to the equivalence
class of (f; X(f1 )(y)) gives the inverse homeomorphism. Mapping x to the equi*
*va_
lence class of (f; x) gives the homeomorphism X(V ) ~=JG (V; W ) ^O(V )X(V ). *
*__
Change of universe appears in several equivalent guises. We could apply the
general theory of prolongation functors left adjoint to forgetful functors, usi*
*ng the
equivariant version of I.2.10 and [18, x3], but we prefer to be more explicit. *
* We
mimic the analogous theory of [6].
Definition 1.2.Let V and V 0be collections of representations as in Definiti*
*on
2.1.0Thus both collections0contain all trivial representations. Define a Gfunc*
*tor
IVV : IGVS ! IGVS by letting
0 0 V 0n 0 n
(IVVX)(V ) = JG (R ; V ) ^O(n)X(R )
if V 02 V 0and dim V 0= n. The evaluation Gmaps of the Gfunctor IVV0X :
JGV 0! TG are given by evaluation Gmaps of0X (restricted to trivial repre
sentations Rn), composition Gmaps of JGV, and canonical Ghomeomorphisms.
69
70 V. "CHANGE" FUNCTORS FOR ORTHOGONAL GSPECTRA
Explicitly, if dimW 0= p, then the evaluation Gmap
0 0 V 0 0
JG (V 0; W 0) ^ (IVVX)(V ) ! (IV X)(W )
is the composite
0 0 0 V 0n 0 n
JGV(V ; W ) ^ JG (R ; V ) ^O(n)X(R )
0n 0 n
! JGV(R ; W ) ^O(n)X(R )
~= JGV(0Rp; W 0) ^O(p)JGV(Rn; Rp) ^O(n)X(Rn)
0p 0 p
! JGV(R ; W ) ^O(p)X(R ):
If V = V (U) and V 0= V (U0) for universes U and U0, then V V 0if and
only if there is a Glinear isometry U ! U0. This is the starting point for t*
*he
change of universe functors in [17]. By inspection (or by [18, 2.1]), the inclu*
*sion0
then induces a full and faithful strong symmetric monoidal functor JGV ! JGV,
and it is in this case that the theory of prolongation functors of [18, x3] app*
*lies.
Here we have a more natural looking but equivalent form of the definition of IV*
*V,0
namely
(1.3) (IVVY0)(V ) = Y (V )
for Y 2 IGVS0and V 2 V . Evaluation Ghomeomorphisms
0n n
JGV(R ; V ) ^O(n)Y (R ) ! Y (V )
give a natural isomorphism comparing the two descriptions of IVVY0. Writing FVVA
to indicate the universe of shift desuspension functors, it follows by inspecti*
*on of
right adjoints that
0 V V 0
(1.4) IVVFV A = FV A
for V 2 V and any based Gspace A.
Write V : TG ! IGVS for the suspension Gspectrum functor. The follow
ing result is analogous to [6, 2.3, 2.4].
Theorem 1.5. Consider collections V , V 0and V 00.
(i)IVV0O V is naturally isomorphic to V 0.
(ii)IVV000O IVV0is naturally isomorphic to IVV00.
(iii)IVVis naturally isomorphic to the identity functor.
(iv)The functor IVV0commutes with smash products with based Gspaces.
(v) The functor IVV0is strong symmetric monoidal.
Therefore IVV0is an equivalence of categories with inverse IVV.0Moreover, IVV0i*
*s ho
motopy preserving, hence IVV0and IVV 0induce inverse equivalences of the homoto*
*py
categories obtained by passing to homotopy classes of maps.
Proof. Evaluation Ghomeomorphisms JG (Rn; V ) ^ Sn ! SV give (i). For
(ii), if dimV 00= n, we have Ghomeomorphisms
00 V 0 00 V 00n 00 V 0n n n
(IVV 0O IV X)(V )= JG (R ; V ) ^O(n)JG (R ; R ) ^O(n)X(R )
~= JGV 00(Rn; V 00) ^O(n)X(Rn)
since JGV(0Rn; Rn) = O(n)+ . Part (iii) is clear from (1.3). Part (iv) is obv*
*ious
and implies the last statement. For (v), consideration of V [ V 0shows that we
1. CHANGE OF UNIVERSE 71
may assume without loss of generality that V V 0, and then (v) holds by the __
equivariant version of [18, 3.3] or I.1.14. _*
*_
We turn to the relationship with model structures. It is important to realize
what Lemma 1.1 does not imply: a map f : X ! Y can be a weak equivalence at
level Rn for all n but still not be a level equivalence. The point is that the *
*Hfixed
point functors do not commute with passage to orbits over O(n).
Similarly, it is important to0realize what the last statement of Thoerem 1.5
does not imply: the functors IVV do not preserve either level equivalences or s*
*s*
isomorphisms in general. Therefore, there is no reason to expect the homotopy
categories associated to the model structures to be equivalent. However, (1.3) *
*and
the characterization of qfibrations and acyclic qfibrations given in III.4.12*
* imply
the following result.
Theorem 1.6. If V V 0, then the functor IVV 0: GIGV 0! GIGV preserves
level equivalences, level fibrations, qfibrations, and acyclic qfibrations,0a*
*nd simi
larly for the positive analogues of these classes of maps. Therefore (IVV; IVV)*
*0is a
Quillen adjoint pair of functors relating the respective level, positive level,*
* stable,
and positive stable model structures.
There is another way to think about change of universe.0For V V 0, we can
define new V model structures on the category of IGVspectra. For the V level
model structure (or positive V level model structure), we define weak equivale*
*nces
and fibrations by restricting attention to levels in V ; equivalently, the V l*
*evel
equivalences and fibrations are created by the forgetful functor IVV.0We define*
* the
V cofibrations of GI VS0to be the Gmaps that satisfy the LLP with respect to
the V level acyclic fibrations. Compare [18, 6.10]. We then let the V stable *
*equiv
alences and the V fibrations be created by IVV.0Thus the V stable equivalence*
*s are
the V ss*isomorphisms, that is, the maps that induce isomorphisms of the homo
topy groups defined using only those V 2 V in the relevant colimits. Arguing as
for the stable model structure and using the characterization of a Quillen equi*
*va
lence given in [18, A.2], we obtain the following result. It is a rare example *
*of an
interesting "Quillen equivalence" of model categories that is an actual equival*
*ence
of underlying categories.
Theorem 1.7. For V V 0, the category GI VS0of IGV0spectra and natural
Gmaps has a V stable model structure in which the functor IVV 0creates the V 
stable equivalences and the V fibrations. The acyclic V fibrations coincide w*
*ith the
V level acyclic firations, and the V cofibrations are the0maps that satisfy t*
*he LLP
with respect to the acyclic V fibrations. The pair (IVV; IVV)0is a Quillen equ*
*ivalence
between GI VS with its stable model structure and GI VS0with its V stable model
structure. The analogous statements for positive V stable model structures hol*
*d.
There is another observation to make along the same lines.
Corollary 1.8. For V V 0, the identity functor Id: GI VS0! GI VS0
is the right adjoint0of a Quillen adjoint pair relating the (positive) stable0m*
*odel
structure on GI VS to the (positive) V stable model structure on GI VS .
Thus the forgetful functor IVV 0: GI VS0 ! GI VS relating the (original)
stable model structures factors through the V stable model structure on GI VS0.
72 V. "CHANGE" FUNCTORS FOR ORTHOGONAL GSPECTRA
That is, the Quillen adjoint pair of Theorem 1.6 is the composite of the Quillen
adjoint pair of Corollary 1.8 and the Quillen adjoint equivalence of Theorem 1.*
*7.
Remark 1.9. There is yet another way to think about change of universe. Fix
IG = IGA ``. Then, for any V , the (positive) V stable model structure on the
category GI S is Quillen equivalent to the (positive) stable model structure on
GI VS , and similarly for the various categories of rings and modules. However,*
* to
make sense of some of the constructions in the following sections, we must work*
* with
the more general categories of IGVspectra, with their intrinsic model structur*
*es.
Remark 1.10. In addition to changes of V , we must deal with changes of the
choice of "indexing Gspaces" within a given V , as in II.2.2. Thus let W V be*
* a
cofinal set of Grepresentations that is closed under finite direct sums and co*
*ntains
the Rn. We have a forgetful functor IWV : IGVS ! IGWS specified as in (1.3).
It can also be specified as in Definition 1.2 and, arguing as in that definitio*
*n and
Theorem 1.5, IWV is an equivalence of categories with inverse equivalence IVW. *
*We
can carry out all of our model category theory in the more general context. The
functor IWV : GI VS ! GI WS preserves qfibrations, and cofinality ensures
that IWV creates the stable equivalences in GI VS . We conclude that (IVW; IWV)*
* is
a Quillen equivalence.
2.Change of groups
Let H be a subgroup of G and write : H ! G for the inclusion. For a
Gspace A, let *A denote A regarded as an Hspace via . We want analogues for
orthogonal Gspectra of such space level observations as (III.1.2)  (III.1.5).
This involves change of universe as well as change of groups. If V = {V }
is a collection of representations of G, then *V = {*V } is a collection of re*
*p
resentations of H. According to our conventions in Definition 2.1, Gsummands
of representations in V are in V , but this need not be true of Hsummands of
representations in *V . For example, not all Hrepresentations are in *A ``(G).
However, we can let W be the collection of Hrepresentations that are isomorphic
to summands of representations in *V . Since *V is cofinal in W and closed un
der finite direct sums, Remark 1.10 applies. For example, if V = A ``(G), then
W = A ``(H) since any Hrepresentation is a summand of a Grepresentation.
To fix ideas and simplify notation, we work with A ``(H) when defining or
thogonal Hspectra, and we do not introduce notation for the change of universe
functor that passes from orthogonal Hspectra indexed on *A ``(G) to orthogonal
Hspectra indexed on A ``(H).
Definition 2.1.For an orthogonal Gspectrum X, let *X be the orthogonal
Hspectrum that is specified by (*X)(*V ) = *X(V ) for representations V of G
and is then extended to all representations of H by Remark 1.10.
Lemma 2.2. The functor * preserves level fibrations, level equivalences, q*
*co
fibrations, ss*isomorphisms, and qfibrations.
Proof. Most of this is clear from the characterizations of the various clas*
*ses
of maps. An exception is the assertion that * preserves qcofibrations, which *
*is
less obvious. Clearly * preserves colimits and satisfies *(FV (A)) = F*V (*A).
When A = G=K+ ^ Sn, *A = *(G=K)+ ^ Sn. If G is finite, then *(G=K) is
isomorphic to a disjoint union of Horbits, the choice of isomorphism depending*
* on
2. CHANGE OF GROUPS 73
a double coset decomposition of H\G=K. Fixing such choices gives a decomposition
of the underlying Hspace of a Gcell complex as an Hcell complex. For a gener*
*al
compact Lie group G, *(G=K) can be decomposed, noncanonically, as a finite
HCW complex [16]. Again, fixing choices of such decompositions allows us to __
decompose the underlying Hspace of a Gcell complex as an Hcell complex. _*
*_
We claim that the functor * has both a left and a right adjoint. On the space
level, for Hspaces B, the left adjoint of * is given by G+ ^H B and the right
adjoint is given by the Gspace of Hmaps FH (G+ ; B). For Gspaces A and A0, we
have obvious identifications of Hspaces
*F (A; A0) = F (*A; *A0) and *(A ^ A0) = *A ^ *A0:
On passage to left and right adjoints, respectively, these formally imply natur*
*al
isomorphisms of Gspaces
(G+ ^H B) ^ A ~=G+ ^H (B ^ *A)
and
F (A; FH (G+ ; B)) ~=FH (G+ ; F (*A; B));
and it is easy to write down explicit isomorphisms.
Proposition 2.3. Let X be an orthogonal Gspectrum and Y be an orthogonal
Hspectrum. Let G+ ^H Y be the orthogonal Gspectrum specified by
(G+ ^H Y )(V ) = G+ ^H Y (*V )
for representations V of G. Then there is an adjunction
GI S (G+ ^H Y; X) ~=HI S (Y; *X);
which is a Quillen adjoint pair relating the respective (positive) level and st*
*able
model structures. Moreover, there is a natural isomorphism
(G+ ^H Y ) ^ X ~=G+ ^H (Y ^ *X):
In particular,
G=H+ ^ X ~=G+ ^H *X:
Proof. The evaluation Gmaps of the Gfunctor G+ ^H Y : JG ! TG are
induced from the evaluation Hmaps of the Hfunctor Y via
JG (V; W ) ^ (G+ ^H Y (*V ))~= G+ ^H (*JG (V; W ) ^ Y (*V ))
~= G+ ^H (JH (*V; *W ) ^ Y (*V ))
 ! G+ ^H Y (*W ):
Lemma 2.2 implies the statement about model structures, and the rest is clear. *
* ___
Proposition 2.4. Let X be an orthogonal Gspectrum and Y be an orthogonal
Hspectrum. Let FH (G+ ; Y ) be the orthogonal Gspectrum specified by
FH (G+ ; Y )(V ) = FH (G+ ; Y (*V ))
for representations V of G. Then there is an adjunction
GI S (X; FH (G+ ; Y )) ~=HI S (*X; Y );
which is a Quillen adjoint pair relating the respective (positive) level and st*
*able
model structures. Moreover, there is a natural isomorphism
F (X; FH (G+ ; Y )) ~=FH (G+ ; F (*X; Y )):
74 V. "CHANGE" FUNCTORS FOR ORTHOGONAL GSPECTRA
In particular,
F (G=H+ ; X) ~=FH (G+ ; *X):
Proof. The adjoints of the evaluation Gmaps of the Gfunctor FH (G+ ; Y )
are induced from the adjoints of the evaluation Hmaps of Y via
FH (G+ ; Y (*V ))! FH (G+ ; F (JH (*V; *W ); Y (*W )))
~= FH (G+ ; F (*JG (V; W ); Y (*W )))
~= F (JG (V; W ); FH (G+ ; Y (*W ))): ___
3. Fixed point and orbit spectra
We relate orthogonal Gspectra to orthogonal spectra via fixed point and orb*
*it
functors, just as for Gspaces.
Definition 3.1.For an orthogonal Gspectrum X, define XG (V ) = X(V )G
for an inner product space V regarded as a trivial representation of G. Regardi*
*ng a
linear isometry f : V ! W as a Glinear isometry between trivial representatio*
*ns,
we see that X(f) is a Gmap since X is a Gfunctor. Therefore XG defines a func*
*tor
J ! T : More formally, let GI trivS denote the category of orthogonal G
spectra indexed only on trivial Grepresentations. We call the objects of GI tr*
*ivS
"naive" orthogonal Gspectra, in contrast to the genuine orthogonal Gspectra of
GI S . The Gfixed point functor is the composite of the change of universe fun*
*ctor
GI S = GI A ``S ! GI trivS
and the Gfixed point functor
GI trivS ! I S :
For H G, define XH = (*X)H .
The following fundamental result relating equivariant and nonequivariant ho
motopy groups is immediate from the definitions.
Proposition 3.2. Let E be an orthogonal Gspectrum. Then
ssH*(E) ~=ss*(EH ):
For any orthogonal Gspectrum X, ssH*(X) ~= ssH*(RX), where RX is a fibrant
approximation of X in the stable or positive stable model structure.
Giving spaces trivial Gaction, we obtain a functor
(3.3) "* : I S ! GI trivS :
We then have the following fixedpoint adjunction and its composite with the
evident change of universe adjunction.
Proposition 3.4. Let X be a naive orthogonal Gspectrum and Y be a non
equivariant orthogonal spectrum. There is a natural isomorphism
GI trivS ("*Y; X) ~=I S (Y; XG ):
For (genuine) orthogonal Gspectra X, there is a natural isomorphism
GI S (i*"*Y; X) ~=I S (Y; (i*X)G );
where i* = IAt``rivand i* = ItrivA.``Both of these adjunctions are Quillen adjo*
*int pairs
relating the respective (positive) level and stable model structures.
3. FIXED POINT AND ORBIT SPECTRA 75
The last statement means that passage to fixed points preserves qfibrations
and acyclic qfibrations. We have the following observation about qcofibratio*
*ns.
In the following two results, we agree to be less pedantic and to write ()G fo*
*r the
composite of i* and passage to Gfixed points. With this notation, the counit of
the second adjunction is a natural Gmap i*"*XG ! X.
Proposition 3.5. For a representation V and a Gspace A, (FV A)G = * un
less G acts trivially on V , when (FV A)G ~=FV (AG ) as a nonequivariant orthog*
*onal
spectrum. The functor ()G preserves qcofibrations, but not acyclic qcofibrat*
*ions.
Proof. For a trivial representation W , (FV A)G (W ) = GJ (V; W ) ^ AG .
If V is nontrivial, there are no nontrivial Glinear isometries V ! W and
GJ (V; W ) = *, whereas GJ (V; W ) = J (V; W ) if V is trivial. This gives the
first statement. Since the functor ()G preserves the colimits used to constru*
*ct
relative cell orthogonal Gspectra, by IV.1.6, it follows that it preserves pre*
*serves
qcofibrations. For nontrivial representations V of G, the maps k0;Vof III.4.6*
* are_
acyclic qcofibrations, whereas kG0;Vis equivalent to * ! S. *
*__
Warning 3.6. The last statement of Proposition 3.5 implies that the functor
()G is not a Quillen left adjoint. This functor does not behave homotopically *
*as
one might expect from the results of [17]. The reason is that it does not commu*
*te
with fibrant replacement (whereas all objects are fibrant in the context of [17*
*]), and
we must replace orthogonal Gspectra by weakly equivalent orthogonal Gspectra
before passing to fixed points in order to obtain the correct homotopy groups.
The following two results are in marked contrast to the situation in [17, 25*
*],
where the (categorical) fixed point functor does not satisfy analogous commutat*
*ion
relations. The point is that these results do not imply corresponding commutati*
*on
results on passage to homotopy categories, in view of Warning 3.6.
Taking V = 0, Proposition 3.5 has the following implication.
Corollary 3.7. For based Gspaces A,
(1 A)G ~=1 (AG ):
Note that the functors i* and "* are strong symmetric monoidal.
Proposition 3.8. For orthogonal Gspectra X and Y , there is a natural map
of (nonequivariant) orthogonal spectra
ff : XG ^ Y G ! (X ^ Y )G ;
and ff is an isomorphism if X and Y are cofibrant.
Proof. The map ff is adjoint to the evident natural Gmap
i*"*(XG ^ Y G) ~=(i*"*XG ) ^ (i*"*Y G) ! X ^ Y:
Using the properties of ()G given in III.1.6, the second statement follows from
Proposition 3.5 and the natural isomorphism
FV A ^ FW B ~=FV W (A ^ B)
of [18, 1.8]. ___
76 V. "CHANGE" FUNCTORS FOR ORTHOGONAL GSPECTRA
We can obtain a sharper version of Proposition 3.4. Let NH denote the nor
malizer of H in G and let W H = NH=H. We can obtain an Hfixed point functor
from orthogonal Gspectra to W Hspectra. It factors as a composite
GI S ! NHI S ! NHI HtrivS ! W HI S
of a change of group functor as in Definition 2.1, a change of universe functor*
*, and
a fixed point functor, all three of which are right adjoints.
It is useful to be more general about the last two functors. Thus let N be
any normal subgroup of G, let J = G=N, and let " : G ! J be the quotient
homomorphism. In the situation above, we are thinking of the normal subgroup H
of NH with quotient group W H.
Definition 3.9.Let GI NtrivS be the category of orthogonal Gspectra in
dexed on Ntrivial representations of G. Define "* : JI S  ! GI NtrivS by
regarding Jspaces as Ntrivial Gspaces. Define ()N : GI NtrivS ! JI S by
passage to Nfixed points spacewise, (XN )(V ) = X(V )N for a Jrepresentation V
regarded as an Ntrivial Grepresentation.
Proposition 3.10. Let X 2 GI NtrivS and Y 2 JI S . There is a natural
isomorphism
GI NtrivS ("*Y; X) ~=JI S (Y; XN ):
For (genuine) orthogonal Gspectra X, there is a natural isomorphism
GI S (i*"*Y; X) ~=JI S (Y; (i*X)N );
where i* = IAN``trivand i* = INtrivA.``Both of these adjunctions are Quillen *
*adjoint
pairs relating the respective (positive) level and stable model structures.
Similarly, we can define orbit spectra. Here again, we must first restrict *
*to
trivial representations. However, since this change of universe functor is a r*
*ight
adjoint and passage to orbits is a left adjoint, the composite functor appears *
*to be
of no practical value (just as in [17]).
Definition 3.11.For X 2 GI trivS , define X=G by (X=G)(V ) = X(V )=G
for an inner product space V . More generally, for X 2 GI NtrivS , define X=N 2
JI S by (X=N)(V ) = X(V )=N for a Jrepresentation V regarded as an Ntrivial
Grepresentation.
Proposition 3.12. Let X 2 GI NtrivS and Y 2 JI S . There is a natural
isomorphism
GI NtrivS (X; "*Y ) ~=JI S (X=N; Y ):
This adjunction is a Quillen adjoint pair relating the respective (positive) le*
*vel and
stable model structures.
Remark 3.13. The left and right adjoints of "* in this section and of * in *
*the
previous section can be regarded as special cases of a composite construction t*
*hat
applies to an arbitrary homomorphism ff : H ! G of compact Lie groups. Let
N = Ker(ff) and K = H=N. We have a quotient homomorphism " : H ! K and
an inclusion : K ! G induced by ff. Since ff = O ", ff* = "* O *. Therefore,*
* if
X 2 GI S and Y 2 HI NtrivS , we have the composite adjunctions
GI S (G+ ^K Y=N; X) ~=HI NtrivS (Y; ff*X)
and
GI S (X; FK (G+ ; Y N)) ~=HI NtrivS (ff*X; Y ):
4. GEOMETRIC FIXED POINT SPECTRA 77
4.Geometric fixed point spectra
There are actually two Gfixed point functors on orthogonal Gspectra, just
as there are on Gspectra [17, IIx9] and [25, XVIx3], namely the "categorical"
one already defined and another "geometric" one. Because the categorical fixed
point functor here seems to enjoy some of the basic properties that motivated
the introduction of the geometric fixed point functor in the classical setting,*
* the
discussion requires some care. We want a version of the Gfixed point functor f*
*or
which the commutation relations of Corollary 3.7 and Proposition 3.8 are true, *
*but
which also preserves acyclic qcofibrations, so that these properties remain tr*
*ue
after passage to fibrantcofibrant approximation of cofibrant objects.
In this section, we work from the beginning in the general context of a norm*
*al
subgroup N of G with quotient group J. The reader may wish to focus on the
special case N = G, in which case J is the trivial group. However, G plays two
quite different roles in that case, and the general case clarifies issues of eq*
*uivariance.
We need some categorical preliminaries.
Definition 4.1.Let E denote the extension
e_____//N____//_G_"_//_J____//e:
We define a category JE enriched over the category JT of based Jspaces. The
objects of JE are the Ginner product spaces V . The Jspace JE (V; W ) of mor
phisms V  ! W is the Nfixed point space JG (V; W )N . Thus, if we ignore the
Jaction, then
JE (V; W ) = NJ (*V; *W ):
A nonbasepoint map (f; x) : V ! W is an Nlinear isometry f : V ! W
together with a point x 2 W N f(V N). Observe that JE = GJ when N = G
and JE = JG when N = e. Let
OE : JE ! JJ
be the Nfixed point Jfunctor. It sends the Ginner product space V to the Ji*
*nner
product space V N and sends a morphism (f; x) : V  ! W to the Nfixed point
morphism (fN ; x) 2 JJ(V N; W N). Let
: JJ ! JE
be the Jfunctor that sends a Jinner product space V to V regarded as a Ginner
product space by pullback along " and is given on morphism spaces JJ(V; W ) by
the identity function; this makes sense since every linear isometry V ! W is an
Nmap. Observe that
OE O = Id: JJ ! JJ:
Definition 4.2.Let JE T denote the category of JE spaces, namely (con
tinuous) Jfunctors JE ! TJ. Note that a JE space Y has structural Jmaps
N V N
Y (V ) ^ SW ! Y (W )
for V W . Let
UOE: JJT  ! JE T and U : JE T  ! JJT
be the forgetful functors induced by OE and . By [18, 3.2], left Kan extension *
*along
OE and gives prolongation functors
POE: JE T  ! JJT and P : JJT  ! JE T
78 V. "CHANGE" FUNCTORS FOR ORTHOGONAL GSPECTRA
left adjoint to UOEand U . Since OEO = Id, U OUOE= Idand therefore POEOP ~=Id:
With these definitions in place, we can define the geometric fixed point fun*
*ctors.
Definition 4.3.Define a fixed point functor FixN : JG T ! JE T by
sending an orthogonal Gspectrum X to the JE space FixNX with (FixN X)(V ) =
X(V )N and with evaluation Jmaps
X(V )N ^ JG (V; W )N ! X(W )N
obtained by passage to Nfixed points from the evaluation Gmaps of X. Define
the geometric fixed point functor
N : JG T  ! JJT
to be the composite POEO FixN. Define a natural Jmap fl : XN ! N X of
orthogonal Jspectra by observing that the categorical fixed point functor can *
*be
reinterpreted as XN = U FixNX and letting fl be the map
(4.4) U j : XN = U FixNX ! U UOEPOEFixNX = POEFixNX = N X;
where j : Id! UOEPOEis the unit of the prolongation adjunction.
We have the following analogue of Proposition 3.5.
Proposition 4.5. For a representation V of G and a Gspace A,
N (FV A) ~=FV NAN :
The functor N preserves qcofibrations and acyclic qcofibrations.
Proof. By the definitions, we have
(FixN FV A)(W ) = JG (V; W )N ^ AN :
Thus FixNFV A = FV AN where FV on the right is left adjoint to the V th Jspace
evaluation functor on the category JE T . We have POEO FV = FV Nby the equality
of their right adjoints, and the first statement follows. By inspection from De*
*fini
tion 4.6, the functor N preserves generating qcofibrations and generating acyc*
*lic
qcofibrations. It also preserves the colimits used to construct relative cell*
* or
thogonal Gspectra, by III.1.6, and it therefore preserves qcofibrations_and a*
*cyclic
qcofibrations. __
Analogues of Corollary 3.7 and Proposition 3.8 follow readily.
Corollary 4.6. For based Gspaces A,
N 1 A ~=1 (AN );
where 1 on the left and right are the suspension spectrum functors from Gspaces
to Gspectra and from Jspaces to Jspectra.
The functors POEand P are strong symmetric monoidal, by [18, 3.3].
Proposition 4.7. For orthogonal Gspectra X and Y , there is a natural Jmap
ff : N X ^ N Y ! N (X ^ Y )
of orthogonal Jspectra, and ff is an isomorphism if X and Y are cofibrant.
4. GEOMETRIC FIXED POINT SPECTRA 79
Proof. By the definition of internal smash products [18, 21.4], there are
canonical maps of Gspaces
X(V ) ^ Y (W ) ! (X ^ Y )(V W ):
Passing to Nfixed point spaces, we obtain a natural Jmap
FixN X Z FixNY ! FixN(X ^ Y ) O
of (JE x JE )spaces. We obtain ff by applying POEto the adjoint Jmap
FixNX ^ FixNY ! FixN(X ^ Y )
of JE spaces. It follows easily from Proposition 4.5 that ff is an isomorphism*
* when
X = FV A and Y = FW B, and it follows inductively that ff is an isomorphism_when
X and Y are cofibrant. __
In the previous section, we interpreted the homotopy groups of the categoric*
*al
fixed points of a fibrant approximation of X as the homotopy groups of X. We
now interpret the homotopy groups of the geometric fixed points of a cofibrant
approximation of X as a different kind of homotopy groups of X. For this, we
introduce homotopy groups of JE spaces.
Definition 4.8.Let Y be a JE space and X be an orthogonal Gspectrum.
Let K J and write K = H=N, where N H G.
(i)Define
N
ssKq(Y ) = colimVssKqV Y (V ) ifq 0;
where V runs over the Ginner product spaces in the universe U, and
NRq
ssKq(Y ) = colimV RqssK0V Y (V ) ifq > 0:
(ii)Define a natural homomorphism
i : ssK*(U Y ) ! ssK*(Y )
by restricting colimit systems to Nfixed Ginner product spaces.
(iii)Define
aeKq(X) = ssKq(FixN X);
so that aeKq(X) = colimVssKqV NX(V )N for q 0, and similarly for q < 0.
(iv)Define a natural homomorphism
: ssK*(XN ) ! ssH*(X)
by restricting colimit systems to Nfixed Ginner product spaces W , using
(W X(W )N )K ~=(W X(W ))H :
(v) Define a natural homomorphism
! : ssH*(X) ! aeK*(X)
by sending an element of ssHq(X) represented by an Hmap f : Sq ^ SV !
X(V ) to the element of aeKq(X) represented by the Kmap fN : Sq^SV N !
X(V )N , q 0, and similarly for q < 0.
Define ss*isomorphisms of JE spaces and ae*isomorphisms of orthogonal Gspec*
*tra
in the evident way.
80 V. "CHANGE" FUNCTORS FOR ORTHOGONAL GSPECTRA
If X is an orthogonal Gspectrum, then is a natural isomorphism. In this
case, we may identify i and ! in view of the following immediate observation.
Lemma 4.9. The homomorphism
i : ssK*(XN ) = ssK*(U FixNX) ! ssK*(FixN X) = aeK*(X)
is the composite of : ssK*(XN ) ! ssH*(X) and ! : ssH*(X) ! aeK*(X).
We also have the following observation.
Lemma 4.10. For orthogonal Jspectra Z, the homomorphism
i : ssK*(Z) = ssK*(U UOEZ) ! ssK*(UOEZ)
is an isomorphism.
Proof. We may rewrite the colimits in Definition 4.8 as iterated colimits by
first considering indexing spaces W in UN and then considering indexing spaces*
* V
in U such that V N = W . Thus, if q 0,
ssKq(Y ) = colim colim ssKqW Y (V ):
WUN V U; V N=W
When Y = UOEZ, Y (V ) = Z(V N) and this colimit reduces to
ssKq(Y ) ~=colimWUN ssKqW Z(W ) = ssKq(Z):
The proof for q < 0 is similar. ___
Via the naturality of i, this leads to the following identification of fl*, *
*where
fl = U j as in (4.4). Observe that the unit j of the prolongation adjunction fo*
*r OE
induces a natural map
1
j* : aeK*(X) = ssK*(FixN X) ! ssK*(UOEPOEFixNX) i!ssK*(POEFixNX) = ssK*(N X):
Lemma 4.11. Let K = H=N, where N H. For orthogonal Gspectra X,
the map fl* : ssK*(XN ) ! ssK*(N X) is the composite
ssK*(XN ) ~=ssH*(X) !!aeK*(X) j*!ssK*(N X):
We have the following basic identification of homotopy groups.
Proposition 4.12. The map j* : aeK*(X) ! ssK*(N X) is an isomorphism
for cofibrant orthogonal Gspectra X.
Proof. The functor N preserves cofiber sequences, wedges, and colimits of
sequences of hcofibrations. Therefore both functors aeK*and ssK*ON convert cof*
*iber
sequences to long exact sequences, convert wedges to direct sums, and convert
colimits of sequences of hcofibrations to colimits of groups. Thus to show th*
*at
j* is an isomorphism on all cofibrant objects, it suffices to show that it is an
isomorphism on objects X = FZA, where Z is a Ginner product space and A is a
GCW complex. We treat the case q 0, the case q < 0 being similar. Here j* is
the map
colimWcolimssKqW (JE (Z; V ) ^ AN ) ! colimWUN ssKqW (JJ(ZN ; W ) ^ AN )
V N=W
induced by the functor OE, where W UN and V U. It suffices to prove that, for
fixed W , the map
hocolimJE (Z; V ) ! JJ(ZN ; W )
V N=W
4. GEOMETRIC FIXED POINT SPECTRA 81
is a Jhomotopy equivalence. Via II.4.1, Definition 4.1 leads to explicit descr*
*iptions
of the relevant Jspaces. Write Z = ZN Z0 and V = W V 0, where V N = W .
The space JE (Z; V ) is the Nfixed point space of the Thom complex of a certain
Gbundle. It can be identified as the Thom complex of an Nfixed point Jbundle
over the base Jspace
I (Z; V )N ~=I (ZN ; W ) x I (Z0; V 0)N :
This bundle is just the product of the Jspace I (Z0; V 0)N with the Jbundle o*
*ver
I (ZN ; W ) whose Thom complex is the Jspace JJ(ZN ; W ). Using this, we see
that the map
JE (Z; V ) ! JJ(ZN ; W )
can be identified with the projection
JJ(ZN ; W ) ^ I (Z0; V 0)N+! JJ(ZN ; W ):
Thus it suffices to prove that the space hocolimI (Z0; V 0)N is Jcontractible.*
* This
is standard. The maps of the colimit system are hcofibrations of Jspaces, and
colimI (Z0; V 0)N ~=I (Z0; colimV 0)N
is the Jspace of Nlinear isometries Z0 ! colimV 0. It is Jcontractible by_t*
*he
proof of [17, II.1.5]. __
Corollary 4.13. If f : X ! Y is a ss*isomorphism of orthogonal Gspectra,
then f is a ae*isomorphism.
Proof. If X and Y are qcofibrant, this is immediate from Propositions 4.5
and 4.12. Since a level weak equivalence is a ae*isomorphism, the general cas*
*e __
follows by use of cofibrant approximation in the level model structure. *
* __
To see that the geometric fixed point functor bears the same homotopical rel*
*a
tionship to the categorical fixed point functor as in the classical case [17, I*
*Ix3], we
need the following notations and lemmas.
Notations 4.14. Let F = F [N] be the family of subgroups of G that do
not contain N; when N = G, this is the family of proper subgroups of G. let
EF be the universal F space, and let E"F be the cofiber of the quotient map
EF+ ! S0 that collapses EF to the nonbasepoint. Then (E"F )H = S0 if
H N and (E"F )H is contractible if H 2 F . The map S0 ! "EF induces a
natural map : X ! X ^ "EF of orthogonal Gspectra.
Although trivial to prove, the following lemma is surprisingly precise.
Lemma 4.15. For orthogonal Gspectra X, the map
N : N X ! N (X ^ "EF )
is a natural isomorphism of orthogonal Jspectra.
Proof. For Gspaces A, FixN(X ^ A) ~=(FixN X) ^ AN . Since (E"F )N = S0,_
the conclusion follows. __
Lemma 4.16. Let K = H=N, where N H. For cofibrant orthogonal G
spectra X, the map ! : ssH*(X ^ "EF ) ! aeK*(X ^ "EF ) is an isomorphism.
82 V. "CHANGE" FUNCTORS FOR ORTHOGONAL GSPECTRA
Proof. For based GCW complexes A and B, the inclusion AN  ! A and
the map : B ! B ^ "EF induce bijections
[A; B ^ "EF ]G ! [AN ; B ^ "EF ]G  [AN ; B]G
(e.g. [17, II.9.3]). Here [AN ; B]G ~= [AN ; BN ]J and BN ~=(B ^ "EF )N . T*
*he
composite isomorphism
[A; B ^ "EF ]G ! [AN ; BN ]J
sends a Gmap f to the Jmap fN . Using that G=H ~= J=K is Nfixed, this
specializes to show that, for q 0,
N N
! : colimVssHqV (X(V ) ^ "EF ) ! colimVssKqV (X(V ) ^ "EF )
is a colimit of isomorphisms. The argument for q < 0 is similar. *
*___
The following analogue of [17, II.9.8] gives an isomorphism in the homotopy
category HoJS between the geometric Nfixed point functor and the composite
of the categorical Nfixed point functor with the smash product with E"F . Let
: X ! RX be a fibrant replacement functor on orthogonal Gspectra, so that
is an acyclic cofibration and RX is an orthogonal Gspectrum.
Proposition 4.17. For cofibrant orthogonal Gspectra X, the diagram
fl N() N
R(X ^ "EF )N_____//N R(X ^ "EFo)o__ (X)
displays a pair of natural ss*isomorphisms of orthogonal Jspectra.
Proof. Since N is an isomorphism by Lemma 4.15 and N is an acyclic
cofibration by Proposition 4.5, we need only consider fl. Let K = H=N and consi*
*der
the diagram
ssH*(X ^ "EF_)!_//aeK*(X ^ "EF )
*  *
fflffl fflffl
ssK*R(X ^ "EF )N~=ssH*R(X ^ "EF_)!//_aeK*R(X ^ "EFj)*//_ssK*N R(X ^ "EF ):
The maps * are isomorphisms since is an acyclic cofibration. The top map !
is an isomorphism by Lemma 4.16, hence the bottom map ! is an isomorphism.
Since R(X ^ "EF ) is cofibrant, j* is an isomorphism by Proposition 4.12. Since*
*_the_
bottom composite is fl*, by Lemma 4.11, this proves the result. *
*__
CHAPTER VI
"Change" functors for SG modules and
comparisons
We explain the analogues for Gspectra and SG modules of the functors on
orthogonal Gspectra that we discussed in Chapter V. It turns out that passage *
*from
the definitions for Gspectra in [17, IIxx14] to definitions for SG modules i*
*s not at
all automatic. We show further that the comparisons among Gspectra, orthogonal
Gspectra, and SG modules respect the change of universe, change of group, fix*
*ed
point spectra, and orbit spectra functors. Technically, these comparisons are t*
*he
heart of our work. They imply that such fundamental homotopical results as the
Wirthm"uller isomorphism and the Adams isomorphism, which are proven for G
spectra in [17], apply verbatim to orthogonal Gspectra and SG modules.
1. Comparisons of change of group functors
Let : H ! G be an inclusion of a closed subgroup in G and write * for
functors that assign Haction to an object with Gaction. We fix a Guniverse U
throughout this section, and we have the Huniverse *U. We may regard Gspectra
indexed on U as Hspectra indexed on *U, thus obtaining a forgetful functor
* : GS ! HS
(where we omit the implicit fixed choice of universes from the notation). This
functor has a left and a right adjoint [17, II.4.1]. Because of the action of t*
*he groups
on the universes, these functors are given by suitably twisted halfsmash produ*
*ct
and function spectra functors, which were denoted by G nH () and F [H; )
in [17]. We change notation and call these functors G+ ^H () and FH (G+ ; )
here. This is consistent with the usual notation for these functors on the spac*
*e level
and with the notation we have used for these functors on the orthogonal spectrum
level. Write SH for the sphere Hspectrum indexed on *U; it may be identified
with *SG . The corresponding functors relating SG modules and SH modules have
not yet been defined in the literature. We first show that the functors relati*
*ng
Gspectra and Hspectra induce corresponding functors relating SG modules and
SH modules, and we then compare these change of group functors on the categori*
*es
of Gspectra, SG modules, and orthogonal Gspectra.
The monad L used in the definition of SG modules is given by the twisted
halfsmash product I (U; U) n () on Gspectra, and it has an adjoint comonad
L# given by the twisted function spectrum functor F [I (U; U); ) [5, Ix4]. The
category SG [L] of Lspectra is defined to be the category of Lalgebras, and it
can be identified with the category of L# coalgebras [5, I.4.3]. Recall that a*
*n SG 
module is an Lspectrum M whose unit map : JM ! M is an isomorphism.
By an abuse of notation, we let *L denote the monad I (*U; *U) n () on H
spectra and let *L# denote its adjoint comonad F [I (*U; *U); ). We continue
83
84 VI. "CHANGE" FUNCTORS FOR SGMODULES AND COMPARISONS
to write J for its *Lspectrum version SH ^L (), so that an SH module N is an
*Lspectrum whose unit map : JN ! N is an isomorphism.
Proposition 1.1. The functor * : GS ! HS and its left and right adjoints
G+ ^H () and FH (G+ ; ) induce a functor * : GM ! HM and its left and
right adjoints G+ ^H () and FH (G+ ; ).
Proof. By [17, VI.1.8], we have commutation isomorphisms relating the mon
ads L and *L to the functors *, G+ ^H (), and FH (G+ ; ). That is, for Gspec*
*tra
D and Hspectra E, we have
*(LD) ~=(*L)(*D)
G+ ^H (*LE) ~=L(G+ ^H E)
FH (G+ ; *L# E) ~=L# FH (G+ ; E):
Diagram chases show that these isomorphisms are compatible with the monad and
comonad structures, so that * carries Lalgebras to *Lalgebras, G+ ^H () car
ries *Lalgebras to Lalgebras, and FH (G+ ; ) carries *L# coalgebras to L# 
coalgebras. For Lalgebras M and *Lalgebras N, we have an isomorphism
*JM ~=J*M
under which * agrees with and an isomorphism
G+ ^H (JN) ~=J(G+ ^H N)
under which G+ ^H agrees with . Therefore * carries SG modules to SH modules
and G+ ^H () carries SH modules to SG modules. As is typical in the theory of
SG modules [5, IIx2], the right adjoint to * : GM  ! HM is obtained as the
composite of the functor FH (G+ ; ) from SH modules to L# coalgebras and the
functor J from L# coalgebras to SG modules, the latter being right adjoint to*
*_the
evident forgetful functor by [5, II.2.5]. _*
*_
These functors are compatible with the Quillen equivalence (F; V) relating G
spectra and SG modules described in x1. The following result holds for either *
*the
cellular or the generalized cellular model structures.
Theorem 1.2. Consider the following diagram:
_____G+^H()_____//
HSOO_________________//GS*_oo_OO
 FH(G+;) 
 
V F V F
 
 G ^ () 
fflffl_+_H_______//_fflffl
HM ________________//_GM*:_oo_
FH(G+;)
Each square of left adjoints and each square of right adjoints commutes up to n*
*atural
isomorphism, and the (G+ ^H (); *) and (*; FH (G+ ; ) are pairs of Quillen
adjoints. Therefore the induced adjoint pairs on homotopy categories agree under
the induced adjoint equivalences HoHS ' HoHM and HoGS ' HoGM .
1. COMPARISONS OF CHANGE OF GROUP FUNCTORS 85
Proof. It is clear from the previous proof that F* ~=*F and F(G+ ^H ()) ~=
(G+ ^H ())F, and it follows by adjunction that FH (G+ ; )V ~=VFH (G+ ; ) and
*V ~=V*. It is clear from the definitions that the functors * preserve weak equ*
*iv
alences and qfibrations. As in V.2.2, because orbit spaces *G=K are triangulab*
*le__
as finite HCW complexes [16], the functors * also preserves qcofibrations. *
* __
Turning to the comparison with orthogonal Gspectra, we have the following
precise analogue of the previous theorem. Recall IV.3.8. We assume that our giv*
*en
universe U is closed under tensor products. Then *U is also closed under tensor
products. We use the collections V (U) and *V (U) to define the categories GI S
and HI S of orthogonal Gspectra and orthogonal Hspectra; for H, this entails
a change of indexing spaces isomorphism that is discussed in Vx2.
Theorem 1.3. Consider the following diagram:
____G+^H()_____//_
HSOSO ________________//_GI*S_oo_OO
 FH(G+;) 
 
N#N N#N
 
 G ^ () 
fflffl_+_H________//_fflffl
HM _________________//_GM*:_oo_
FH(G+;)
Each square of left adjoints and each square of right adjoints commutes up to n*
*atural
isomorphism, and the (G+ ^H (); *) and (*; FH (G+ ; )) are pairs of Quillen
adjoints. Therefore the induced adjoint pairs on homotopy categories agree under
the induced adjoint equivalences HoHI S ' HoHM and HoGI S ' HoGM .
Proof. By inspection, *N*(V ) ~=N*(*V ), naturally in V . It follows that
*N ~= N* and N# * ~= *N# . Therefore FH (G+ ; )N# ~=N# FH (G+ ; ) and
N(G+ ^H ()) ~=(G+ ^H ())N. Again, it is clear that the functors * preserve we*
*ak
equivalences and (restricted) qfibrations, and it follows from the triangulabi*
*lity_of
orbits that * preserves (generalized) qcofibrations. *
*__
The Wirthmuller isomorphism explains the homotopical behavior of the func
tors FH (G+ ; Y ). On homotopy categories, there is a natural isomorphism
(1.4) FH (G+ ; Y ) ' G+ ^H L(H)Y;
where L(H) is the tangent Hrepresentation at the identity coset of G=H. This
is proven for Hspectra Y in [17, IIx6], use of HCW spectra being convenient
in the proof. By the results above, it follows for SH modules and for orthogon*
*al
Hspectra. Writing [; ]G for morphisms in homotopy categories, we have
(1.5) [G+ ^H Y; X]G ~=[Y; *X]H
and
(1.6) [*X; Y ]H ~=[X; G+ ^H L(H)Y ]G
for Gobjects X and Hobjects Y .
86 VI. "CHANGE" FUNCTORS FOR SGMODULES AND COMPARISONS
2. Comparisons of change of universe functors
Change of universe functors are studied on SG modules in [6] and [25, XXIVx*
*3].
Let U and U0 be any two Guniverses and let V = V (U) and V 0= V (U0). Using
superscripts to identify categories, we have strong symmetric monoidal equivale*
*nces
of categories 0 0
IVV : IGVS ! IGVS
and 0 0
IUU : MGU ! MGU:
Since these functors enjoy virtually identical formal properties, it is to be e*
*xpected
that the adjoint pairs (N; N# ) connecting their sources and targets commute wi*
*th
them. Unfortunately, this expectation is overoptimistic, and the precise compa*
*ri
son is one of the most subtle aspects of the entire theory. In fact, despite th*
*e formal
similarity, we have no direct comparison between these change of universe funct*
*ors
in general.
We focus on the special case that is relevent to the applications. We consid*
*er
an inclusion of universes i : U ! U0, so that V V 0. We write i* for the forg*
*etful
functor IVV.0It is specified by
0
(i*X)(V ) = X(V ) forX 2 IGUS and V 2 V :
We have similar forgetful functors
0 U * U0 U
i* : PUG ! PG and i : SG ! SG
specified by 0
(i*T )(V ) = T (iV ) forT 2 PUG and V U:
It is clear that the following diagram commutes up to natural isomorphism:
SGU0__`__//PU0GoUo_IGV 0
i* i* i*
fflffl fflffl fflffl
SGU __`__//_PUGoUo_IGV:
Since the forgetful functors U and ` are the right adjoints of Quillen equivale*
*nces,
this implies the compatibility of the three functors i* under the induced equiv*
*alences
of homotopy categories. 0
The0analogue IUU0of i* on MGU0is not induced by restriction of the functor i*
on SGU. Its inverse functor IUU is compared homotopically with the left adjoint*
* i*
of i* in [6] and the comparison suggests adjointly that the functor IUU0is comp*
*atible
with the functors i* in the diagram above after passage to homotopy categories.
However, it is awkward to construct this derived functor in a way that allows a
direct comparison between IUU0and the functors i*, and we need such a comparison
in our study of fixed point functors in the next section.0
We solve this0problem by showing that i* : SGU ! SGUinduces0a new forgetful
functor i* : MGU ! MGU that has a left adjoint i* :0MGU ! MGU and giving a
direct comparison between the new functor i* and IUU. To do this, we require th*
*at
the universe U0 be the direct sum of U and the orthogonal complement U? of U
in U0. This holds in all cases of interest. The following result holds for eith*
*er the
cellular or the generalized cellular model structures.
2. COMPARISONS OF CHANGE OF UNIVERSE FUNCTORS 87
Theorem 2.1. Let U0 = U U? .0Then there is an adjoint pair0of Gfunctors
with left adjoint i* : MGU ! MGU and right adjoint i* : MGU ! MGU. On
passage to Gfixed categories, the functors i* and i* give a Quillen adjoint pa*
*ir
such that the following diagrams of left and of right adjoints commute up to na*
*tural
isomorphism:
_____i*____// 0
GSOUOoo___i*____GSOUO
 
 
V F VF
 
 
fflffl_i*___//_fflffl0
GM U oo___i*____GM U:
The adjoint pair (i*; i*) relating MGUand MGU0is defined in0terms of an adjo*
*int
pair (i*; i*) relating the respective categories SUG[L] and SUG[L] of Lspectra*
*.0Here
the right adjoint i* is defined as the restriction of the functor i* : SGU ! S*
*GU of
the spectrum level adjoint pair (i*; i*) discussed in [17, II.1.3]. We first sh*
*ow that
this restriction makes sense.
Lemma 2.2. The functor i* : SGU0! SGUcarries0LU0algebras to LU algebras
and thus restricts to a functor i* : SGU[L] ! SGU[L].
Proof. Define a map of monoids
ff : I (U; U) ! I (U0; U0)
by ff(f) = f idU?. By the theory0of twisted halfsmash products [5, App], there
results a0monad ff n () on SGU and an induced map fffrom this monad to the
monad LU = I (U0; U0)n(). The following diagram of spaces commutes trivially:
I (U; U)_______ff________//I (U0; U0)
LLL qqq
LLL qqq
I(id;i)&&LLLLxxqI(i;id)qqq
I (U; U0):
By [5, A.6.2], this implies an isomorphism of functors i*LU ~=ffni*(). Composi*
*ng
with ffthere results a natural transformation
0
fi : i*LU ! LU i*
of functors SGU ! SGU0. Taking adjoints, we obtain a natural transformation
"fi: i*L#U0! L#Ui*
of functors SGU0 ! SGU. If M0 is an LU0algebra and thus an L#U0coalgebra,
say with structure map 0 : M0 ! L#U0M0, then i*M0 is an L#Ucoalgebra with
structure map = "fiO i*0. This proves the result. __*
*_
Definition 2.3.We define the left adjoint i* : SGU[L] ! SGU0[L] of i*. For
an LU spectrum M with action : LU M ! M, let i*M be the coequalizer of
LU0i* U0
LU0i*LUM_____//L i*M
88 VI. "CHANGE" FUNCTORS FOR SGMODULES AND COMPARISONS
and the composite
LU0fi U0 U0 0 U0
LU0i*LUM _____//L L i*M___//L i*M;
where 00is0the product0of the monad LU0. Now easy diagram chases show that
0 : LU LU i*M ! LU i*M passes to coequalizers to define an action of L0 on
i*M and that the resulting functor i* is left adjoint to i*.
To define the functors i* and i* relating MGUand MGU0, we need the following
lemma. Since the proof is technical and distracting, we defer it to x6.
Lemma 2.4. For any LU spectrum E, the unit maps
i*JEoo___Ji*JE_Ji*_//Ji*E
are isomorphisms.
This has the following immediate consequence.
Lemma 2.5. The functor i*: SGU[L] ! SGU0[L] carries0SUGmodules to SU0G
modules and thus restricts to a functor i*: MGU ! MGU.
Formally, i*U = Ui*, where U : MGU ! SGU[L] is the forgetful functor.
Definition 2.6.Define i*: MGU0! MGUto be Ji*U. Since U is an embedding
of a full subcategory and J is the right adjoint of U [5, II.1.3], we see that *
*i* is the
right adjoint of i*.
Proof of Theorem 2.1. We now pass to fixed point categories. Lemma 2.4
implies the commutativity of the diagrams of left and right adjoints displayed *
*in the
theorem. The adjoint pair (i*; i*) on Gspectra is a Quillen pair since the fun*
*ctor i*
preserve weak equivalences and qfibrations. It follows that the adjoint pair_(*
*i*;_i*)
on SG modules is also a Quillen adjoint pair. __
The strong symmetric monoidal analogue IU0Uof i* defined in [6] is the funct*
*or
appropriate to the study of highly structured ring and module spectra, and we
have the following comparison. There is a class ESUGof SUGmodules that contains
all of the cofibrant objects in all of our categories of highly structured ring*
* and
module spectra over cofibrant commutative SG algebras and enjoys especially go*
*od
homotopy theoretic properties. It was defined equivariantly [6, 3.1], following*
* [5,
VII.6.4], but it is best to reinterpret ESUGto mean the equivariant analogue of*
* the
larger class defined in [2, 9.3], which enjoys the same good properties.
Proposition 2.7. There is a natural map ff : i*M ! IU0UM of SU0Gmodules
that is a weak equivalence for all M 2 ESUG. Moreover, both maps in the diagram
i*M ^ i*N ff^ff//_IU0UM ^ IU0UN ~=IU0U(Mo^fN)i*(Mf^oN)_
are weak equivalences for all M; N 2 ESUG. Therefore the induced functor i* :
HoGM U ! HoGM U0is strong symmetric monoidal.
Proof. The definition of i* on Smodules M can be written concisely as
(2.8) i*M = I (U0; U0) nI (U;U)M;
2. COMPARISONS OF CHANGE OF UNIVERSE FUNCTORS 89
where the twisted halfsmash product is defined with respect to the map
I (i; id) : I (U0; U0) ! I (U; U0):
According to [6, 2.3], the functor IU0U: MGU ! MGU0is defined by
0 0
(2.9) IUUM = I (U; U ) nI (U;U)M:
This makes sense because0the corresponding functor S U[L] ! S U0[L] carries
SUGmodules0to SUGmodules [6, 2.4]. Therefore I (i; id) induces a natural map *
*of
SUGmodules
0
(2.10) ff : i*M ! IUUM:
We have an evident commutative diagram of underlying Gspectra
I (U0; U0) n M____//_I (U0; U0) nI (U;U)M
 
 ff
fflffl fflffl
I (U; U0) n M______//I (U; U0) nI (U;U)M
in which the horizontal arrows are quotient maps. The left vertical arrow is a
homotopy equivalence of Gspectra for all "tame" Gspectra M, in particular for
all SG modules M 2 ESUG, by [5, I.2.5]. The bottom horizontal arrow is proven
to be a weak equivalence for all M 2 ESUGin [6, p.148], and essentially the same
argument shows that the top horizontal arrow and thus also ff is a weak equival*
*ence.
The proof that ff ^ ff is a weak equivalence when applied to SUGmodules in ESU*
*Gis
similar; compare [5, VIIx6] or [9, x9]. __*
*_
We now assume that our universes U and U0 = U U? are closed under tensor
products and consider the resulting Quillen equivalences (N; N# ). We have a cl*
*ass
EIG of orthogonal Gspectra parallel to the class ESG of SG modules. Precisely*
*, let
EIG be the subclass of objects of IG S consisting of SG together with the ortho*
*go
nal Gspectra of the form Xj=, where X is a positive cell orthogonal Gspectrum
and is a subgroup of j. Let EIG be the smallest class of orthogonal Gspectra
that contains EIG and is closed under wedges, pushouts along hcofibrations, se
quential colimits of hcofibrations, finite smash products, and homotopy equiva
lences. It is clear from the arguments in IIIxx5, 7, 8 that the class EIG conta*
*ins all
of the positive cofibrant objects in all of our categories of highly structured*
* ring and
module spectra over cofibrant commutative orthogonal ring spectra. The following
analogue of Theorem 2.1 is technically parallel to Proposition 2.7.
Theorem 2.11. Write i* = IVV0: IGV ! IGV 0and i* = IVV 0: IGV 0! IGV.
There are natural maps of SU0Gmodules
i*NX __ff//_IU0UNXfioNi*Xo_
that are weak equivalences for all X in EIG. Passing to homotopy categories, the
left and right adjoints in the following diagram commute up to natural isomorph*
*ism.
90 VI. "CHANGE" FUNCTORS FOR SGMODULES AND COMPARISONS
_____i*____// 0
HoGIOVO oo___i*____HoGIOVO
 
 
N#N N#N
 
 
fflffl__i*___//_ fflffl0
HoGM U oo___i*____HoGM U:
Proof. Since N takes positive cell orthogonal Gspectra to cell SG modules
and is strong symmetric monoidal and a topological left adjoint, N carries the *
*class
EIG into the class ESG. The map ff is given by Proposition 2.7, and that result
gives that ff is a weak equivalence on objects in EIG.
We construct fi by use of0observations on "right exact" functors from Ix2.
Recall that i*FVVS0 ~=FVVS0 (V.1.4) and that the functors N are defined in term*
*s0
of the functors0N* of (IV.3.4). We have the contravariant functors N*U0i*, IUUN*
**U:
JGV ! MGU. The first satisfies
0 0
(2.12) NU0i*(FV S0) = N*U0i*(V ) = JI (V U0; U0) n VVU S :
By [6, 2.2], composition induces a homeomorphism of Gspaces
fl : I (U; U0) xI (U;U)I (V U; U) ! I (V U; U0):
Via (2.9), this implies that the second satisfies
0 0 U0 * 0 V U 0
(2.13) IUUNU (FV S ) = IU NU (V ) ~=JI (V U; U ) n V S :
The functors NU0i* and IU0UNU are left adjoints and therefore right exact. Sin*
*ce
their0composites with the functor that sends V 2 JGV to FV S0 are NU0i* and
IUUNU , we obtain natural isomorphisms
0 U0 *
NU0i*X = NU0i* JGV X and IUUNU X = IU NU JGV X
For0V 2 V , let i(V ) = idi : V U ! V U0. The functors i(V )*VVU and
VVU are isomorphic since both are left adjoint to the V th space functor. By [*
*5,
A.6.2], it follows that the map
I (i(V ); id) : I (V U0; U0) ! I (V U; U0)
induces a natural map
0 *
fi* : N*U0i*(V ) ! IUUNU (V ):
In turn, fi* induces the required natural map fi : Ni*X ! IU0UNX. When X =
FV A, where V G 6= 0, we see by use of (2.12), (2.13), and the "untwisting theo*
*rem"
of [5, A.5.5] that fi is obtained by applying J to a map of Lspectra whose und*
*erlying
map of Gspectra is isomorphic to the map
0 0 U0
I (i(V ); id)+ ^ id: I (V U0; U0)+ ^ UVA ! I (V U; U )+ ^ V A:
Since I (V U0; U0) and I (V U; U0) are Gcontractible Gspaces [5, XI.1.5], t*
*his
map is a homotopy equivalence of Gspectra, hence fi is a weak equivalence. By
passage to wedges, pushouts, sequential colimits, and retracts, it follows that*
* fi is
a weak equivalence when X is positive cofibrant. By III.2.6, finite smash produ*
*cts
of positive cofibrant orthogonal Gspectra are positive cofibrant. We pass to o*
*rbit
3. COMPARISONS OF FIXED POINT AND ORBIT GSPECTRA FUNCTORS 91
orthogonal Gspectra Xj= by use of III.8.4, and we pass to general orthogonal
Gspectra X 2 EIG by an analysis of their structure that is similar to, but sim*
*pler_
than, the analogous analysis in the category of Smodules given in [5, VIIx6]. *
* __
Remark 2.14. On homotopy categories HoGS , HoGM , and HoGI S , the
functors i* are all strong symmetric monoidal. For HoGS this is implicit in [17,
II.3.14], for HoGM this is part of Proposition 2.7, and for HoGI S the functor
i* on GI S is already strong symmetric monoidal. Diagram chases show that the
isomorphisms i*X ^ i*Y ! i*(X ^ Y ) in these three settings agree under the
equivalences induced by functors F and N.
3. Comparisons of fixed point and orbit Gspectra functors
First assume given a trivial Guniverse. We denote it UG and let V G = V (UG*
* ),
so that V G is just the collection of all finite dimensional inner product spac*
*es,
with trivial action by G. For Gspectra and SG modules indexed on UG and for
orthogonal Gspectra indexed on V G, we pass to fixed points and orbits levelwi*
*se.
That is, in all three settings, for an object X,
(3.1) XG (V ) = X(V )G and (X=G)(V ) = X(V )=G:
In the case of SG modules, since G acts trivially on I (UG ; UG ), it is clear*
* that
XG and X=G inherit Smodule structures from the SG module structure on X.
Formally, for Gspectra X and Lspectra Y , we have isomorphisms of functors
(LX)G ~=L(XG ); (LX)=G ~=L(X=G); (JY )G ~=J(Y G); (JY )=G ~=J(Y=G):
Write "* for functors that assign trivial Gaction to nonequivariant objects. *
*For
SG modules X and Smodules Y , we then have adjunctions
(3.2) GM ("*Y; X) ~=M (Y; XG ) and GM (X; "*Y ) ~=M (X=G; Y ):
Since G acts trivially on UG , the cellular and generalized cellular model stru*
*ctures
coincide on GM , and these are Quillen adjoint pairs since in both cases the ri*
*ght
adjoints are easily seen to preserve weak equivalences and qfibrations. The sa*
*me
holds for Gspectra, and orthogonal Gspectra work similarly by Vx3. Inspections
of definitions give the following elementary comparisons.
Theorem 3.3. Consider the following diagram, in which all spectra are index*
*ed
on a trivial universe:
______()=G_______//
GSOO__________________//S"*_oo_OO
 ()G 
 
VF V F
 
 ()=G 
fflffl_____________/fflffl/
GMOO__________________//M"*_oo_OO
 ()G 
 
NN# N N#
 
 ()=G 
fflffl____________//fflffl_
GI S ________________//_I"S*:_oo_
()G
92 VI. "CHANGE" FUNCTORS FOR SGMODULES AND COMPARISONS
Each square of left adjoints and each square of right adjoints commutes up to n*
*atural
isomorphism, and the ("*; ()G ) and (()=G; "*) are pairs of Quillen adjoints.
Therefore the induced adjoint pairs on homotopy categories agree under the indu*
*ced
adjoint equivalences HoGS ' HoGM ' HoGI S and HoS ' HoM ' HoI S .
Now let U be a complete Guniverse, or any Guniverse closed under tensor
products, and let V = V (U). We have the Gfixed universe UG , and the theory of
the previous section applies to the inclusion i : UG ! U.
Definition 3.4.For Gspectra, SG modules, or orthogonal Gspectra X in
dexed on the universe U (or on V (U)), define XG = (i*X)G .
We do not define orbit Gspectra this way, preferring to regard them as defi*
*ned
only on Gfixed universes. WeGwrite "#G= i*"*Gin all three contexts;Gthat is, *
*we
interpret "# as a functor S U ! GS U , M U ! GM U, or I V ! GI V.
In all three contexts, "# is left adjoint to ()G . For example, for SUGmodule*
*s X
and SUG modules Y ,
G G
(3.5) GM U("# Y; X) ~=M U (Y; X ):
In view of Theorem 2.1, this is a Quillen adjoint. Deleting the universes from *
*the
notation and combining Theorems 2.1, 2.11, and 3.3, we obtain somewhat more
precise information than is given in the statement of the following result.
Theorem 3.6. The fixed point functors
HoGS ! HoS ; HoGM ! HoM ; HoGI S ! HoI S
and their left adjoints "# agree under the equivalences of their domain and tar*
*get
categories induced by Quillen equivalences (F; V) and (N; N# ).
Remark 3.7. In all three contexts, the counit of the adjunction ("# ; ()G *
*) is
a natural Gmap "# XG  ! X. It is the analogue of the inclusion AG ! A for
based Gspaces A. Working in the trivial universe, "* clearly commutes with sma*
*sh
products, and i* commutes with smash products up to natural isomorphism in the
respective homotopy categories by Remark 2.14. For Gspectra, SG modules, or
orthogonal Gspectra X and Y , there results a natural map
"# (XG ^ Y G) ~=("# XG ) ^ ("# Y G) ! X ^ Y:
in the respective homotopy category. Its adjoint is a natural map
XG ^ Y G ! (X ^ Y )G :
These maps agree under the equivalences induced by the pairs (F; V) and (N; N# *
*).
We have similarly compatible natural maps
XG ^ AG ! (X ^ A)G
and
1 (AG ) ! (1 A)G
where 1 on the left and right refer to the universes UG and U. These maps
are not equivalences. They are discussed on Gspectra in [5, 3.14], and we have
nothing new to say about their analogues for SG modules; we could just as well*
* lift
along the equivalence induced by (F; V). The analogues for orthogonal Gspectra
are surprisingly wellbehaved on the pointset level, as explained in Vx3.
3. COMPARISONS OF FIXED POINT AND ORBIT GSPECTRA FUNCTORS 93
If W H = NH=H, we can obtain Hfixed point functors from Gobjects to
W Hobjects by forgetting down to NHobjects, changing to an Htrivial universe,
and taking Hfixed points. The first step is discussed in x1, and we discuss t*
*he
second and third steps more generally. Thus let N be a normal subgroup of G, let
J = G=N, and let " : G ! J be the quotient homomorphism. We are thinking of
the normal subgroup H of NH with quotient group WH.
Fix a complete Guniverse U and consider its Nfixed subuniverse UN . Write
V N = V (UN ). The universe U is the direct sum of UN and its orthogonal comple
ment, which is the sum of all irreducible sub Gspaces V of U such that UN 6= *
*0.
Moreover both universes UN and U are closed under tensor products. Therefore,
using IV.3.8, the results of the previous section apply to the change of univer*
*se
functors associated to the inclusion i : UN ! U. Write "* for any functor that
assigns to Jobjects the same objects regarded as Nfixed Gobjects.
Definition 3.8.Define "# = i*"*; this specifies functors
N U UN U V N V
JS U ! GS ; JM ! GM ; and JI S ! GI S :
For Gobjects X indexed on the Nfixed universe UN (or on V N), define Jobjects
XN indexed on UN by (XN )(V ) = X(V )N . For Gobjects X indexed on the
complete Guniverse U, define XN = (i*X)N . This specifies Nfixed point functo*
*rs
N U UN V V N
GS U ! JS U ; GM  ! JM ; and GI S ! JI S :
In all three cases, this functor ()N is right adjoint to "# , and ("# ; ()N *
*) is a
Quillen adjoint pair. For example, for SG modules X and SJmodules Y ,
N N
(3.9) GM U("# Y; X) ~=JM U (Y; X ):
Theorem 3.6 generalizes to this situation.
Theorem 3.10. The Nfixed point functors
HoGS ! JHoS ; HoGM ! JHoM ; HoGI S ! JHoI S
and their left adjoints "# agree under the equivalences of their domain and tar*
*get
categories induced by Quillen equivalences (F; V) and (N; N# ).
If we start in the Nfixed point universe, we can be more precise. Here we
define orbit Jspectra levelwise, just as we defined fixed point Jspectra.
Definition 3.11.For Gobjects X indexed on the Nfixed universe UN , define
the Jobject X=N indexed on UN by (X=N)(V ) = X(V )=N.
We have the evident adjunctions, and Theorem 3.3 generalizes directly.
Theorem 3.12. Consider the following diagram, in which all spectra are in
dexed on the Nfixed universe UN :
94 VI. "CHANGE" FUNCTORS FOR SGMODULES AND COMPARISONS
_______()=N______//
GSO__________________//JS"*_oo_OOO
 ()N 
 
VF VF
 
 ()=N 
fflffl____________//_fflffl
GMOO _________________//_JM"*_oo_OO
 ()N 
 
NN# NN#
 
 ()=N 
fflffl___________//_fflffl
GI S ________________//_JI"S*:_oo_
()N
Each square of left adjoints and each square of right adjoints commutes up to n*
*atural
isomorphism, and the ("*; ()N ) and (()=N; "*) are pairs of Quillen adjoints.
The induced adjoint pairs on homotopy categories agree under the induced adjoint
equivalences HoGS ' HoGM ' HoGI S and HoJS ' HoJM ' HoJI S .
4. Nfree Gspectra and the Adams isomorphism
Following [17, IIx2], which is clarified by our present model theoretic fram*
*e
work, we relate families to change of universe and use this relation to describ*
*e and
compare Nfree Gspectra, SG modules, and orthogonal Gspectra. This allows us
to transport the Adams isomorphism, which is perhaps the deepest foundational
result in equivariant stable homotopy theory, from Gspectra to SG modules and
orthogonal Gspectra.
Theorem 4.1. Let i : U0 ! U be an inclusion of Guniverses and consider
the family F = F (U; U0) of subgroups H of G such that there exists an Hlinear
isometry U ! U0.
(i)H 2 F if and only if U is Hisomorphic to U0.
(ii)I (U; U0) is a0universal F space.
(iii)i* : HoF S U ! HoF S U is an equivalence of categories.
Proof. Parts (i) and (ii) are [17, II.2.4 and II.2.11].0Part (iii) is0[17, *
*II.2.6].
We give the idea. First, for an F cofibrant X02 GS U and any Y 02 GS U,
i* : [X0; Y 0]F ~=[X0; Y 0]G ! [i*X0; i*Y 0]G ~=[i*X0; i*Y 0]F
is a bijection. To see this, one uses (i) to prove that the unit Y 0! i*i*Y 0o*
*f the
(i*; i*) adjunction is an F equivalence [17, II.1.9].0Second, for an F cell c*
*omplex
X in GS U, there is an F cell complex X02 GS U and an equivalence i*X0 ! X.
In fact, using ordinary rather than generalized cell structures, we can constru*
*ct X
inductively so that its cells are in bijective correspondence with those of X0.*
* Other
choices of X such that i*X is weakly equivalent to X0 are EF+ ^ i*X0 and, using_
(ii), I (U; U0) n X0; see [17, II.2.14]. _*
*_
Now return to the consideration of a normal subgroup N of G with quotient
group J. Let U be a complete Guniverse and let U0 = UN . Using these universes,
5. THE GEOMETRIC FIXED POINT FUNCTOR AND QUOTIENT GROUPS 95
the results of the x2 allow us to transport the conclusion of the previous theo*
*rem
to both SG modules and orthogonal Gspectra.
Definition 4.2.Define F (N) to be the family of subgroups H of G such that
H \ N = e. For Gspectra, SG modules, or orthogonal Gspectra indexed on either
U or UN , an F (N)object is called an Nfree Gobject, and an F (N)cell compl*
*ex
is called an Nfree Gcell complex.
Thus an Nfree Gcell complex is built up out of cells of orbit types G=H su*
*ch
that H \ N = e. This correctly captures the intuition. Note that we are free to
use either the cellular or the generalized cellular interpretation of a Gcell *
*complex
here. The following elementary observation [17, II.2.4] ties things together.
Lemma 4.3. The families F (U; UN ) and F (N) are the same.
Theorem 4.4. For a normal subgroup N, i* : HoF (N)S UN ! HoF (N)S U
is an equivalence of categories, and similarly for SG modules and orthogonal G
spectra.
In either universe, we can identify HoF (N)S with the full subcategory of
Nfree Gspectra in HoGS . The previous result is summarized by the slogan that
"Nfree Gspectra live in the Ntrivial universe". Using Theorem 3.12, it gives
(4.5) [X=N; Y ]J ~=[X; "*Y ]G ~=[i*X; "# Y ]G
for an Nfree Gobject X and any Gobject Y , both indexed on UN . We can ask
about the behavior with the order of variables reversed, and the Adams isomor
phism relating the orbit and fixed point functors gives the answer. On homotopy
categories, there is a natural isomorphism
(4.6) X=N ~=(A i*X)N
for an Nfree Gobject X indexed on UN , where A is the Grepresentation given
by the tangent space of N at e. Use of i* to pass to the complete universe befo*
*re
taking fixed points is essential. This result is proven for Gspectra X in [17,*
* IIx7].
Here the cellular model structure has a considerable advantage over the general*
*ized
cellular model structure, but the conclusion carries over to our other categori*
*es.
Using (3.9), this implies that
(4.7) [Y; X=N]J ~=["# Y; A i*X]G :
5.The geometric fixed point functor and quotient groups
The geometric fixed point functor was defined on Gspectra in [17, IIx9], wh*
*ere
it was shown to commute up to equivalence with smash products and the suspension
spectrum functor; see also [25, XVIxx3,6]. Recall that F [N] denotes the family
of subgroups H such that H does not contain N. Note that F (N) F [N], with
equality only if N = e. For Gspectra X, there is an equivalence
(5.1) N X ' (E"F [N] ^ X)N :
In the case of SG modules, it seems best to define N X = (X ^E"F [N])N , altho*
*ugh
this obscures the simple space level intuition behind the notion. On orthogonal
Gspectra, we have given a natural geometric definition of N and have derived
(5.1) from that definition. Since (5.1) holds in all cases, the N agree under o*
*ur
equivalences between homotopy categories.
96 VI. "CHANGE" FUNCTORS FOR SGMODULES AND COMPARISONS
An important role of the original geometric fixed point functor is its use to
prove an equivalence between the homotopy category of Jspectra indexed on UN
and the homotopy category of Gspectra indexed on U that are concentrated over
N, namely Gspectra X such that ssH*(X) = 0 unless H contains N. Specialization
of Theorems 6.13 and 6.14 gives the following starting point.
Theorem 5.2. A Gspectrum X is concentrated over N if and only if the map
: X  ! E"F [N] ^ X is a weak equivalence. Smashing with E"F [N] defines
an equivalence of categories from HoF 0[N]S to the full subcategory of Gspectra
concentrated over N in HoGS .
Theorem 5.3. There is an adjoint equivalence from HoJS to the full subcat
egory of Gspectra concentrated over N in HoGS .
By the last statement of Theorem 6.13, for a Gspectrum X concentrated over
N and any Jspectrum Y ,
(5.4) * : [E"F [N] ^ "# Y; X]G ! ["# Y; X]G ~=[Y; XN ]J
is an isomorphism. This gives the required adjunction, and its unit and counit
are proven to be equivalences in [17, II.x9]. By the comparisons we have given,
Theorems 5.2 and 5.3 apply verbatim to SG modules and orthogonal Gspectra.
6. Technical results on the unit map : JE ! E
Finally, we return to the proof of Lemma 2.4. The argument relies on the
following lemmas, which are in essence special cases of the fundamental lemmas
of [5] that make the smash product of Smodules associative and unital, namely
"Hopkins' lemma" [5, I.5.4] and "the accidental isomorphism lemma" [5, I.8.1].
Lemma 6.1. Let U, U1, U2 be Guniverses, and let W be a Ginner product
space (either finite or countably infinite dimensional). The Gmap
ffi : I (U W; U1) xI (U;U)I (U2; U) ! I (U2 W; U1)
specified by ffi(g; f) = g O (f idW) is an isomorphism.
Proof. We only need to show that ffi is a nonequivariant isomorphism of spa*
*ces.
Choose (nonequivariant) isometric isomorphisms between U and U1 and U2. Under
these isomorphisms, we can identify ffi with the analogous map
I (U W; U) xI (U;U)I (U; U) ! I (U W; U);
which is clearly an isomorphism. ___
Lemma 6.2. Let U, U0, U1 be Guniverses, and let W be a Ginner product
space (either finite or countably infinite dimensional). The Gmap
I (U U0 W; U1)=(I (U; U) x I (U0; U0)) ! I (U0 W; U1)=I (U0; U0)
induced by restriction of isometries is an isomorphism.
Proof. The previous lemma gives an isomorphism
ffi : I (U0 W; U1) xI (U0;U0)I (U U0; U0) ~=I (U U0 W; U1)
of right (I (U; U) x I (U0; U0))spaces. We claim that
I (U U0; U0)=(I (U; U) x I (U0; U0))
6. TECHNICAL RESULTS ON THE UNIT MAP : JE ! E 97
is the onepoint space. Indeed, it suffices to show this nonequivariantly, and *
*after
choosing an isometric isomorphism U ~=U0, this is just [5, I.8.1]. Therefore, a*
*fter
passing to orbits over I (U; U) x I (U0; U0), ffi induces an isomorphism
I (U0 W; U1)=I (U0; U0) ! I (U U0 W; U1)=(I (U; U) x I (U0; U0))
Its inverse is induced by restriction of isometries. *
*___
The proof of Lemma 2.4 requires a clarification of the definitions of JE and*
* the
unit map : JE ! E given in [5, I.8.3]. We work in a given universe U, using
the linear isometries operad L such that L (j) = I (Uj; U). By definition,
JE = S ^L E = L (2) nL (1)xL (1)SG Z E;
where Z is the external smash product. Here SG ~=L (0) n S0. The structure map
fl : L (2) x L (0) x L (1) ! L (1) of L induces a map
^fl: ^L(1) = L (2) xL (1)xL (1)L (0) x L (1) ! L (1);
which is a Ghomotopy equivalence [5, XI.2.2]. Form the orbit space L (2)=L (1)
with respect to the right action of L (1) on L (2) given by (g; e) ! g O (e i*
*d),
and let L (1) act on the right of L (2)=L (1) by ([g]; f) ! [g O (idf)]. Then *
*^fl
factors as the composite of the homeomorphism
fl: ^L(1) ! L (2)=L (1)
given by fl(g; 0; f) = [g O (idf)] and the map
i2 : L (2)=L (1) ! L (1)
obtained by restricting isometries U U to the second summand U. These maps are
both G and L (1)equivariant. Using [5, I.2.2(ii)], which describes iterated tw*
*isted
halfsmash products, and the isomorphism fl, we obtain an identification
JE ~=L (2)=L (1) nL (1)E:
Under this identification, : JE ! E coincides with the map
i2 nL (1)id: L (2)=L (1) nL (1)E ! L (1) nL (1)E ~=E:
The proof of Lemma 2.4. Recall that we are given universes U0 = U U? .
Using 6.1, the description (2.8) of i*, and [5, A.6.2], we obtain natural isomo*
*rphisms
of i*JE, Ji*JE, and Ji*E with A1nL U(1)E, A2nL U(1)E, and A3nL U(1)E, where
A1, A2, and A3 are the (G n L U(1))spaces over I (U; U0) specified by
A1 = I (U U U? ; U0)=L U(1)
0 U
A2 = I (U0 U U U? ; U0)=(L U(1) x L (1)); and
0
A3 = I (U0 U U? ; U0)=L U(1):
The maps
i*JEoo___Ji*JE_Ji*_//Ji*E
are induced by appropriate restriction maps A1  A2 ! A3. By Lemma 6.2,_
these maps are isomorphisms. __
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Index of Notation
n, 14 IG, 28
U;U0, 17
A ``, 28 IV;V*0, 72, 83
D, 5 i*, 74, 76, 8789
DT , 5 i*, 74, 76, 8789, 95
d*, 5 J, 16, 61
EvV, 32 J , 8
ESU, 88 JG, 31
G J V, 31
EIG, 89 JG, 77
"EF, 67 JEET , 77
"*, 74, 76, 91, 93
"# , 92, 93 K, 44
K+ , 47
FI, 38 KG, 33
FJ, 38 KGV, 33
F+ I, 40
F+ J, 40 L, 11
Fd, 5 L, 15, 57
FV, 12, 32 L# , 83
FixN, 78 L , 15
N , 78, 95 `, 11
F, 12, 16, 57 , 15, 57, 97
F(N), 95 V;W, 43
F[N], 81, 95
OE : JE ! JJ, 77 M, 19, 63
M, 7
GI S , 29 MG, 57
GI T , 28
GI NtrivS , 76 NH, 76
GI V, 28 N,*8, 62
GI trivS , 74 N#, 8, 60
GM, 57 N , 8, 62
GP, 27 N S , 11
GS , 27 : JJ ! JE, 77
GT , 26 U, 15
1 , 33
IU0U, 86 1
IUU0, 89 V , 33
IVV0, 69, 70, 86 P, 6, 33
I , 7 P , 77
I (U; U0), 15 POE, 77
I S , 7 P, 11
IGS , 29 PU , 15
IGT , 28 PUG, 27
IGV, 28 PG, 27
100
INDEX OF NOTATION 101
PG[L], 63
ssK*, 79
ssH*, 40
aeK*, 79
SU , 15
SG, 28, 57
SVG, 28
SI , 8
UV, 15
1 , 33
1V, 12, 33, 57
V , 70
S , 11
S [L], 15
S U, 15
SG, 27
SG[L], 63
TU;U0V;V,017
, 17
T , 5
TG, 26
U, 6, 29, 33
U , 77
UOE, 77
V *, 32
V, 12, 57
V , 28
V (U), 27
V G, 91
WH, 76
Index
acyclic positive qcofibration, 47 Gspectrum, 27
acyclic positive qfibration, 47 Gtopological model category, 36
acyclic qcofibration, 43 Guniverse
acyclic qfibration, 43 complete, 27
Adams isomorphism, 95 trivial, 27
universe, 27
bicomplete, 5, 62 generalized qcofibration, 58
Bousfield Fmodel structure, 65 generalized sphere Gspectrum, 57
Bousfield localization, 65 generalized sphere SGmodule, 58
cellular approximation theorem, 56 hcofibration, 34
cellular model structure, 55 homotopy groups, 12, 40, 58, 79
change of universe functor, 15 homotopy pullback, 45
closed under tensor products, 61
cofamily, 67 I*prefunctor, 4
cofibrant, 8 I*prespectra, 4
Cofibration Hypothesis, 34 IGFSP, 29
compactly generated model category, 34IGspace, 28
complete Guniverse, 27 IGspectrum, 29
continuous Gfunctor, 27 inclusion prespectrum, 14
cotensor, 32 positive, 14
create, 8 indexing Gspace, 27
CW SGmodule, 56 internal function IGspace, 31
Eequivalence, 64 internal smash product, 6, 30
Elocal, 64 Lprespectrum, 19, 63
Elocalization, 64
Emodel structure, 65 Lspectrum, 15, 20, 57, 63
E1 ring spectrum, 4 level acyclic fibration, 38
exact, right, 5, 62 level acyclic qcofibration, 38
external function IGspace, 29 level equivalence, 38
external smash product, 6, 29 levellfibration,e38vel model structure, 37
F0equivalence, 67 linear isometries operad, 15
F0object, 67
Fcofibration, 65 model category
Fequivalence, 64 compactly generated, 34
Fmodel structure, 65 Gtopological, 36
Fobject, 64 model structure
family, 54, 64 Bousfield F, 65
function IGspace cellular, 55
external, 29 E, 65
internal function IGspace F, 65
internal, 31 level, 37
function spectrum, 30 positive level, 39
positive stable, 7, 47
Gequivalence, 64 stable, 43
Gprespectrum, 27 V , 71
102
INDEX 103
monoid axiom, 50 V model structure, 71
Nfree Gcell complex, 95 weak equivalence, 58
N spectrum, 11 Wirthmuller isomorphism, 85
naive orthogonal Gspectrum, 74
natural Gmap, 27
nondegenerately based, 37
Gspectrum, 40
spectrum, 11
operadic smash product, 15
orthogonal Gspectrum, 29
orthogonal Gspectrum, 40
orthogonal ring Gspectrum, 30
orthogonal space, 7
orthogonal spectrum, 8
ss*isomorphism, 41, 79
positive cofibrant, 8
positive inclusion prespectrum, 14
positive level model structure, 39
positive Gspectrum, 47
positive spectrum, 11
positive qfibration, 47
positive stable model structure, 7, 47
prespectrum, 11
prolongation functor, 6
pushoutproduct axiom, 50
qcofibration, 34, 38, 58
qfibration, 34, 43, 58
restricted qfibration, 58
ae*isomorphism, 79
right exact, 5, 62
Smodule, 15
SGalgebra, 59
SGmodule, 57
shift desuspension spectrum, 32
smash product, 6, 30
external, 6, 29
internal, 6, 30
operadic, 15
twisted half, 14
spectrum, 11
sphere Gspectrum, 57
sphere SGmodule, 58
stable equivalence, 48
stable model structure, 43
symmetric spectrum, 10
tame spectrum, 14
tensor, 32
Thom category, 17
topological Gcategory, 26
trivial Guniverse, 27
twisted halfsmash product, 14
unit map, 15, 57
universe, see Guniverse