DUALITY AND PROSPECTRA
J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN
Abstract.Cofiltered diagrams of spectra, also called prospectra, have ar*
*isen in di
verse areas, and to date have been treated in an ad hoc manner. The purpo*
*se of this
paper is to systematically develop a homotopy theory of prospectra and t*
*o study its
relation to the usual homotopy theory of spectra, as a foundation for fut*
*ure appli
cations. The surprising result we find is that our homotopy theory of pro*
*spectra is
Quillen equivalent to the opposite of the homotopy theory of spectra. Thi*
*s provides
a convenient duality theory for all spectra, extending the classical noti*
*on of Spanier
Whitehead duality which works well only for finite spectra. Roughly speak*
*ing, the new
duality functor takes a spectrum to the cofiltered diagram of the Spanier*
*Whitehead
duals of its finite subcomplexes. In the other direction, the duality fun*
*ctor takes a
cofiltered diagram of spectra to the filtered colimit of the SpanierWhit*
*ehead duals of
the spectra in the diagram. We prove the equivalence of homotopy theories*
* by showing
that both are equivalent to the category of indspectra (filtered diagram*
*s of spectra).
To construct our new homotopy theories, we prove a general existence theo*
*rem for
colocalization model structures generalizing known results for cofibrantl*
*y generated
model categories.
1. Introduction
In recent years there have been many areas in which cofiltered diagrams of sp*
*ectra have
naturally arisen as a way to organize homotopical information. For example, see*
* the work
of Cohen, Jones and Segal [6] and Hurtubise [13] on Floer homology theory, Ando*
* and
Morava [1] on formal groups and free loop spaces, and unpublished work of Dwyer*
* and
Rezk and of Arone on Goodwillie calculus. Prospectra are also likely to be the*
* target of
an 'etale realization functor on stable motivic homotopy theory [16].
A cofiltered diagram is called a prospectrum (see Section 3) and in general *
*it contains
more information than its homotopy limit. Thus it is necessary to develop a hom*
*otopy
theory of prospectra, and our goal is to do this systematically and to study i*
*ts relation to
the usual homotopy theory of spectra. We expect that this framework will be use*
*ful for
many applications.
SpanierWhitehead duality was one of the reasons for which the stable homotop*
*y cate
gory was invented [4] [19] [22]. The idea is that there is a contravariant func*
*tor from the
stable homotopy category to itself that induces an equivalence between the homo*
*topy cat
egory of finite spectra and its own opposite. This functor is defined by taking*
* a spectrum
X to the function spectrum F (X, S0), where S0 is the sphere spectrum.
____________
1991 Mathematics Subject Classification. 55P42 (Primary); 55P25, 18G55, 55U35*
*, 55Q55 (Secondary).
Key words and phrases. Spectrum, prospectrum, SpanierWhitehead duality, clo*
*sed model category,
colocalization.
The authors thank the SFB 343 at Universität Bielefeld, Germany. They also t*
*hank Greg Arone
for originally motivating the project and Stefan Schwede for useful conversatio*
*ns. The first author was
supported by an NSERC Research Grant and the second author was supported by an *
*NSF Postdoctoral
Research Fellowship.
1
2 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN
One important property of SpanierWhitehead duality is that the double dual o*
*f an
infinite spectrum is not weakly equivalent to the original spectrum. In some co*
*ntexts, this
is a useful property because it gives a method for producing new interesting sp*
*ectra. On
the other hand, it is sometimes inconvenient that SpanierWhitehead duality doe*
*s not give
an equivalence between the whole stable homotopy category and its opposite.
The same situation arises in linear algebra over a field k. The functional du*
*al induces
an equivalence of the category of finite dimensional kvector spaces with its o*
*wn opposite,
but it does not extend to an equivalence on the whole category of kvector spac*
*es. One
solution is to pass to the category of profinite kvector spaces. In fact, the*
* category of
kvector spaces is equivalent to the opposite of the category of profinite kv*
*ector spaces.
Because of the strong analogy between the stable homotopy category and catego*
*ries of
chain complexes, it is natural to ask whether the same kind of solution works f*
*or spectra.
The main result of this paper is that it does. We define a homotopy theory for *
*the category
of prospectra and show that its opposite is equivalent to the usual homotopy t*
*heory of
spectra. However, the situation is significantly more complicated because of th*
*e intricacies
of homotopy categories.
The most naive approach is to consider proobjects in the homotopy category o*
*f spectra.
The unstable version of this approach appears in [3] and [23]. Homotopy theoris*
*ts have
learned through countless examples that considering diagrams in a homotopy cate*
*gory is
usually the wrong viewpoint. Rather, it is better to consider commutative diagr*
*ams in a
geometric category and then study the homotopy theory of these diagrams. Follow*
*ing this
philosophy, we consider the category of proobjects in a geometric category of *
*spectra and
then equip it with a homotopy theory.
More precisely, we construct a model structure on prospectra in which the we*
*ak equiva
lences are detected by cohomotopy groups. This model structure is contravariant*
*ly Quillen
equivalent to a model structure on the category of indspectra (i.e., the categ*
*ory of fil
tered systems of spectra) in which the weak equivalences are detected by homoto*
*py groups.
The model structure on indspectra is in turn Quillen equivalent to the usual s*
*table model
structure for spectra.
The cofibrant prospectra in our new model structure are easy to describe. Th*
*ey are
the prospectra that are essentially levelwise cofibrant, that is, they are lev*
*elwise cofibrant
up to isomorphism. The fibrant prospectra are only slightly harder to describ*
*e. They
are the strictly fibrant prospectra that are essentially levelwise homotopyfi*
*nite, that is,
the strictly fibrant prospectra (see Section 3.4) that are also levelwise weak*
*ly equivalent
to a finite complex, up to isomorphism. Dually, the fibrant indspectra are the*
* essentially
levelwise fibrant indspectra, and the cofibrant indspectra are the strictly c*
*ofibrant ind
spectra that are essentially levelwise homotopyfinite. The importance of finit*
*e complexes
is no surprise since we are defining homotopy theories that work well with resp*
*ect to
SpanierWhitehead duality.
The description of fibrant prospectra in terms of homotopyfinite spectra al*
*lows us to
compute the total derived functor Rlim of the limit functor from prospectra to*
* spectra.
For a constant prospectrum X, RlimX is the SpanierWhitehead double dual of X.*
* See
Remark 6.9 for more details.
Our chief tool for establishing the appropriate homotopy theories of prospec*
*tra and
indspectra is a general existence theorem for a certain kind of colocalization*
* of model
categories (see Theorem 2.6). This result is a generalization of [10, Thm. 5.1.*
*1] because
cofibrantly generated model structures satisfy our hypotheses, and our proof is*
* very simi
lar. In our application, we begin with the strict model structure on prospectr*
*a in which
DUALITY AND PROSPECTRA 3
the weak equivalences are, up to isomorphism, the levelwise weak equivalences (*
*see Sec
tion 3.4). Then we use mapping spaces into the spheres to determine the colocal*
* weak
equivalences of prospectra. Dually, for indspectra, we start with the strict *
*structure and
then use mapping spaces out of the spheres to determine the colocal weak equiva*
*lences.
In this paper, we need a model for spectra that has a wellbehaved function s*
*pectrum
defined on the geometric category, not just on the homotopy category. We use sy*
*mmetric
spectra [12] for this model. Section 5 reviews the relevant ideas. It is also p*
*ossible to work
entirely in the category of Smodules [9].
In fact, we do not really need the full power of a general function spectrum *
*construction.
Rather, we only need to define function spectra of the form F (X, S0). It has b*
*een suggested
to us that this is probably possible on more naive categories of spectra such a*
*s the one
described in [5], but we have not checked the details.
Our proofs are written in such a way that they can easily be applied to other*
* situations
involving prospectra. For example, if E is any generalized cohomology theory, *
*then we
can define a model structure on the category of prospectra in which the weak e*
*quivalences
are detected by Ecohomology groups. And the proof of our duality result extend*
*s to a
proof that this model category is Quillen equivalent to the opposite of the mod*
*el category
of End(E)module spectra.
We assume that the reader is familiar with the language and basic results of *
*model
categories. The original reference is [21], but we conform to the notations and*
* terminology
of [10]. See also [7] or [11].
1.1. Organization. The paper is organized as follows. We begin with the general*
* exis
tence theorem for Kcolocal model structures. Next we review the theory of pro*
*categories
and indcategories. Then we study colocal model structures on procategories a*
*nd ind
categories in general.
The second part of the paper begins with a review of some details about symme*
*tric
spectra. Next we construct and study the model structures for prospectra and i*
*ndspectra.
Finally, we prove that the various model structures are Quillen equivalent.
2. Colocalizations of Model Structures
In this section, we prove a general theorem about colocalization of model str*
*uctures.
Much of what appears here is very similar to [10, Ch. 5]. One important differe*
*nce is that
we work with model structures that may not be cofibrantly generated.
Start with a right proper model category C, that is, a model category in whic*
*h the
pullback of a weak equivalence along a fibration is always a weak equivalence. *
*We refer
to the cofibrations, weak equivalences, and fibrations of C as underlying cofib*
*rations,
weak equivalences, and fibrations. For convenience, assume that C is simplicial*
* and write
Map(., .) for the simplicial mapping space. In fact, the results of this sectio*
*n carry over to
the nonsimplicial setting, but one must use the technical machinery of homotop*
*y function
complexes [10, Ch. 17].
Let K be a set of objects of C. Since we shall only use the homotopical prope*
*rties of
the objects in K, we may as well assume that each object in K is underlying cof*
*ibrant.
Definition 2.1. A map f : X ! Y in C is a Kcolocal weak equivalence if for each
A in K, the map
Map(A, ^X) ! Map(A, ^Y)
4 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN
is a weak equivalence of simplicial sets, where ^X! ^Yis a fibrant replacement *
*for X ! Y ,
that is, there is a commuting square
X ____//_^X
 
 
fflffl fflffl
Y ____//_^Y
whose rows are underlying fibrant replacements.
The idea is that we detect Kcolocal weak equivalences by considering maps ou*
*t of
objects in K.
Observe that the choice of fibrant replacements for X and Y does not matter; *
*if the
map
Map(A, ^X) ! Map(A, ^Y)
is a weak equivalence for one choice of fibrant replacements, then it is a weak*
* equivalence
for any other choice of fibrant replacements. Also note that underlying weak eq*
*uivalences
are automatically Kcolocal weak equivalences.
Definition 2.2. A map in C is a Kcolocal fibration if it is an underlying fibr*
*ation. A
map in C is a Kcolocal cofibration if it has the left lifting property with re*
*spect to all
Kcolocal acyclic fibrations.
Because there is no difference between underlying fibrations and Kcolocal fi*
*brations,
we use the term "fibrationü nambiguously for maps in either class.
We are defining a right Bousfield localization of the model category C in the*
* sense of [10,
Defn. 3.3.1]. We add more weak equivalences, keep the fibrations unchanged, and*
* define
the cofibrations to be what they must be.
Theorem 2.6 states that under some general hypotheses on C, our definitions a*
*re a
model structure. However, the twooutofthree axiom and the retract axiom are *
*satisfied
in general. This follows from an inspection of the definitions.
The following results basically appear in [10, Ch. 5] with minor obvious chan*
*ges in the
proofs.
Lemma 2.3.
(a)The class of Kcolocal acyclic cofibrations is the same as the class of unde*
*rlying acyclic
cofibrations.
(b)Let A be any object of K. For n 0, the map i : @ [n] A ! [n] A is a Kcol*
*ocal
cofibration.
(c)Let p : X ! Y be a fibration between fibrant objects X and Y . Then p is a K*
*colocal
acyclic fibration if and only if it has the right lifting property with resp*
*ect to every
underlying cofibration @ [n] A ! [n] A in which A belongs to K and n *
*0.
Proof.For part (a), the proof of [10, Lem. 5.3.2] works word for word. For part*
*s (b) and
(c), the proofs of [10, Prop. 5.2.5] and [10, Prop. 5.2.4] also work. Although *
*we do not
have a set of generating acyclic cofibrations at our disposal, we have avoided *
*this necessity
by assuming that the map in part (c) is already a fibration.
In order to prove the rest of the model structure axioms, we must add hypothe*
*ses on
C.
Hypothesis 2.4. Let C be a right proper simplicial model category, and let K be*
* a set
of cofibrant objects in C. Suppose that there exists a regular cardinal ~ with *
*the following
DUALITY AND PROSPECTRA 5
properties. First, each object of K is ~small relative to the underlying cofi*
*brations.
Second, if
X0 ! X1 ! . .!.Xfi! . . .
is a ~sequence of underlying cofibrations and p : colimfiXfi! Y is a map such *
*that the
composition pfi: Xfi! Y is a fibration for each successor ordinal fi, then p is*
* also a
fibration.
See Section 3.3 for a review of the notions of smallness and ~sequences.
The idea is that fibrations are closed under a certain kind of sufficiently l*
*ong sequential
colimit. If C is cofibrantly generated, then we may choose ~ such that the dom*
*ains of
each of the underlying generating acyclic cofibrations as well as the objects o*
*f K are ~
small relative to the underlying cofibrations. Hence cofibrantly generated mode*
*l categories
always satisfy Hypothesis 2.4. However, in our intended application to prospec*
*tra, C is
not cofibrantly generated, but the above hypothesis is still satisfied.
It is not usually a problem to find a single ~ for which each A in K is ~sma*
*ll. As long
as each A is ~Asmall for some ~A, we may choose ~ to be an upper bound for the*
* ordinals
~A.
The following lemma is not a factorization axiom for the Kcolocal model stru*
*cture that
we are constructing. The problem is that Kcolocal acyclic fibrations are not d*
*etected by
the right lifting property with respect to the maps @ [n] A ! [n] A. See *
*[10,
Ex. 5.2.7] for an example of this problem. Using this lemma, the factorization *
*we want
follows from [10, Prop. 5.3.5].
Lemma 2.5. Under Hypothesis 2.4, every map f : X ! Y has a factorization into a
Kcolocal cofibration i : X ! W followed by a fibration p : W ! Y that has the *
*right
lifting property with respect to all maps @ [n] A ! [n] A for A in K.
Proof.We use a variation on the small object argument [10, x 10.5]. Let J0 be t*
*he set of
all squares
@ [n] A____//_X
 
 f
fflffl fflffl
[n] A_____//_Y
for which A belongs to K. Define Z0 to be the pushout
_ !
a a
[n] A X,
J0 J`@ [n] A
0
and let j0 : X ! Z0 and q0 : Z0 ! Y be the obvious maps.
Now factor the map q0 into an underlying acyclic cofibration i0 : Z0 ! W0 fol*
*lowed by
a fibration p0 : W0 ! Y . This finishes the first stage of the factorization.
We build the whole factorization by a transfinite induction of length ~. If f*
*i is a limit
ordinal, then set Wfito be colimff>
 ""
 "
fflffl"
P 0
in which the dotted arrow exists because P ! P 0is a strict acyclic cofibration*
* and Y is
strictly fibrant. Therefore, Y is also a retract of P 0. The class of proobjec*
*ts that belong
to C essentially levelwise is closed under retract [15, Thm. 5.5], so it suffic*
*es to consider
P 0.
For each s, there is a zigzag
Zs As ! Ps ! Ps0
of weak equivalences. Since Zs belongs to C, the assumption on C implies that e*
*ach Ps0
belongs to C.
Remark 3.8. Let C be a proper simplicial model category, and let C be any class*
* of objects
of C that is closed under weak equivalences. Then strict weak equivalences pres*
*erve the
class of objects in proC that belong to C essentially levelwise. The proof of *
*this fact is
basically the same as the proof of Proposition 3.7 but slightly shorter.
4.Colocalizations of ProCategories
In this section, we apply Theorem 2.6 to get a general colocalization result *
*for homotopy
theories of procategories. Let C be a proper simplicial model category. Let *
*K be any
set of fibrant objects in C, and let cK be the set of constant proobjects cA s*
*uch that A
belongs to K.
Definition 4.1. A map in proC is a cofibration if it is an essentially levelwi*
*se cofibration.
These cofibrations are exactly the strict cofibrations of prospectra.
Definition 4.2. A map f : X ! Y in proC is a cKcolocal weak equivalence if it
induces a weak equivalence
Map pro(X~, cA) = colimsMapC(X~s, A) ! colimtMapC(Y~t, A) = Mappro(Y~, cA)
for all A in K, where ~Xand ~Yare strictly cofibrant replacements for X and Y .
Definition 4.3. A map in proC is a cKcolocal fibration if it has the right li*
*fting
property with respect to all cKcolocal acyclic cofibrations.
12 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN
Theorem 4.4. Let C be a proper simplicial model category, and let K be any set *
*of fibrant
objects in C. Definitions 4.1, 4.2, and 4.3 define a left proper simplicial mod*
*el structure
on proC.
Proof.This is an application of a dual version of Theorem 2.6 in which the obje*
*cts of
K are required to be cosmall and cofibrations are preserved by sequential limit*
*s. These
hypotheses are proved in Proposition 3.3 and [15, Cor. 5.4].
We emphasize that it is not necessary that the model category C satisfies Hyp*
*othesis 2.4
in Theorem 4.4. It is important that proC does satisfy this hypothesis. This h*
*appens by
general arguments about procategories, not by using specific properties of C.
Theorem 4.4 could be stated even more generally. There is no need to colocali*
*ze with
respect only to constant proobjects, since every proobject is cosmall for som*
*e cardinal
(see Corollary 3.5).
Theorem 6.5 below is one example of the situation in Theorem 4.4. See [18] fo*
*r other
examples.
We really do need the dual of Theorem 2.6 in order to establish the cKcoloca*
*l model
structure of Theorem 4.4; the dual of [10, Thm. 5.1.1] is not strong enough. Th*
*e problem is
that the strict model structure for proC is not fibrantly generated in general*
*. See [17, x5]
for a proof that the strict model structure for prosimplicial sets is not fibr*
*antly generated.
Remark 4.5. Suppose that C is a stable model category in the sense that the loo*
*ps and
suspension functors are inverse Quillen equivalences of C with itself. Let K be*
* a set of
fibrant objects of C such that for every A in K, A and A~are weakly equivalen*
*t to
elements of K, where ~Ais a cofibrant replacement for A. In other words, K is c*
*losed, up
to homotopy, under suspensions and loops. Then it can be proved that the cKco*
*local
model structure is also stable. We will not need this result, but the model str*
*ucture of
Theorem 6.5 is an example of this situation.
We do not know whether the cKcolocal model structure on proC is always righ*
*t proper,
even though we are always assuming that C is proper. If we assume in addition t*
*hat C is
stable, then we can show that the model structure is right proper.
Proposition 4.6. If C is a stable proper simplicial model category and K is a s*
*et of fibrant
objects of C, then the cKcolocal model structure is right proper.
Proof.Suppose given a pullback square
q
W ____//_Z
g f
fflffl fflffl
X __p_//_Y
in proC in which p is a cKcolocal fibration and f is a cKcolocal weak equiva*
*lence. We
want to show that g is also a cKcolocal weak equivalence.
Let F be the homotopy fibre of p with respect to the strict model structure, *
*which is
also the homotopy fibre of q. Now F~is the homotopy cofibre of both p and q, w*
*here ~F
is a cofibrant replacement for F . We have a diagram
Map (W~, cA)___//_Map(Z~, cA)_//_Map( F~, cA)
  
  
fflffl fflffl fflffl
Map (X~, cA)___//_Map(Y~, cA)_//_Map( F~, cA)
DUALITY AND PROSPECTRA 13
of simplicial sets in which the rows are fibre sequences. Here ~W, ~Z, ~X, and *
*~Yare cofibrant
replacements for W , Z, X, and Y , and A is any object of K. The second and thi*
*rd vertical
maps are weak equivalences, which means that the first vertical map is also.
4.1. Fibrant ProObjects. Later we shall need a more explicit description of cK*
*colocal
fibrant proobjects. This section contains this description. We are still assum*
*ing that C is
a proper simplicial model category and that each A in K is fibrant.
Definition 4.7. The class of Knilpotent objects of C is the smallest class of *
*fibrant
objects such that:
(1)the terminal object of C is Knilpotent;
(2)weak equivalences between fibrant objects preserve Knilpotence;
(3)and if X is Knilpotent, A belongs to K, and X ! A@ [n]is any map, then t*
*he
fibre product X xA@ [n]A [n]is again Knilpotent.
In other words, an object is Knilpotent if and only if it can be built, up t*
*o weak
equivalence, from the terminal object by a finite sequence of base changes of m*
*aps of
the form A [n]! A@ [n]with A in K. The terminology arises from the connection
with nilpotent spaces when C is the category of simplicial sets and K is the co*
*llection of
EilenbergMac Lane spaces. See [18] for details.
In this paper, the only important example occurs with C the category of spect*
*ra and
K the set of spheres. In this specific case, we give in Proposition 6.6 a more*
* concrete
description of Knilpotent spectra.
Lemma 4.8. Let f : X ! Y be any cKcolocal weak equivalence between cofibrant p*
*ro
objects. Then the map Map (f, cZ) : Map(Y, cZ) ! Map(X, cZ) is a weak equivalen*
*ce for
all Knilpotent objects Z of C.
Proof.The map Map (f, cZ) is a weak equivalence (even an isomorphism) when Z = *
**.
Since X and Y are cofibrant and cZ is strictly fibrant, the weak homotopy type*
*s of
Map(Y, cZ) and Map (X, cZ) do not depend on the choice of Z up to weak equivale*
*nce.
It only remains to consider condition (3) of Definition 4.7. Suppose for indu*
*ction that
Z is Knilpotent and that Map(Y, cZ) ! Map(X, cZ) is a weak equivalence. Let Z0*
*be the
fibre product Z xA@ [k]A [k]for some object A in K. Since A [k]! A@ [k]is a fib*
*ration,
this fibre product is actually a homotopy fibre product, which means that Map (*
*Y, cZ0) is
the homotopy fibre product of the top row in the diagram
Map (Y, cZ)____//Map(@ [k] Y, cA)___//Map( [k] Y, cA)
  
  
fflffl fflffl fflffl
Map (X, cZ)____//Map(@ [k] X, cA)__//_Map( [k] X, cA),
and Map (X, cZ0) is the homotopy fibre product of the bottom row. The left vert*
*ical map
is a weak equivalence by the induction assumption, and the other two vertical m*
*aps are
weak equivalences because the cKcolocal model structure is simplicial and beca*
*use f is a
cKcolocal weak equivalence. Thus, the induced map on homotopy fibre products i*
*s also
a weak equivalence.
Proposition 4.9. An object X of proC is cKcolocal fibrant if and only if it i*
*s strictly
fibrant and essentially levelwise Knilpotent.
That X is essentially levelwise Knilpotent means that X is isomorphic to a p*
*roobject
Y such that each Ys is Knilpotent.
14 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN
Proof.First suppose that X is cKcolocal fibrant. Every cKcolocal fibration is*
* a strict
fibration, so X is strictly fibrant. It remains to show that X is essentially l*
*evelwise K
nilpotent. By applying a functorial cofibrant replacement construction levelwis*
*e to X, we
get a map ~X! X which is a levelwise weak equivalence such that ~Xis levelwise *
*cofibrant
(and, in particular, strictly cofibrant). If we can show that ~Xis essentially *
*levelwise K
nilpotent, then we can use Proposition 3.7 to conclude that X is also essential*
*ly levelwise
Knilpotent. In other words, we might as well assume that X is strictly cofibra*
*nt.
Consider the factorization X ! W ! c* of the map X ! c* as described in the d*
*ual
to the proof of Lemma 2.5, so W is cKcolocal fibrant. Note that each cA is !c*
*osmall by
Proposition 3.3. Therefore, we may take ~ to be !, and there are no limit ordin*
*als in the
construction of W .
Since X is strictly cofibrant, the dual of Lemma 2.3(c) tells us that the map*
* X ! W
is a cKcolocal acyclic cofibration. Hence X is a retract of W because X is cK*
*colocal
fibrant. The class of proobjects having any property essentially levelwise is *
*closed under
retracts [15, Thm. 5.5], so it suffices to consider W . But the class of proob*
*jects having
any property essentially levelwise is also closed under cofiltered limits [15, *
*Thm. 5.1], so
it suffices to consider each Wfi.
Assume for induction that the proobject Wfi1is levelwise Knilpotent. We ma*
*y take
a level representation for the diagram
Q [n]
JficA


Q fflffl
Wfi1____//_JficA@ [n],
and we construct Zfiby taking the levelwise fibre product. In fact, it is possi*
*ble to construct
the level representation in such a way that the replacement for Wfi1is a diagr*
*am of
objects that already appeared in the original Wfi1. This means that the new Wf*
*i1is
still levelwise Knilpotent.
The construction of arbitrary products in procategories [14, Prop. 11.1] sho*
*ws that the
map Y Y
cA [n]! cA@ [n]
Jfi Jfi
is levelwise a finite product of maps of the form
A [n]! A@ [n].
It follows immediately that Zfiis levelwise Knilpotent. Now Wfi! Zfiis a leve*
*lwise
weak equivalence, so Wfiis also levelwise Knilpotent. This finishes one implic*
*ation.
Now suppose that X is essentially levelwise Knilpotent and strictly fibrant.*
* We may
assume that each Xs is Knilpotent. Using the lifting property characterization*
* of cK
colocal fibrant proobjects, we must show that the map
f : Map(B, X) ! Map(A, X)
is an acyclic fibration of simplicial sets for every cKcolocal acyclic cofibra*
*tion i : A ! B
in proC. In fact, by [10, Prop. 13.2.1], we may further assume that A and B ar*
*e cofibrant
proobjects (using that the cKcolocal model structure is left proper). We alre*
*ady know
that f is a fibration because of the strict model structure. Since X is strictl*
*y fibrant, f is
weakly equivalent to holimsfs, where fs is the map
fs : Map(B, cXs) ! Map(A, cXs).
DUALITY AND PROSPECTRA 15
Therefore, we need only show that each fs is a weak equivalence. This is true *
*by
Lemma 4.8.
Remark 4.10. In Definition 4.7, one might also require that the class of Knilp*
*otent objects
is closed under retracts. Strangely, this makes no difference in Proposition 4*
*.9. The
statement of that proposition is true exactly as worded, whether or not Knilpo*
*tent objects
are closed under retracts. This surprising phenomenon arises from the surprisin*
*g way in
which retracts interact with essentially levelwise properties of proobjects [1*
*5, Thm. 5.5].
4.2. IndCategories. All the results of this section dualize to indcategories.*
* More specif
ically, let K be a set of cofibrant objects in a proper simplicial model catego*
*ry C. A map
f : X ! Y in indC is a cKcolocal weak equivalence if it induces a weak equiva*
*lence
Map ind(cA, ^X) = colimsMapC(A, ^Xs) ! colimtMapC(A, ^Yt) = Mapind(cA, ^Y)
for all A in K, where ^Xand ^Yare strictly fibrant replacements for X and Y . F*
*ibrations
are just strict fibrations, and cKcolocal cofibrations are defined by a liftin*
*g property.
These definitions give a right proper cKcolocal model structure on indC. If C*
* is stable,
then this model structure is also left proper. An indobject is cKcolocal cofi*
*brant if and
only if it is strictly cofibrant and, up to isomorphism, it is levelwise weakly*
* equivalent
to an object that can be built out of the initial object by a finite sequence o*
*f cofibrant
homotopy pushouts of maps of the form @ [n] A ! [n] A.
5.Preliminaries on Spectra
Now we review some definitions and results about spectra. We work in the cate*
*gory of
symmetric spectra [12]. This category has a proper simplicial cofibrantly gener*
*ated stable
model structure. We take this structure as the "standard" model structure for s*
*pectra.
Whenever we write "spectrum", we always mean "symmetric spectrum". We write Sn *
*for
a fixed cofibrant and fibrant model for the nth sphere spectrum.
The category of symmetric spectra is closed symmetric monoidal. This means th*
*at it
has an associative commutative unital smash product ^ and an internal function *
*object
F (., .) such that F (Z, .) is right adjoint to .^Z. That is, there is a biject*
*ion between maps
X ! F (Z, Y ) and maps X ^ Z ! Y .
We shall use the following model theoretic property of the functor F (., Y ) *
*when Y is
an arbitrary fixed fibrant spectrum [12, Cor. 5.3.9]. Namely, F (., Y ) takes c*
*ofibrations to
fibrations. More precisely, if i : A ! B is a cofibration, then
F (i, Y ) : F (B, Y ) ! F (A, Y )
is a fibration.
The weak equivalences in the category of spectra are defined in [12, Defn. 3.*
*1.3]. The
stable homotopy category is the category obtained by inverting these maps, whic*
*h are
also called stable equivalences. We will not repeat the definition of weak equi*
*valence here,
but using the following definition of homotopy groups we will state an equivale*
*nt condition
below.
Definition 5.1. For any spectrum X, let ßnX be the set [Sn, X] of maps in the s*
*table
homotopy category.
When X is a fibrant spectrum, ßnX is isomorphic to the traditional nth stable*
* homotopy
group colimkßn+kXk. More generally, we can calculate ßnX by considering the tra*
*ditional
nth stable homotopy group of a fibrant replacement for X. Weak equivalences ar*
*e not
defined in terms of homotopy groups because the definition of homotopy groups d*
*epends
16 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN
on the prior existence of the stable homotopy category. Nevertheless, the homot*
*opy groups
do detect stable weak equivalences of symmetric spectra in the sense that a map*
* f : X ! Y
is a stable equivalence if and only if ßnf is an isomorphism for every n 2 Z.
Filtered colimits preserve fibrant spectra since the generating acyclic cofib*
*rations have
compact domains [12, Defn. 3.4.9] (recall that an object C of a category is com*
*pact
if Hom (C, .) commutes with filtered colimits). Since filtered colimits also p*
*reserve the
traditional stable homotopy groups, it follows that
ßn(colimsXs) ~=colims(ßnXs)
for every filtered diagram X of spectra.
6. ß*Model Structure on ProSpectra
In this section we specialize Theorem 4.4 to describe a model structure on th*
*e category
of prospectra that involves cohomotopy. Later we compare the associated homot*
*opy
theory to ordinary stable homotopy theory.
Definition 6.1. A map of prospectra is a cofibration if it is an essentially l*
*evelwise
cofibration.
These cofibrations are identical with strict cofibrations of prospectra.
Recall that the cohomotopy group ßnX of a spectrum X is the group of stable w*
*eak
homotopy classes [X, Sn]. Thus ß* is the cohomology theory represented by the s*
*phere
spectrum S0.
Definition 6.2. A map f : X ! Y of prospectra is a ß*weak equivalence if it i*
*nduces
an isomorphism colimsßnYs ! colimtßnXt for every n 2 Z.
It is important that we are not requiring that ßnX ! ßnY be an indisomorphis*
*m. Non
isomorphic indabelian groups may have isomorphic colimits when one allows infi*
*nitely
generated abelian groups. It is also important that we are not using the groups*
* ßn limsXs
or ßn holimsXs. In general, very little can be said about these groups in term*
*s of the
groups ßnXs.
Proposition 6.3. Let f : X ! Y be a map of prospectra, and let ~f: ~X! ~Ybe a *
*strictly
cofibrant replacement for f. The following conditions are equivalent:
(1)f is a ß*weak equivalence;
(2)Map (f~, cSn) : Map (Y~, cSn) ! Map (X~, cSn) is a weak equivalence of si*
*mplicial
sets for every n 2 Z.
(3)colimtF (Y~t, S0) ! colimsF (X~s, S0) is a weak equivalence of spectra.
Proof.For any cofibrant prospectrum Z and any k 0,
ßkMap (Z, cSn) = ßkcolimtMap(Zt, Sn) = colimt[Zt, Snk].
The case k = 0 tells us that condition (2) implies condition (1).
Now suppose that f is a ß*weak equivalence. From the computation of the prev*
*ious
paragraph, we know that Map (f~, cSn) induces an isomorphism on all homotopy gr*
*oups
at the canonical basepoints. Since Map (f~, cSn) is weakly equivalent to the l*
*oop space
Map (f~, cSn+1), we only need to compute homotopy groups at one basepoint. Th*
*is
shows that condition (1) implies condition (2).
For condition (3), note that ßkcolimtF (X~, S0) is isomorphic to colimtßkXt *
*(and
similarly for ~Y). This shows that condition (3) is equivalent to condition (1).
DUALITY AND PROSPECTRA 17
Definition 6.4. A map of prospectra is a ß*fibration if it has the right lift*
*ing property
with respect to all ß*acyclic cofibrations.
Theorem 6.5. Definitions 6.1, 6.2, and 6.4 define a proper simplicial model str*
*ucture on
the category of prospectra.
We call this the ß*model structure on prospectra.
Proof.Proposition 6.3 tells us that we are discussing a cKcolocalization, wher*
*e K is the
set of spheres. Therefore, Theorem 4.4 gives us everything but right properness*
*. Since the
model category of spectra is stable (see the last paragraph of Section 3), Prop*
*osition 4.6
gives us right properness.
Now we will identify the ß*fibrant prospectra. We say that a spectrum is ho*
*motopy
finite if it is weakly equivalent to a finite complex, i.e. if its image in the*
* stable homotopy
category is in the thick subcategory generated by S0.
Proposition 6.6. Let K be the set of spheres. A spectrum is Knilpotent (see D*
*efini
tion 4.7) if and only if it is fibrant and homotopyfinite.
Proof.First suppose that X is Knilpotent. We work by induction over the number*
* of
pullbacks in the construction of X, noting that the terminal object is homotopy*
*finite and
weak equivalences preserve homotopyfiniteness.
To do the inductive step, assume that X equals Y x(Sk)@ [n](Sk) [n], where Y *
* is
homotopyfinite. We have to show that X is also homotopyfinite. Each of the sp*
*ectra
Y , (Sk)@ [n], and (Sk) [n]is homotopyfinite, and the map (Sk) [n]! (Sk)@ [n]i*
*s a
fibration. Therefore, X is a homotopy fibre product of three homotopyfinite sp*
*ectra, so
X is also homotopyfinite.
Now assume that X is fibrant and homotopyfinite. We have to show that X is *
*K
nilpotent. We induct on the number of cells in X. If X is weakly contractible, *
*then it is
Knilpotent by definition. To do the inductive step, suppose that there is a fi*
*bre sequence
X ! Y ! Sk,
where X has one more cell than Y and Y is Knilpotent. (This is dual to the usu*
*al way
of attaching cells, but produces the same class of finite complexes because of *
*Spanier
Whitehead duality.)
We claim that there is a homotopy pullback square
*______//(Sk) [1]

 
 
fflffl fflffl
Sk ____//_(Sk)@ [1],
where the bottom horizontal map is inclusion into the first factor. See the fol*
*lowing lemma
for the proof. In the diagram
X _____//_*____//(Sk) [1]
 
  
  
fflffl fflffl fflffl
Y ____//_Sk___//(Sk)@ [1],
the left square is also a homotopy pullback square since X is the homotopy fibr*
*e of Y ! Sk.
Thus the composite square is also a homotopy pullback square, which means that *
*X is
Knilpotent.
18 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN
Lemma 6.7. Let f : Sk ! (Sk)@ [1]= Sk x Sk be the inclusion into the first fact*
*or.
Then there is a homotopy pullback square
*______//(Sk) [1]

 
 
fflffl fflffl
Sk __f_//_(Sk)@ [1].
Proof.Consider the pushout diagram
@ [1]+_____// [1]+
 
 
fflffl fflffl
[0]+ _____//_ [1]
of pointed simplicial sets, where the top horizontal arrow is the obvious inclu*
*sion and the
left vertical arrow takes 0 to 0 and takes both 1 and the basepoint to the base*
*point. Note
that [1] is pointed at 1.
If we apply the functor Map ( 1 (), Sk) to this diagram, we obtain a pullbac*
*k square
Map ( 1 [1], Sk)___//(Sk) [1]
 
 
fflffl fflffl
(Sk) [0]________//(Sk)@ [1]
of spectra in which the upper right corner is contractible. This diagram is a h*
*omotopy
pullback diagram because the right vertical map is a fibration.
Proposition 6.8. A prospectrum X is ß*fibrant if and only if it is essentiall*
*y levelwise
homotopyfinite and strictly fibrant.
This means that each Xs has the weak homotopy type of a finite complex. There*
* is
no compatibility requirement for the weak equivalences between the spectra Xs a*
*nd the
finite complexes.
Proof.This follows immediately from Proposition 4.9 together with Proposition 6*
*.6.
Remark 6.9. Using the above description of ß*fibrant objects, it is possible t*
*o describe
explicitly the total derived functor Rlimof the limit functor from prospectra *
*to spectra.
This functor is computed by taking the limit of a ß*fibrant replacement. With *
*the strict
structure on prospectra, Rlim is just the homotopy limit functor. However, wi*
*th the
ß*model structure, Rlimis related to SpanierWhitehead duality.
Let X be a spectrum; we shall calculate Rlim(cX). Take a ß*fibrant replaceme*
*nt ^Xfor
cX. From Proposition 4.9, we know that each spectrum in ^Xis homotopyfinite; t*
*hus, ^Xis
levelwise weakly equivalent to F (F (X^, S0), S0). Therefore, Rlim(cX) is weakl*
*y equivalent
to holimsF (F (X^s, S0), S0), which is equivalent to F (colimsF (X^s, S0), S0).*
* A computa
tion of homotopy groups shows that ßn colimsF (X^s, S0) equals ßnX, so colimsF*
* (X^s, S0)
is weakly equivalent to the ordinary SpanierWhitehead dual F (X, S0) of X. Thu*
*s, we
have shown that Rlim(cX) is equivalent to F (F (X, S0), S0).
A similar analysis shows that if X is an arbitrary prospectrum, then RlimX i*
*s equiv
alent to F (colimsF (Xs, S0), S0), i.e., the SpanierWhitehead dual of the coli*
*mit of the
levelwise SpanierWhitehead dual of X.
DUALITY AND PROSPECTRA 19
7. ß*Model Structure on IndSpectra
Now we proceed to indspectra. All of the following definitions and results a*
*re dual to
analogous results in the previous section. We skip the proofs because they are *
*no different.
Definition 7.1. A map of indspectra is a fibration if it is an essentially lev*
*elwise fibra
tion.
These fibrations are identical with strict fibrations of indspectra.
Definition 7.2. A map of indspectra X ! Y is a ß*weak equivalence if for every
n 2 Z, the map colimsßnXs ! colimtßnYt is an isomorphism.
Proposition 7.3. Let f : X ! Y be a map of indspectra, and let ^f: ^X! ^Ybe a *
*strictly
fibrant replacement for f. The following conditions are equivalent:
(1)f is a ß*weak equivalence;
(2)Map (cSn, ^f) : Map (cSn, ^Y) ! Map (cSn, ^X) is a weak equivalence of si*
*mplicial
sets for every n 2 Z;
(3)colimsXs ! colimtYt is a stable weak equivalence of spectra.
Definition 7.4. A map of indspectra is a ß*cofibration if it has the left lif*
*ting property
with respect to all ß*acyclic fibrations.
Theorem 7.5. Definitions 7.1, 7.2, and 7.4 define a proper simplicial model str*
*ucture on
the category of indspectra.
We call this the ß*model structure on indspectra.
Proof.Everything but left properness is an application of the dual version of T*
*heorem 4.4.
Left properness follows in a manner dual to the proof of Proposition 4.6.
Proposition 7.6. Consider the smallest class of cofibrant spectra such that:
(1)* belongs to the class;
(2)the class is closed under weak equivalences between cofibrant spectra;
(3)and if X belongs`to the class and @ [n] Sk ! X is any map, then the pus*
*hout
[n] Sk @ [n] SkX also belongs to the class.
This class coincides with the class of cofibrant homotopyfinite spectra.
Proposition 7.7. An indspectrum X is ß*cofibrant if and only if it is essenti*
*ally level
wise homotopyfinite and strictly cofibrant.
8.Comparison of Homotopy Theories
This section contains the main results of this paper. Namely, the homotopy ca*
*tegory
of prospectra is the opposite of the ordinary stable homotopy category. First,*
* we study
the homotopy theory of indspectra.
Theorem 8.1. The constant functor c from spectra to indspectra is right adjoin*
*t to the
functor colim. These functors form a Quillen equivalence when considering the ß*
**model
structure on indspectra.
Proof.Let X be an indspectrum, and let Y be a spectrum. By direct calculatio*
*n,
Hom ind(X, cY ) ~=Hom (colimX, Y ). Thus c and colimare adjoint.
The functor c preserves fibrations and weak equivalences. Therefore, c and co*
*limare a
Quillen pair.
By Proposition 7.3, X ! cY is a ß*weak equivalence if and only if colimsXs !*
* Y is
a weak equivalence of spectra. Hence, c and colimform a Quillen equivalence.
20 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN
According to Theorem 8.1, the homotopy category of indspectra is equivalent *
*to the
ordinary stable homotopy category. We suspect that this model structure is not *
*cofibrantly
generated, but we have not been able to prove it.
By Proposition 7.7, every indspectrum X is ß*weakly equivalent to an indsp*
*ectrum
whose objects are finite cell complexes. Therefore, the ordinary stable homotop*
*y category
is equivalent to a homotopy category of ind(finite cell complexes), but this l*
*atter homotopy
category does not arise from a model structure.
The next step is to compare the categories of prospectra and indspectra.
Lemma 8.2. The contravariant functor F (., Y ) from spectra to spectra is its o*
*wn adjoint.
Proof.A map X ! F (Z, Y ) corresponds to a map X ^ Z ! Y . This corresponds to a
map Z ! F (X, Y ) because X ^ Z and Z ^ X are isomorphic.
Let Y be a fixed spectrum. By acting levelwise, the functor F (., Y ) induce*
*s a con
travariant functor from prospectra to indspectra. It also induces a contravar*
*iant functor
from indspectra to prospectra.
Proposition 8.3. Let Y be an arbitrary fixed spectrum. The contravariant func*
*tors
F (., Y ) from prospectra to indspectra and from indspectra to prospectra a*
*re adjoint
in the sense that
Hom pro(X, F (Z, Y )) ~=Hom ind(Z, F (X, Y ))
for every prospectrum X and every indspectrum Z.
Proof.This follows from direct computation and Lemma 8.2.
Proposition 8.4. Let Y be a homotopyfinite fibrant spectrum. Then the contrava*
*riant
functors F (., Y ) from prospectra to indspectra and from indspectra to pro*
*spectra are a
Quillen pair between the ß*model structure on prospectra and the opposite of *
*the ß*model
structure on indspectra.
Proof.We already know that the functors are an adjoint pair by Proposition 8.3.*
* In
order to show that they are a Quillen pair, we must prove that F (., Y ) takes *
*cofibrations
(resp., ß*acyclic cofibrations) of prospectra to fibrations (resp., ß*acycli*
*c fibrations) of
indspectra.
Since Y is fibrant, F (., Y ) takes cofibrations of spectra to fibrations of *
*spectra. There
fore, F (., Y ) takes levelwise cofibrations of prospectra to levelwise fibrat*
*ions of ind
spectra. It follows that F (., Y ) takes essentially levelwise cofibrations of*
* prospectra to
essentially levelwise fibrations of indspectra.
Now let i : A ! B be a ß*acyclic cofibration of prospectra. Fix k 2 Z and l*
*et Z be a
fibrant model for the spectrum kY . Since Z is again homotopyfinite, the cons*
*tant pro
spectrum cZ is ß*fibrant. Therefore, the map Map (B, cZ) ! Map (A, cZ) is an a*
*cyclic
fibration of simplicial sets. In particular, ß0colimsMap (Bs, Z) ! ß0colimtMap(*
*At, Z) is
an isomorphism. This means that ß0colimsF (Bs, Z) ! ß0colimtF (At, Z) is also *
*an
isomorphism (because ß0F (C, D) = ß0Map (C, D) for any spectra C and D and be
cause ß0 commutes with filtered colimits). Since ß0colimsF (Bs, Z) is isomorph*
*ic to
ßkcolimsF (Bs, Y ) (and similarly for A), it follows that colimsF (Bs, Y ) ! co*
*limtF (At, Y )
is a weak equivalence of spectra. Proposition 7.3 tells us that the map F (i, Y*
* ) : F (B, Y ) !
F (A, Y ) is a ß*weak equivalence of indspectra.
Theorem 8.5. The contravariant functors F (., S0) from prospectra to indspect*
*ra and
from indspectra to prospectra form a Quillen equivalence between the ß*model*
* structure
on prospectra and the opposite of the ß*model structure on indspectra.
DUALITY AND PROSPECTRA 21
Proof.We already showed in Proposition 8.4 that the functors are a Quillen pair*
*. Note
that the hypothesis of Proposition 8.4 is satisfied because S0 is homotopyfini*
*te.
Let X be a cofibrant prospectrum, let Z be a cofibrant indspectrum, and let*
* f : X !
F (Z, S0) be a map of prospectra. Our goal is to show that f is a ß*weak equi*
*valence if
and only if the adjoint map Z ! F (X, S0) is a ß*weak equivalence of indspect*
*ra.
By Proposition 6.3, the map f is a ß*weak equivalence if and only if the map
colimsF (F~(Zs, S0), S0) ! colimtF (X, S0)
is a weak equivalence of spectra. Here ~F(C, D) refers to a cofibrant replaceme*
*nt for the
function spectrum F (C, D).
By Proposition 7.7, we may assume that each Zs is homotopyfinite. Since the *
*Spanier
Whitehead double dual of a finite complex is itself, the map Z ! F (F~(Z, S0), *
*S0) is a
levelwise weak equivalence. In particular, the map
colimsZs ! colimsF (F~(Zs, S0), S0)
is a weak equivalence of spectra.
The previous two paragraphs imply that f is a weak equivalence if and only if*
* the
composition
colimsZs ! colimsF (F~(Zs, S0), S0) ! colimtF (X, S0)
is a weak equivalence of spectra. By Proposition 7.3, this last map is a weak e*
*quivalence
if and only if the map Z ! F (X, S0) is a ß*weak equivalence of indspectra.
Corollary 8.6 (Main result). The category of prospectra with its ß*model stru*
*cture is
Quillen equivalent to the opposite of the category of symmetric spectra with it*
*s usual stable
model structure. The equivalence is a composite of two Quillen pairs going in *
*opposite
directions:
prospectra! indspectraop spectraop.
We have indicated the directions of the left adjoints.
22 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN
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Department of Mathematics, University of Western Ontario, London, Ontario, Ca*
*nada
Email address: jdc@uwo.ca
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
Email address: isaksen@math.wayne.edu