DUALITY AND PRO-SPECTRA J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN Abstract. Cofiltered diagrams of spectra, also called pro-spectra, have arisen in di- verse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of pro-spectra and to study its relation to the usual homotopy theory of spectra, as a foundation for future 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. This provides a convenient duality theory for all spectra, extending the classical notion of Spanier- Whitehead duality which works well only for finite spectra. Roughly speaking, 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 functor takes a cofiltered diagram of spectra to the filtered colimit of the Spanier-Whitehead duals of the spectra in the diagram. We prove the equivalence of homotopy theories by showing that both are equivalent to the category of ind-spectra (filtered diagrams of spectra). To construct our new homotopy theories, we prove a general existence theorem for colocalization model structures generalizing known results for cofibrantly generated model categories. 1. Introduction In recent years there have been many areas in which cofiltered diagrams of spectra 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Segal [6] and Hurtubise [13 ] on Floer homology theory, Ando and Morava [1] on formal groups and free loop spaces, and unpublished work of Dwyer and Rezk and of Arone on Goodwillie calculus. Pro-spectra are also likely to be the target of an 'etale realization functor on stable motivic homotopy theory [16 ]. A cofiltered diagram is called a pro-spectrum (see Section 3) and in general it contains more information than its homotopy limit. Thus it is necessary to develop a homotopy theory of pro-spectra, and our goal is to do this systematically and to study its relation to the usual homotopy theory of spectra. We expect that this framework will be useful for many applications. Spanier-Whitehead duality was one of the reasons for which the stable homotopy cate- gory was invented [4] [19 ] [22 ]. The idea is that there is a contravariant functor from the stable homotopy category to itself that induces an equivalence between the homotopy 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, pro-spectrum, Spanier-Whitehead duality, closed model category, colocalization. The authors thank the SFB 343 at Universit"at Bielefeld, Germany. They also thank Greg Arone for originally motivating the project and Stefan Schwede for useful conversations. 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 Spanier-Whitehead duality is that the double dual of an infinite spectrum is not weakly equivalent to the original spectrum. In some contexts, this is a useful property because it gives a method for producing new interesting spectra. On the other hand, it is sometimes inconvenient that Spanier-Whitehead duality does 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 dual induces an equivalence of the category of finite dimensional k-vector spaces with its own opposite, but it does not extend to an equivalence on the whole category of k-vector spaces. One solution is to pass to the category of pro-finite k-vector spaces. In fact, the category of k-vector spaces is equivalent to the opposite of the category of pro-finite k-vector spaces. Because of the strong analogy between the stable homotopy category and categories of chain complexes, it is natural to ask whether the same kind of solution works for spectra. The main result of this paper is that it does. We define a homotopy theory for the category of pro-spectra and show that its opposite is equivalent to the usual homotopy theory of spectra. However, the situation is significantly more complicated because of the intricacies of homotopy categories. The most naive approach is to consider pro-objects in the homotopy category of spectra. The unstable version of this approach appears in [3] and [23 ]. Homotopy theorists have learned through countless examples that considering diagrams in a homotopy category is usually the wrong viewpoint. Rather, it is better to consider commutative diagrams in a geometric category and then study the homotopy theory of these diagrams. Following this philosophy, we consider the category of pro-objects in a geometric category of spectra and then equip it with a homotopy theory. More precisely, we construct a model structure on pro-spectra in which the weak equiva- lences are detected by cohomotopy groups. This model structure is contravariantly Quillen equivalent to a model structure on the category of ind-spectra (i.e., the category of fil- tered systems of spectra) in which the weak equivalences are detected by homotopy groups. The model structure on ind-spectra is in turn Quillen equivalent to the usual stable model structure for spectra. The cofibrant pro-spectra in our new model structure are easy to describe. They are the pro-spectra that are essentially levelwise cofibrant, that is, they are levelwise cofibrant up to isomorphism. The fibrant pro-spectra are only slightly harder to describe. They are the strictly fibrant pro-spectra that are essentially levelwise homotopy-finite, that is, the strictly fibrant pro-spectra (see Section 3.4) that are also levelwise weakly equivalent to a finite complex, up to isomorphism. Dually, the fibrant ind-spectra are the essentially levelwise fibrant ind-spectra, and the cofibrant ind-spectra are the strictly cofibrant ind- spectra that are essentially levelwise homotopy-finite. The importance of finite complexes is no surprise since we are defining homotopy theories that work well with respect to Spanier-Whitehead duality. The description of fibrant pro-spectra in terms of homotopy-finite spectra allows us to compute the total derived functor Rlim of the limit functor from pro-spectra to spectra. For a constant pro-spectrum X, Rlim X is the Spanier-Whitehead double dual of X. See Remark 6.9 for more details. Our chief tool for establishing the appropriate homotopy theories of pro-spectra and ind-spectra 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 pro-spectra in which DUALITY AND PRO-SPECTRA 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 pro-spectra. Dually, for ind-spectra, we start with the strict structure and then use mapping spaces out of the spheres to determine the colocal weak equivalences. In this paper, we need a model for spectra that has a well-behaved function spectrum defined on the geometric category, not just on the homotopy category. We use symmetric spectra [12 ] for this model. Section 5 reviews the relevant ideas. It is also possible to work entirely in the category of S-modules [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 been suggested to us that this is probably possible on more naive categories of spectra such as 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 pro-spectra. For example, if E is any generalized cohomology theory, then we can define a model structure on the category of pro-spectra in which the weak equivalences are detected by E-cohomology groups. And the proof of our duality result extends to a proof that this model category is Quillen equivalent to the opposite of the model 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 K-colocal model structures. Next we review the theory of pro-categories and ind-categories. Then we study colocal model structures on pro-categories and ind- categories in general. The second part of the paper begins with a review of some details about symmetric spectra. Next we construct and study the model structures for pro-spectra and ind-spectra. 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 structures. Much of what appears here is very similar to [10 , Ch. 5]. One important difference 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 which 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 cofibrations, 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 section carry over to the non-simplicial setting, but one must use the technical machinery of homotopy function complexes [10 , Ch. 17]. Let K be a set of objects of C. Since we shall only use the homotopical properties of the objects in K, we may as well assume that each object in K is underlying cofibrant. Definition 2.1. A map f : X ! Y in C is a K-colocal 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 K-colocal weak equivalences by considering maps out 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 equivalences are automatically K-colocal weak equivalences. Definition 2.2. A map in C is a K-colocal fibration if it is an underlying fibration. A map in C is a K-colocal cofibration if it has the left lifting property with respect to all K-colocal acyclic fibrations. Because there is no difference between underlying fibrations and K-colocal fibrations, we use the term "fibration" unambiguously 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 are a model structure. However, the two-out-of-three 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 changes in the proofs. Lemma 2.3. (a) The class of K-colocal acyclic cofibrations is the same as the class of underlying acyclic cofibrations. (b) Let A be any object of K. For n 0, the map i : @ [n] A ! [n] A is a K-colocal 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 respect 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 parts (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 hypotheses 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 PRO-SPECTRA 5 properties. First, each object of K is ~-small relative to the underlying cofibrations. 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 long sequential colimit. If C is cofibrantly generated, then we may choose ~ such that the domains of each of the underlying generating acyclic cofibrations as well as the objects of K are ~- small relative to the underlying cofibrations. Hence cofibrantly generated model categories always satisfy Hypothesis 2.4. However, in our intended application to pro-spectra, 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 ~-small. As long as each A is ~A -small 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 K-colocal model structure that we are constructing. The problem is that K-colocal acyclic fibrations are not detected 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 K-colocal 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 the 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 followed 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 fi 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 pro-objects that belong to C essentially levelwise is closed under retract [15 , Thm. 5.5], so it suffices to consider P 0. For each s, there is a zig-zag Zs As ! Ps ! Ps0 of weak equivalences. Since Zs belongs to C, the assumption on C implies that each 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 preserve the class of objects in pro-C 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 Pro-Categories In this section, we apply Theorem 2.6 to get a general colocalization result for homotopy theories of pro-categories. 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 pro-objects cA such that A belongs to K. Definition 4.1. A map in pro-C is a cofibration if it is an essentially levelwise cofibration. These cofibrations are exactly the strict cofibrations of pro-spectra. Definition 4.2. A map f : X ! Y in pro-C is a cK-colocal weak equivalence if it induces a weak equivalence Map pro(X", cA) = colimsMap C (X"s, A) ! colimtMap C (Y"t, A) = Map pro(Y", cA) for all A in K, where X" and "Y are strictly cofibrant replacements for X and Y . Definition 4.3. A map in pro-C is a cK-colocal fibration if it has the right lifting property with respect to all cK-colocal 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 model structure on pro-C. Proof. This is an application of a dual version of Theorem 2.6 in which the objects of K are required to be cosmall and cofibrations are preserved by sequential limits. 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 Hypothesis 2.4 in Theorem 4.4. It is important that pro-C does satisfy this hypothesis. This happens by general arguments about pro-categories, not by using specific properties of C. Theorem 4.4 could be stated even more generally. There is no need to colocalize with respect only to constant pro-objects, since every pro-object is cosmall for some cardinal (see Corollary 3.5). Theorem 6.5 below is one example of the situation in Theorem 4.4. See [18 ] for other examples. We really do need the dual of Theorem 2.6 in order to establish the cK-colocal model structure of Theorem 4.4; the dual of [10 , Thm. 5.1.1] is not strong enough. The problem is that the strict model structure for pro-C is not fibrantly generated in general. See [17 , x5] for a proof that the strict model structure for pro-simplicial sets is not fibrantly generated. Remark 4.5. Suppose that C is a stable model category in the sense that the loops 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 equivalent to elements of K, where A" is a cofibrant replacement for A. In other words, K is closed, up to homotopy, under suspensions and loops. Then it can be proved that the cK-colocal model structure is also stable. We will not need this result, but the model structure of Theorem 6.5 is an example of this situation. We do not know whether the cK-colocal model structure on pro-C is always right proper, even though we are always assuming that C is proper. If we assume in addition that 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 set of fibrant objects of C, then the cK-colocal model structure is right proper. Proof. Suppose given a pullback square q W ______//Z g|| |f| fflffl| fflffl| X __p___//Y in pro-C in which p is a cK-colocal fibration and f is a cK-colocal weak equivalence. We want to show that g is also a cK-colocal 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, where 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 PRO-SPECTRA 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 third vertical maps are weak equivalences, which means that the first vertical map is also. 4.1. Fibrant Pro-Objects. Later we shall need a more explicit description of cK-colocal fibrant pro-objects. This section contains this description. We are still assuming that C is a proper simplicial model category and that each A in K is fibrant. Definition 4.7. The class of K-nilpotent objects of C is the smallest class of fibrant objects such that: (1) the terminal object of C is K-nilpotent; (2) weak equivalences between fibrant objects preserve K-nilpotence; (3) and if X is K-nilpotent, A belongs to K, and X ! A@ [n] is any map, then the fibre product X xA@ [n] A [n] is again K-nilpotent. In other words, an object is K-nilpotent if and only if it can be built, up to weak equivalence, from the terminal object by a finite sequence of base changes of maps 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 collection of Eilenberg-Mac Lane spaces. See [18 ] for details. In this paper, the only important example occurs with C the category of spectra and K the set of spheres. In this specific case, we give in Proposition 6.6 a more concrete description of K-nilpotent spectra. Lemma 4.8. Let f : X ! Y be any cK-colocal weak equivalence between cofibrant pro- objects. Then the map Map (f, cZ) : Map (Y, cZ) ! Map (X, cZ) is a weak equivalence for all K-nilpotent 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 types of Map (Y, cZ) and Map (X, cZ) do not depend on the choice of Z up to weak equivalence. It only remains to consider condition (3) of Definition 4.7. Suppose for induction that Z is K-nilpotent 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 fibration, 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 vertical map is a weak equivalence by the induction assumption, and the other two vertical maps are weak equivalences because the cK-colocal model structure is simplicial and because f is a cK-colocal weak equivalence. Thus, the induced map on homotopy fibre products is also a weak equivalence. Proposition 4.9. An object X of pro-C is cK-colocal fibrant if and only if it is strictly fibrant and essentially levelwise K-nilpotent. That X is essentially levelwise K-nilpotent means that X is isomorphic to a pro-object Y such that each Ys is K-nilpotent. 14 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN Proof. First suppose that X is cK-colocal fibrant. Every cK-colocal fibration is a strict fibration, so X is strictly fibrant. It remains to show that X is essentially levelwise K- nilpotent. By applying a functorial cofibrant replacement construction levelwise to X, we get a map X" ! X which is a levelwise weak equivalence such that X" is levelwise cofibrant (and, in particular, strictly cofibrant). If we can show that X" is essentially levelwise K- nilpotent, then we can use Proposition 3.7 to conclude that X is also essentially levelwise K-nilpotent. In other words, we might as well assume that X is strictly cofibrant. Consider the factorization X ! W ! c* of the map X ! c* as described in the dual to the proof of Lemma 2.5, so W is cK-colocal fibrant. Note that each cA is !-cosmall by Proposition 3.3. Therefore, we may take ~ to be !, and there are no limit ordinals 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 cK-colocal acyclic cofibration. Hence X is a retract of W because X is cK-colocal fibrant. The class of pro-objects having any property essentially levelwise is closed under retracts [15 , Thm. 5.5], so it suffices to consider W . But the class of pro-objects 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 pro-object Wfi-1 is levelwise K-nilpotent. We may take a level representation for the diagram Q [n] JficA | | Q fflffl| Wfi-1 ______// JficA@ [n], and we construct Zfiby taking the levelwise fibre product. In fact, it is possible to construct the level representation in such a way that the replacement for Wfi-1 is a diagram of objects that already appeared in the original Wfi-1. This means that the new Wfi-1 is still levelwise K-nilpotent. The construction of arbitrary products in pro-categories [14 , Prop. 11.1] shows 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 Zfi is levelwise K-nilpotent. Now Wfi ! Zfi is a levelwise weak equivalence, so Wfiis also levelwise K-nilpotent. This finishes one implication. Now suppose that X is essentially levelwise K-nilpotent and strictly fibrant. We may assume that each Xs is K-nilpotent. Using the lifting property characterization of cK- colocal fibrant pro-objects, we must show that the map f : Map (B, X) ! Map (A, X) is an acyclic fibration of simplicial sets for every cK-colocal acyclic cofibration i : A ! B in pro-C. In fact, by [10 , Prop. 13.2.1], we may further assume that A and B are cofibrant pro-objects (using that the cK-colocal model structure is left proper). We already know that f is a fibration because of the strict model structure. Since X is strictly fibrant, f is weakly equivalent to holim sfs, where fs is the map fs : Map (B, cXs) ! Map (A, cXs). DUALITY AND PRO-SPECTRA 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 K-nilpotent 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 K-nilpotent objects are closed under retracts. This surprising phenomenon arises from the surprising way in which retracts interact with essentially levelwise properties of pro-objects [15 , Thm. 5.5]. 4.2. Ind-Categories. All the results of this section dualize to ind-categories. More specif- ically, let K be a set of cofibrant objects in a proper simplicial model category C. A map f : X ! Y in ind-C is a cK-colocal weak equivalence if it induces a weak equivalence Map ind(cA, ^X) = colimsMap C (A, ^Xs) ! colimtMap C (A, ^Yt) = Map ind(cA, ^Y) for all A in K, where X^ and ^Y are strictly fibrant replacements for X and Y . Fibrations are just strict fibrations, and cK-colocal cofibrations are defined by a lifting property. These definitions give a right proper cK-colocal model structure on ind-C. If C is stable, then this model structure is also left proper. An ind-object is cK-colocal cofibrant 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 of 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 category of symmetric spectra [12 ]. This category has a proper simplicial cofibrantly generated stable model structure. We take this structure as the "standard" model structure for spectra. 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 that 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 bijection 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 cofibrations 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, which are also called stable equivalences. We will not repeat the definition of weak equivalence here, but using the following definition of homotopy groups we will state an equivalent condition below. Definition 5.1. For any spectrum X, let ssn X be the set [Sn , X] of maps in the stable homotopy category. When X is a fibrant spectrum, ssn X is isomorphic to the traditional nth stable homotopy group colimk ssn+k Xk. More generally, we can calculate ssn X by considering the traditional nth stable homotopy group of a fibrant replacement for X. Weak equivalences are not defined in terms of homotopy groups because the definition of homotopy groups depends 16 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN on the prior existence of the stable homotopy category. Nevertheless, the homotopy 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 ssn f is an isomorphism for every n 2 Z. Filtered colimits preserve fibrant spectra since the generating acyclic cofibrations have compact domains [12 , Defn. 3.4.9] (recall that an object C of a category is compact if Hom (C, .) commutes with filtered colimits). Since filtered colimits also preserve the traditional stable homotopy groups, it follows that ssn (colimsXs) ~=colims(ssn Xs) for every filtered diagram X of spectra. 6. ss*-Model Structure on Pro-Spectra In this section we specialize Theorem 4.4 to describe a model structure on the category of pro-spectra that involves cohomotopy. Later we compare the associated homotopy theory to ordinary stable homotopy theory. Definition 6.1. A map of pro-spectra is a cofibration if it is an essentially levelwise cofibration. These cofibrations are identical with strict cofibrations of pro-spectra. Recall that the cohomotopy group ssn X of a spectrum X is the group of stable weak homotopy classes [X, Sn ]. Thus ss* is the cohomology theory represented by the sphere spectrum S0. Definition 6.2. A map f : X ! Y of pro-spectra is a ss* -weak equivalence if it induces an isomorphism colims ssn Ys ! colimt ssn Xt for every n 2 Z. It is important that we are not requiring that ssn X ! ssn Y be an ind-isomorphism. Non- isomorphic ind-abelian groups may have isomorphic colimits when one allows infinitely generated abelian groups. It is also important that we are not using the groups ssn lims Xs or ssn holim sXs. In general, very little can be said about these groups in terms of the groups ssn Xs. Proposition 6.3. Let f : X ! Y be a map of pro-spectra, 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 simplicial sets for every n 2 Z. (3) colimtF (Y"t, S0) ! colims F (X"s, S0) is a weak equivalence of spectra. Proof. For any cofibrant pro-spectrum Z and any k 0, sskMap (Z, cSn ) = ssk colimtMap (Zt, Sn ) = colimt[Zt, Sn-k ]. 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 previous paragraph, we know that Map (f", cSn ) induces an isomorphism on all homotopy groups at the canonical basepoints. Since Map (f", cSn ) is weakly equivalent to the loop space Map (f", cSn+1 ), we only need to compute homotopy groups at one basepoint. This shows that condition (1) implies condition (2). For condition (3), note that ssk colimtF (X", S0) is isomorphic to colim tss-k Xt (and similarly for "Y). This shows that condition (3) is equivalent to condition (1). DUALITY AND PRO-SPECTRA 17 Definition 6.4. A map of pro-spectra is a ss* -fibration if it has the right lifting property with respect to all ss*-acyclic cofibrations. Theorem 6.5. Definitions 6.1, 6.2, and 6.4 define a proper simplicial model structure on the category of pro-spectra. We call this the ss* -model structure on pro-spectra. Proof. Proposition 6.3 tells us that we are discussing a cK-colocalization, where 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), Proposition 4.6 gives us right properness. Now we will identify the ss*-fibrant pro-spectra. We say that a spectrum is homotopy- 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 K-nilpotent (see Defini- tion 4.7) if and only if it is fibrant and homotopy-finite. Proof. First suppose that X is K-nilpotent. 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 homotopy-finiteness. To do the inductive step, assume that X equals Y x(Sk)@ [n](Sk ) [n], where Y is homotopy-finite. We have to show that X is also homotopy-finite. Each of the spectra Y , (Sk )@ [n], and (Sk ) [n] is homotopy-finite, and the map (Sk ) [n] ! (Sk )@ [n] is a fibration. Therefore, X is a homotopy fibre product of three homotopy-finite spectra, so X is also homotopy-finite. Now assume that X is fibrant and homotopy-finite. 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 K-nilpotent by definition. To do the inductive step, suppose that there is a fibre sequence X ! Y ! Sk , where X has one more cell than Y and Y is K-nilpotent. (This is dual to the usual 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 following 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 fibre of Y ! Sk . Thus the composite square is also a homotopy pullback square, which means that X is K-nilpotent. 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 factor. 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 inclusion and the left vertical arrow takes 0 to 0 and takes both 1 and the basepoint to the basepoint. Note that [1] is pointed at 1. If we apply the functor Map ( 1 (-), Sk ) to this diagram, we obtain a pullback 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 homotopy pullback diagram because the right vertical map is a fibration. Proposition 6.8. A pro-spectrum X is ss*-fibrant if and only if it is essentially levelwise homotopy-finite 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 and 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 to describe explicitly the total derived functor Rlim of the limit functor from pro-spectra to spectra. This functor is computed by taking the limit of a ss*-fibrant replacement. With the strict structure on pro-spectra, Rlim is just the homotopy limit functor. However, with the ss*-model structure, Rlim is related to Spanier-Whitehead duality. Let X be a spectrum; we shall calculate Rlim (cX). Take a ss*-fibrant replacement X^ for cX. From Proposition 4.9, we know that each spectrum in ^X is homotopy-finite; thus, X^ is levelwise weakly equivalent to F (F (X^, S0), S0). Therefore, Rlim (cX) is weakly equivalent to holim sF (F (X^s, S0), S0), which is equivalent to F (colim sF (X^s, S0), S0). A computa- tion of homotopy groups shows that ssn colims F (X^s, S0) equals ss-n X, so colims F (X^s, S0) is weakly equivalent to the ordinary Spanier-Whitehead dual F (X, S0) of X. Thus, we have shown that Rlim (cX) is equivalent to F (F (X, S0), S0). A similar analysis shows that if X is an arbitrary pro-spectrum, then Rlim X is equiv- alent to F (colim sF (Xs, S0), S0), i.e., the Spanier-Whitehead dual of the colimit of the levelwise Spanier-Whitehead dual of X. DUALITY AND PRO-SPECTRA 19 7. ss*-Model Structure on Ind-Spectra Now we proceed to ind-spectra. All of the following definitions and results are dual to analogous results in the previous section. We skip the proofs because they are no different. Definition 7.1. A map of ind-spectra is a fibration if it is an essentially levelwise fibra- tion. These fibrations are identical with strict fibrations of ind-spectra. Definition 7.2. A map of ind-spectra X ! Y is a ss*-weak equivalence if for every n 2 Z, the map colims ssn Xs ! colimt ssn Yt is an isomorphism. Proposition 7.3. Let f : X ! Y be a map of ind-spectra, 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 simplicial sets for every n 2 Z; (3) colimsXs ! colimt Yt is a stable weak equivalence of spectra. Definition 7.4. A map of ind-spectra is a ss*-cofibration if it has the left lifting property with respect to all ss*-acyclic fibrations. Theorem 7.5. Definitions 7.1, 7.2, and 7.4 define a proper simplicial model structure on the category of ind-spectra. We call this the ss*-model structure on ind-spectra. Proof. Everything but left properness is an application of the dual version of Theorem 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 pushout [n] Sk @ [n] Sk X also belongs to the class. This class coincides with the class of cofibrant homotopy-finite spectra. Proposition 7.7. An ind-spectrum X is ss*-cofibrant if and only if it is essentially level- wise homotopy-finite and strictly cofibrant. 8. Comparison of Homotopy Theories This section contains the main results of this paper. Namely, the homotopy category of pro-spectra is the opposite of the ordinary stable homotopy category. First, we study the homotopy theory of ind-spectra. Theorem 8.1. The constant functor c from spectra to ind-spectra is right adjoint to the functor colim . These functors form a Quillen equivalence when considering the ss*-model structure on ind-spectra. Proof. Let X be an ind-spectrum, and let Y be a spectrum. By direct calculation, Hom ind(X, cY ) ~=Hom (colim X, Y ). Thus c and colim are adjoint. The functor c preserves fibrations and weak equivalences. Therefore, c and colim are a Quillen pair. By Proposition 7.3, X ! cY is a ss*-weak equivalence if and only if colim sXs ! Y is a weak equivalence of spectra. Hence, c and colim form a Quillen equivalence. 20 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN According to Theorem 8.1, the homotopy category of ind-spectra 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 ind-spectrum X is ss*-weakly equivalent to an ind-spectrum whose objects are finite cell complexes. Therefore, the ordinary stable homotopy category is equivalent to a homotopy category of ind-(finite cell complexes), but this latter homotopy category does not arise from a model structure. The next step is to compare the categories of pro-spectra and ind-spectra. Lemma 8.2. The contravariant functor F (., Y ) from spectra to spectra is its own 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 ) induces a con- travariant functor from pro-spectra to ind-spectra. It also induces a contravariant functor from ind-spectra to pro-spectra. Proposition 8.3. Let Y be an arbitrary fixed spectrum. The contravariant functors F (., Y ) from pro-spectra to ind-spectra and from ind-spectra to pro-spectra are adjoint in the sense that Hom pro(X, F (Z, Y )) ~=Hom ind(Z, F (X, Y )) for every pro-spectrum X and every ind-spectrum Z. Proof. This follows from direct computation and Lemma 8.2. Proposition 8.4. Let Y be a homotopy-finite fibrant spectrum. Then the contravariant functors F (., Y ) from pro-spectra to ind-spectra and from ind-spectra to pro-spectra are a Quillen pair between the ss*-model structure on pro-spectra and the opposite of the ss*-model structure on ind-spectra. 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 pro-spectra to fibrations (resp., ss*-acyclic fibrations) of ind-spectra. Since Y is fibrant, F (., Y ) takes cofibrations of spectra to fibrations of spectra. There- fore, F (., Y ) takes levelwise cofibrations of pro-spectra to levelwise fibrations of ind- spectra. It follows that F (., Y ) takes essentially levelwise cofibrations of pro-spectra to essentially levelwise fibrations of ind-spectra. Now let i : A ! B be a ss*-acyclic cofibration of pro-spectra. Fix k 2 Z and let Z be a fibrant model for the spectrum kY . Since Z is again homotopy-finite, the constant pro- spectrum cZ is ss*-fibrant. Therefore, the map Map (B, cZ) ! Map (A, cZ) is an acyclic fibration of simplicial sets. In particular, ss0 colimsMap (Bs, Z) ! ss0 colimtMap (At, Z) is an isomorphism. This means that ss0 colimsF (Bs, Z) ! ss0 colimtF (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 ss0 colimsF (Bs, Z) is isomorphic to ssk colims F (Bs, Y ) (and similarly for A), it follows that colims F (Bs, Y ) ! colimt F (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 ind-spectra. Theorem 8.5. The contravariant functors F (., S0) from pro-spectra to ind-spectra and from ind-spectra to pro-spectra form a Quillen equivalence between the ss*-model structure on pro-spectra and the opposite of the ss*-model structure on ind-spectra. DUALITY AND PRO-SPECTRA 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 homotopy-finite. Let X be a cofibrant pro-spectrum, let Z be a cofibrant ind-spectrum, and let f : X ! F (Z, S0) be a map of pro-spectra. Our goal is to show that f is a ss*-weak equivalence if and only if the adjoint map Z ! F (X, S0) is a ss*-weak equivalence of ind-spectra. 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 replacement for the function spectrum F (C, D). By Proposition 7.7, we may assume that each Zs is homotopy-finite. 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. 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Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. (2) 100 (1974), 1-79. Department of Mathematics, University of Western Ontario, London, Ontario, Canada E-mail address: jdc@uwo.ca Department of Mathematics, Wayne State University, Detroit, MI 48202, USA E-mail address: isaksen@math.wayne.edu