COMPLETIONS OF PRO-SPACES DANIEL C. ISAKSEN Abstract.For every ring R, we present a pair of model structures on the category of pro-spaces. In the first, the weak equivalences are detecte* *d by cohomology with coefficients in R. In the second, the weak equivalences * *are detected by cohomology with coefficients in all R-modules (or equivalent* *ly by pro-homology with coefficients in R). In the second model structure, fib* *rant replacement is essentially just the Bousfield-Kan R-tower. When R = Z=p,* * the first homotopy category is equivalent to a homotopy theory defined by Mo* *rel but has some convenient categorical advantages. 1.Introduction The notion of R-completion has been a valuable tool to homotopy theorists. The basic idea is to start with a space X and then construct another space X^Rwhose homotopy type is entirely determined by the singular cohomology H*(X; R) of X with coefficients in R. In other words, X^Rremembers the R-cohomology of X but forgets all other information. Bousfield and Kan [BK ] constructed R-completions for a large class of spaces that they called R-good spaces. The basic construction goes as follows. Start w* *ith a space X. Then define a cosimplicial space ~R*X. This cosimplicial space giv* *es rise to a tower . .!.R2X ! R1X ! R0X of fibrations. Finally, X^Ris the homotopy limit of this tower. Unfortunately, this process only works for the R-good spaces. In fact, the to* *wer described above is correct for all spaces, but the homotopy limit causes proble* *ms when X is not R-good. Thus, one approach to generalizing the construction of Bousfield and Kan to arbitrary spaces is to consider X^Rnot as a single space but rather as the whole tower. In fact, it is best to think of X^Ras a pro-space [D ]. This paper is co* *ncerned with the homotopical foundations for pro-spaces suitable for this viewpoint on * *R- completion. When R = Z=p, Morel [Mo ] constructed a homotopy theory of simplicial pro- finite sets that is suitable for studying Z=p-completions of spaces. Unfortunat* *ely, there are a few problems with the approach in [Mo ]. Namely, it is not true tha* *t the category of simplicial pro-finite sets is equal to the category of pro-simplici* *al finite sets. In fact, the former is only a retract of the latter. This means that we m* *ust be very careful with our intuitive ideas about simplicial pro-finite sets. ____________ 1991 Mathematics Subject Classification. 55P60, 55N10 (Primary); 18G55* *, 55U35 (Secondary). Key words and phrases. pro-space, singular cohomology, completion, Bousfield* *-Kan tower. This work was partially supported by a National Science Foundation Postdocto* *ral Research Fellowship. The author acknowledges useful conversations with Bill Dwyer and Da* *niel Biss. 1 2 DANIEL C. ISAKSEN We prove the following theorem in Section 6. Theorem 1.1. Let R be any ring. There is a model structure on the category of pro-simplicial sets in which the weak equivalences are maps f : X ! Y such that H*(Y ; R) ! H*(X; R) is an isomorphism. One of our main results is that the homotopy theory of pro-simplicial sets fr* *om Theorem 1.1 is equivalent to Morel's homotopy theory of simplicial pro-finite s* *ets when R = Z=p. The advantage of our approach is that one does not have to worry about the unintuitive nature of simplicial pro-finite sets. The proofs of [Mo ] rely in an essential way on notions of finiteness and take great advantage of the fact that Z=p is a finite ring. In fact, finiteness is n* *ot really such an important ingredient, as demonstrated by the proof of Theorem 1.1, which works for any ring R. In some contexts [AM ] [I4], one wants to take R-completions of pro-spaces. A* *l- though [Mo ] handles Z=p-completions of spaces perfectly well, it is not quite * *right for R-completions of arbitrary pro-spaces. Here is the underlying reason. If X * *! Y is a map of pro-spaces such that H*(Y ; R) ! H*(X; R) is an isomorphism, then H*(Y ; M) ! H*(X; M) is not necessarily an isomorphism for all R-modules M (see Example 5.3). These observations motivate the following theorem, which is also proved in Section 6. Theorem 1.2. Let R be any ring. There is a model structure on the category of pro-simplicial sets in which the weak equivalences are maps f : X ! Y such that H*(Y ; M) ! H*(X; M) is an isomorphism for all R-modules M. We will show later that the weak equivalences in the model structure of Theor* *em 1.2 can also be described as the maps f : X ! Y such that Hn(X; R) ! Hn(Y ; R) is an isomorphism of pro-groups for each n 0. For R-completions of pro-spaces, the homotopy theory constructed in Theorem 1.2 is better than the homotopy theory of Theorem 1.1. It retains more informat* *ion, remembering not just the cohomology with coefficients in R but the cohomology with coefficients in all R-modules. Moreover, the homotopy theory of Theorem 1.2 has a close link with the Bousfield-Kan R-tower [BK ]. In particular, the Bousf* *ield- Kan R-tower of a pointed connected space X is basically the same thing as the fibrant replacement of X in the R-homological model structure. We explore this link in Section 7. The main tool in the proof of Theorem 1.1 is a general localization result fr* *om [CI]. The proof of Theorem 1.2 is similar, but we have to be careful about the set-theoretical complications introduced by allowing cohomology with coefficien* *ts in arbitrarily large R-modules. In both model structures, we give an explicit description of fibrant objects.* * This allows for computations, as demonstrated in Section 9, where we compute the R- completion of the classifying space of a finitely-generated free group. In orde* *r to describe the fibrant objects, we must introduce a notion of nilpotence for spac* *es that is related to but distinct from the usual notion of nilpotence. See Sectio* *n 3 for more details. At the end of the paper, we list several questions concerning this subject th* *at remain unanswered. We hope that this will encourage future work on this topic. 1.1. Background. We assume familiarity with model structures. The original ref- erence is [Q ], but [Ho ] and [Hi] are the modern thorough references. Also, [D* *S ] is COMPLETIONS OF PRO-SPACES 3 a good introduction to the subject. We warn the reader about one convention con- cerning the model structure axioms. We will always start with a model category C in which factorizations are functorial. However, when we produce model categori* *es on the pro-category pro-C, we will not necessarily obtain functorial factorizat* *ions. Recent work of Chorny [C ] suggests that functoriality is obtainable, but we wi* *ll ignore the question here. We work exclusively with simplicial sets, rather than topological spaces. From now on, the word space always means simplicial set. It is possible to obtain a* *ll of the results of this paper for topological spaces instead of simplicial sets,* * but simplicial sets are somewhat easier to work with. We also assume a certain amount of familiarity with pro-categories, although * *we give a brief review of the most important points in Section 2. Although there a* *re many established and thorough references for pro-categories such as [AM ], [SGA* *4 ], or [EH ], the reader is encouraged to look also at [I1], [I2], and [I3] for asp* *ects of pro-categories that are particularly relevant to this paper. 1.2. Organization. We begin in Section 2 with a brief review of pro-categories, touching only on the issues that are most important for present purposes. We al* *so recall the strict model structure [EH ] [I3] on the category of pro-spaces and * *a general localization result from [CI] that will allow us to produce new model structure* *s for pro-spaces. In the following section, we study the fibrant objects in these loc* *alized model structures. For this purpose, we need a variation on the standard notion * *of nilpotence for spaces. Section 4 contains some relatively straightforward material on homological al* *ge- bra in pro-abelian categories; this is basically just transferring well-known r* *esults to a new setting. The main part of the paper begins in Section 5 with the study of pro-maps that induce isomorphisms in various kinds of singular cohomology. Section 6 de- scribes the model structures that have these cohomology isomorphisms as weak equivalences. The rest of the paper is dedicated to applications and connections to other established theories. We begin in Section 7 with the link between our construct* *ions and the Bousfield-Kan notion of R-completion. Then in Section 8 we compare our constructions to Morel's theory of Z=p-completion. We warn the reader that Sect* *ion 8 is a bit tricky because it involves a comparison of several categories that f* *eel very similar but are definitely distinct. In Section 9, we compute the Z=p-completio* *n of the classifying space of a finitely-generated free group. Finally, we list some* * open questions in Section 10. 2. Pro-Categories and Homotopy Theories for Pro-Spaces 2.1. Pro-Categories. We begin with a brief overview of pro-categories and the h* *o- motopy theory of pro-spaces. Standard references on pro-categories include [SGA* *4 ], [AM ], and [EH ]. See also [I2] and [I3] for details specifically relevant to t* *he homo- topy theory of pro-categories. Definition 2.1. For a category C, the category pro-C has objects all cofiltering diagrams in C, and Hom pro-C(X, Y ) = limscolimtHomC(Xt, Ys). 4 DANIEL C. ISAKSEN Composition is defined in the natural way. The word pro-object refers to objects of pro-categories. A constant pro-object is one indexed by the category with one object and one (identity) map. A level representation of a map f : X ! Y is: a cofiltered index category I; cofiltered diagrams ~Xand ~Yindexed by I that are pro-isomorphic to X and Y respectively; and a natural transformation ~f: ~X! ~Yrepresenting a pro-map that is isomorphic to f. Every map has a level representation [AM , App. 3.2] [Me ]. A pro-object X satisfies a certain property levelwise if each Xs satisfies th* *at property, and X satisfies this property essentially levelwise if it is isomorph* *ic to another pro-object satisfying this property levelwise. Similarly, a level r* *epre- sentation X ! Y satisfies a certain property levelwise if each Xs ! Ys has this property. A map of pro-objects satisfies this property essentially levelwise if* * it has a level representation satisfying this property levelwise. Let c : C ! pro-C be the functor taking an object X to the constant pro-object with value X. Note that this functor makes C a full subcategory of pro-C. The limit functor lim: pro-C ! C is the right adjoint of c. To avoid confusion, we * *write limprofor limits computed within the category pro-C. We recall the construction of cofiltered limits in pro-C (see, for example, [* *I1, x 4]). The specific details of this construction will be used in several places later.* * Start with a functor X : A ! pro-C : a 7! Xa, where A is a cofiltered index category. The index category I for limproX consists of all pairs (a, s) such that a belon* *gs to A and s belongs to the indexing category of Xa. A morphism (a, s) ! (b, t) cons* *ists of a morphism a ! b in A together with a map Xas! Xbtin C that represents the pro-map Xa ! Xb. Finally, limproX is defined to be the functor I ! C that takes (a, s) to Xas. 2.2. Strict Homotopy Theory of Pro-Spaces and Its Localizations. We now review from [I3] the strict homotopy theory of pro-spaces. The strict model structure was originally defined in [EH ]. The strict weak equivalences (resp., cofibrations) are the essentially levelwise weak equivalences (resp., cofibrati* *ons), and the strict fibrations are defined by the right lifting property. In fact, a* * more explicit description of the fibrations in terms of matching maps is possible [I* *3, x 4]. The strict model structure is proper and simplicial. Recall that the nth singular cohomology group Hn(X; M) of a pro-space X with coefficients in an abelian group M is defined to be colimsHn(Xs; M) [AM , 2.2] * *[S]. In fact, there is an isomorphism between Hn(X; M) and the set [X, cK(M, n)]pro of weak homotopy classes of maps of pro-spaces. The pro-space cK(M, n) is the constant pro-space with value an Eilenberg-Mac Lane space. We recall the following localization result for pro-spaces. The full proof (* *in greater generality), which owes much to [Hi, Ch. 5], appears in [CI]. Theorem 2.2. Let K be a set of fibrant spaces. There exists a left proper sim- plicial model structure on the category of pro-spaces such that the cofibration* *s are the essentially levelwise cofibrations and such that a map f : X ! Y is a weak equivalence if and only if Map pro(Y, cA) = colimsMap(Ys, A) ! colimtMap(Xt, A) = Map pro(X, cA) is a weak equivalence for every object A in K. COMPLETIONS OF PRO-SPACES 5 The weak equivalences in the above theorem are called K-colocal weak equiv- alences. 3.Nilpotent Spaces We will need to understand the fibrant objects in the localized model categor* *ies of Theorem 2.2. Definition 3.1. Let K be any collection of fibrant spaces. The class of K- nilpotent spaces is the smallest class of fibrant spaces such that: (1) the space * is K-nilpotent; (2) the K-nilpotent spaces are closed under weak equivalences between fibrant spaces; k (3) and if X is K-nilpotent, A belongs to K, and X ! A@ is any map, then the fiber product X xA@ k A k is also K-nilpotent. Because the fiber product in (3) above is actually a homotopy fiber product, it is only the weak homotopy types of the spaces in K that matter. Thus, the K-nilpotent spaces are more properly a collection of weak homotopy types rather than a collection of actual spaces. The consequence is that we are allowed to c* *hoose any (fibrant) models for the weak homotopy types in K that are most convenient for our purposes. Lemma 3.2. A space X is K-nilpotent if and only if it is fibrant and weakly equivalent to a space that can be built from * by finitely many pullbacks of ty* *pe (3) in Definition 3.1. In other words, when constructing a K-nilpotent space, it is not necessary to* * use any weak equivalences until the very last step. Proof.Let Cn be the class of K-nilpotent spaces that can be built from * with fewer than n + 1 pullbacks (and possibly also weak equivalences), and let Dn be the class of fibrant spaces that are weakly equivalent to a space that can be b* *uilt from * with fewer than n + 1 pullbacks (but without any weak equivalences). By definition, Dn is contained in Cn. We will show by induction that Cn and Dn are equal. The classes C0 and D0 consist of the fibrant contractible spaces, so they are equal. Now suppose that Cn-1 and Dn-1 are equal. Let X belong to Cn. Then X is weakly equivalent to a space X0xA@ k A k, where X0 belongs to Cn-1. By the induction assumption, X0also belongs to Dn-1, so X0is weakly equivalent to a space X00that can be built from * by fewer than n pullbacks. Now X0xA@ k A k is weakly equivalent to X00xA@ k A k because the fiber products are actually homotopy fiber products. Thus X is weakly equivalent to X00xA@ k A k, which is a space that can be built from * using fewer than n + 1 pullbacks. Therefore* *, X belongs to Dn. The next theorem demonstrates the relevance of K-nilpotent objects. It is pro* *ved in [CI, Prop. 4.9]. Theorem 3.3. Let K be a set of fibrant spaces. In the model structure of Theorem 2.2, the fibrant objects are precisely the pro-spaces that are both strictly fi* *brant and essentially levelwise K-nilpotent. 6 DANIEL C. ISAKSEN Recall that a fibration is principal if it is the base change of a fibration * *with contractible total space. We do not require that the base be connected. Therefo* *re, some of the fibers of a principal fibration may be empty; however, the non-empty fibers are all weakly homotopic. Lemma 3.4. The fibration p : K(A, n) k ! K(A, n)@ k is principal for all n 0, k 0, and every abelian group A. Its non-empty fiber is weakly equivalent to K(A, n - k). Proof.Throughout this proof, we use models for K(A, n) that are simplicial abel* *ian groups [Ma ]. In particular, this means that K(A, n) is fibrant. Also, K(A, n* *) is based at 0. k Only one path-component of K(A, n)@ is in the image of p; it consists of the maps @ k ! K(A, n) that are null-homotopic. If n 6= k - 1, then K(A, n)@ k only has one component, but this is irrelevant. Thus, we only have to computekthe fi* *ber of p over the zero map 0 : @ k ! K(A, n) (which is a point of K(A, n)@ ), and this fiber is kK(A, n), as desired. We still have to check that p is principal. Let v be the 0th vertex of k. Th* *en we have a short exact sequence k {v} 0 ! Xn,k! K(A, n) ! K(A, n) = K(A, n) ! 0 of simplicial abelian groups, where Xn,kis the subspace of K(A, n) k consisting* * of all maps that take v to 0. The projection k ! {v} gives a splitting, so K(A, n* *) k is isomorphic to K(A, n) x Xn,k. Similarly, K(A, n)@ k is isomorphic to K(A, n)* * x Yn,k, where Yn,kis the subspace of K(A, n)@ k consisting of all maps that take v to 0. The fibration Xn,k! Yn,kis principal because Xn,kis contractible (which follows from the fact that k is contractible). The identity map on K(A, n) is * *of course principal, and a product of two principal fibrations is again principal.* * This shows that p is principal. Proposition 3.5. Let C be a class of R-modules, and let K be the class of Eilen* *berg- Mac Lane spaces K(M, n) such that M belongs to C. A space X is K-nilpotent if and only if it is fibrant and there exists a finite tower Xn ! Xn-1 ! . .!.X1 ! X0 = *, where Xn is weakly equivalent to X and each map Xk ! Xk-1 is a principal fibration whose non-empty fibers belong to K. Proof.For onekdirection, Lemmak3.4 tells us that for every M in C, the map p : K(M, n) ! K(M, n)@ is a principal fibration whose non-empty fibers are weakly equivalent to K(M, n - k). In view of Lemma 3.2, this shows that every K-nilpotent space X has a tower of the desired form. Now we will show that if X has a tower of the desired form, then X is K- nilpotent. By induction on the length of the tower, we just need to show that if p : E ! B is a principal fibration whose non-empty fibers are weakly equivalent* * to K(M,`n) and`B is K-nilpotent, then E is K-nilpotent. We can rewrite p in the fo* *rm OE E ! B0 B1, where the fibers over B0 are all empty and the fibers over B1 are all weakly equivalent to K(M, n). By the following lemma, we know that B1 is K-nilpotent. Thus it suffices to replace B with B1 and assume that p is surject* *ive. COMPLETIONS OF PRO-SPACES 7 Let p0: E0! B0be a fibration with E0contractible such that p is a base change* * of p0. Since p is surjective, the image of the map B ! B0lies in the same componen* *t as the image of p0. This means that we may assume that B0is connected. Since every fiber of p0is weakly equivalent to K(M, n), we know that B0 is weakly equivalent to K(M, n + 1). We now know that E is the homotopy pullback of the diagram B ! K(M, n + 1) *. Recall the spaces Xn+1,1and Yn+1,1from the proof of Lemma 3.4. Since @ k is a model for Sk-1, the space Yn,kis weakly equivalent to k-1K(M, n), which is a model for K(M, n - k + 1). Thus E is the homotopy pullback of the diagram B ! Yn+1,1 Xn+1,1, so E is also the homotopy pullback of the diagram B ! K(M, n + 1) x Yn+1,1 K(M, n + 1) x Xn+1,1. This identifies E up to homotopy as a fiber product 1 B xK(M,n+1)@ 1K(M, n + 1) , which implies that E is K-nilpotent. Lemma 3.6. Let C be a class of R-modules, and let K be the class`of Eilenberg- Mac Lane spaces K(M, n) such that M belongs to C. Let X = Y Z. If X is K-nilpotent, then Y and Z are K-nilpotent. We make no assumptions about whether Y and Z are connected. Proof.We choose the model for K(M, 0) that consists simply of the underlying set of M, viewed as a discrete simplicial set. The map K(M, 0) 1 ! K(M, 0)@ 1 then becomes the diagonal M ! M x M. Take any map X ! K(M, 0)@ 1 such that Y maps to the diagonal1and Z maps off the diagonal. Then the pullback X xK(M,0)@ 1K(M, 0) is isomorphic to Y . This shows that Y is K-nilpotent. The same argument shows that Z is K-nilpotent. When C is the collection of all R-modules, the notion of K-nilpotence defined here is related but not equivalent to the notion of R-nilpotence in [BK ]. Howe* *ver, as Proposition 3.5 and [BK , Prop. III.5.3] show, every space that is nilpotent* * in the sense of Bousfield and Kan is also nilpotent in our sense. The converse is not * *true. For example, a K-nilpotent space (such as K(R, 0)) need not be connected. One way to see the difference is to note that the two notions of nilpotence are bas* *ed upon different notions of principality. The question is whether the base spaces* * are required to be connected (or, equivalently, whether empty fibers are allowed). It would be nice to have algebraic conditions on the homotopy groups of a dis- connected space that guarantee that the space is K-nilpotent. However, such a condition has eluded us so far. See Section 10 for an elaboration of this probl* *em. 8 DANIEL C. ISAKSEN 4. Pro-Homological Algebra In this section, we let A be any abelian category. Recall that the category p* *ro-A is again abelian [AM , App. 4.5]. The monomorphisms are the essentially levelwi* *se monomorphisms, and the epimorphisms are the essentially levelwise epimorphisms. We also assume that A has enough injectives. This implies that pro-A also has enough injectives [Z]. Lemma 4.1. Let A be an abelian category, and let I be an injective object of A. Then cI is an injective object of pro-A. Proof.Let i : A ! B be a monomorphism in pro-A; we may assume that i is a level monomorphism. Let f : A ! cI be any map in pro-A. This map is represented by a map fs : As ! I in A for some s. Now fs extends over is since I is injective * *and is is a monomorphism. This extension represents a map g : B ! I, and g extends f over i. Lemma 4.2. Let A be an abelian category, and let A be any object of pro-A. The groups Extnpro-A(A, cB) and colimtExtnA(As, B) are isomorphic for every object B of A and every n 0. Proof.Let B ! I0 ! I1 ! . . . be an injective resolution of B in A. Then cB ! cI0 ! cI1 ! . . . is an injective resolution of cB in pro-A by Lemma 4.1 and the fact that c pres* *erves exactness. Therefore, Ext*pro(A, cB) is the homology of the complex Hom pro(A, cB) ! Hom pro(A, cI0) ! Hom pro(A, cI1) ! . .,. which is equal to the complex colimsHomA (As, B) ! colimsHomA(As, I0) ! colimsHomA(As, I1) ! . ... Since filtered colimits are exact, the homology of the last complex is equal to colimsExt*A(As, B). Now we construct a universal coefficients spectral sequence for pro-spaces. L* *et X be an arbitrary pro-space, and let M be an R-module. Choose an injective resolution 0 ! M ! I0 ! I1 ! . ... Consider the bicomplex K*,*given by the formula Kp,q= Cq(X; Ip) = colimsCq(Xs; Ip) = colimsHom(Cq(Xs; R), Ip). This is a first-quadrant bicomplex with cohomological grading. Now Cq(Xs; R) is a free R-module, so the complex colimsHom (Cq(Xs; R), I*) is exact. Taking cohomology of K*,*with respect to the p-differential gives that E0,q1= colimsHom(Cq(Xs; R), M) = Cq(X; M) and Ep,q1= 0 if p > 0. Therefore, E0,q2= Hq(X; M) and Ep,q2= 0 if p > 0. Thus, the spectral sequence collapses, and the cohomology of the total complex of K*,* is H*(X; M). COMPLETIONS OF PRO-SPACES 9 Now we compute in the other order. Taking cohomology of K*,*with respect to the q-differential gives Ep,q1= Hq(X; Ip) = colimsHq(Xs; Ip). Because Ip is injective, this equals colimsHom (Hq(Xs; R), Ip). Therefore, taki* *ng cohomology with respect to the p-differential gives Ep,q2= colimsExtpR(Hq(Xs; R), M) = Extppro(Hq(X; R), M). The second equality above relies on Lemma 4.2. Hence, we have a convergent first-quadrant cohomological spectral sequence Ep,q2= Extppro(Hq(X; R), cM) =) Hp+q(X; M). This spectral sequence is called the pro-universal coefficients spectral se- quence. 5.Cohomological and Homological Weak Equivalences In this section we collect some results about the cohomology and pro-homology of pro-spaces, emphasizing the differences and similarities with the situation * *of ordinary spaces. The first lemma looks quite strange at first glance, but it becomes plausible* * when one remembers that filtered limits are exact in pro-categories [I1] [AHJM ]. Lemma 5.1. If a 7! Xa is a cofiltered diagram of pro-spaces, then the cohomology group Hn(limproaXa; M) is isomorphic to colimaHn(Xa; M), where limprois the limit internal to the category of pro-spaces. Proof.This follows by direct computation using the construction of cofiltered l* *imits in pro-categories given in Section 2.1. Definition 5.2. Let R be any ring. An R-cohomology weak equivalence is a map of pro-spaces X ! Y inducing isomorphisms Hn(Y ; R) ! Hn(X; R) for every n 0. Suppose that f is a map of ordinary spaces inducing an R-cohomology isomor- phism. Then f induces an isomorphism in cohomology with coefficients in an arbi- trary product of copies of R. But every free R-module is a retract of some prod* *uct of copies of R, so f induces an isomorphism in cohomology with coefficients in * *any free R-module. Now every R-module M belongs to a short exact sequence 0 ! F1 ! F2 ! M ! 0 in which F1 and F2 are free, so the long exact sequence of cohomology groups and the five lemma imply that f induces an isomorphism in cohomology with coefficie* *nts in any R-module. In contrast to the above paragraph, if f is a map of pro-spaces that is an R- cohomology weak equivalence, then f does not necessarily induce a cohomology isomorphism with coefficients in all R-modules. The argument from the previous paragraph breaks down because cohomology of pro-spaces does not commute with arbitrary products of coefficients. Actually, cohomologyQonly commutes with fin* *ite products.QOne explanation for this difference isQthat K( tMt, n) is weakly equ* *iva- lentQto tK(Mt, n) for ordinary spaces, but cK( tMt, n) is not weakly equival* *ent to tcK(Mt, n) as pro-spaces. 10 DANIEL C. ISAKSEN Example 5.3. Let V be an infinite-dimensional Z=p-vector space, and let n 1. We give an example of a pro-space X for which Hn(X; Z=p) = 0 but Hn(X; V ) is non-zero. Consider the pro-vector space W consisting of all subspaces of V with finite codimension; the structure maps are the inclusions of subspaces. Let X be the pro-space K(W, n) obtained by applying levelwise the functor K(-, n) to W . Now Hn(X; Z=p) equals colimsHom Z=p(Ws, Z=p). This colimit is zero because the kernel of any homomorphism Ws ! Z=p is equal to Wt for some t. On the other hand, Hn(X; V ) equals colimsHom Z=p(Ws, V ), which is non-zero because it contains the element represented by any of the inclusions Ws ! V . Because of this phenomenon, we introduce the following definition, which is distinct from Definition 5.2. Definition 5.4. Let R be any ring. An R-homology weak equivalence is a map of pro-spaces X ! Y inducing isomorphisms Hn(X; R) ! Hn(Y ; R) of pro-groups for all n 0. This definition can be reformulated in terms of cohomology. The next result implies that every R-homology weak equivalence is an R-cohomology weak equiva- lence. As shown in Example 5.3, the converse is not true. Proposition 5.5. A map f : X ! Y is an R-homology weak equivalence if and only if it induces an isomorphism f* : Hn(Y ; M) ! Hn(X; M) for all n 0 and all R-modules M. The following proof is inspired by [BK , Prop. III.6.7]. Proof.First suppose that f is an R-homology weak equivalence. For any R-module M, f induces an isomorphism of E2-terms of the pro-universal coefficients spect* *ral sequence (see Section 4). Therefore, the map on abutments is also an isomorphis* *m. For the other implication, suppose that f is a cohomology isomorphism with coefficients in any R-module. If I is an injective R-module, then Hn(X; I) = colimsHom(Hn(Xs; R), I) = Hom pro(Hn(X; R), cI). Similarly, Hn(Y ; I) = Hom pro(Hn(Y ; R), cI). It follows that Hom pro(Hn(Y ; R), cI) ! Hom pro(Hn(X; R), cI) is an isomorphism for every injective R-module I. Now let M be an arbitrary pro-R-module. Choose a monomorphism M ! I such that I is an injective pro-R-module. Then there is a diagram Hom pro(Hn(Y ; R), M)___//_Hompro(Hn(Y ; R), I) | | | | fflffl| fflffl| Hom pro(Hn(X; R), M)____//_Hompro(Hn(X; R), I) in which the horizontal maps are monomorphisms. Since the right vertical map is an isomorphism, the left vertical map must be a monomorphism. We conclude that Hn(X; R) ! Hn(Y ; R) is an epimorphism of pro-abelian groups. Now let K be the pro-group that is the kernel of the map Hn(X; R) ! Hn(Y ; R). Since Hom pro(-, I) is exact for all injective pro-groups I, the first paragrap* *h tells us that Hom pro(K, I) is zero for all injectives I. But the category of pro-abe* *lian COMPLETIONS OF PRO-SPACES 11 groups has enough injectives, so this can only happen if K equals zero. This me* *ans that the map Hn(X; R) ! Hn(Y ; R) is an isomorphism. From now on, we will freely switch between the cohomological and homologi- cal descriptions of R-homology weak equivalences, as given in Definition 5.4 and Proposition 5.5. We will need to rephrase cohomology isomorphisms in terms of weak equivalences of mapping spaces. Proposition 5.6. Let M be any R-module, and let K(M, n) be a fibrant Eilenberg- Mac Lane space. If f : X ! Y is any map of pro-spaces, then Hn(f; M) is an isomorphism for all n 0 if and only if the map Map pro(Y, cK(M, n)) ! Map pro(X, cK(M, n)) is a weak equivalence for all n 0. Proof.First suppose that the maps between mapping spaces are weak equivalences. We get isomorphisms after taking ß0, which gives us the desired cohomology iso- morphisms. Now suppose that the maps Hn(f; M) are isomorphisms for all n 0. Since all pro-spaces are cofibrant, the homotopy types of the mapping spaces do not change if we alter X or Y up to strict weak equivalence. This means that we may assume that f is a levelwise cofibration; let Z be its cofiber, which is computed leve* *lwise. Recall that Z is pointed canonically. Using the long exact sequence in cohomolo* *gy, note that the reduced cohomology group ~Hn(Z; M) is zero for all n 0. Now we want to show that the fibration Map pro(Y, cK(M, n)) ! Map pro(X, cK(M, n)) is actually an acyclic fibration of simplicial sets. We can do this by showing * *that it has the right lifting property with respect to all generating cofibrations @ k * *! k. After the usual adjointness arguments, we need to show that every diagram ` @ k Y @ k X k X _____//cK(M, n) | | fflffl| k Y has a lift. Since K(M, n) is fibrant, formal model category arguments show that we only need obtain a lift up to homotopy; then we can adjust this lift to get * *an actual lift. In other words, we need to show that the map ` k Hn( k Y ; M)_____//Hn(@ k Y @ k X X; M) is surjective. The cofiber of the vertical map above is Sk ^ Z, where the smash product is constructed levelwise. Using the long exact sequence in cohomology, * *it suffices to show that the reduced cohomology group ~Hn+1(Sk ^ Z; M) is zero. But this group is equal to ~Hn+1-k(Z; M), which we already computed to be zero. From now on, we will frequently express cohomology isomorphisms in terms of weak equivalences of mapping spaces as in Proposition 5.6. For example, we have the following corollary. Corollary 5.7. 12 DANIEL C. ISAKSEN (1) A map f is an R-cohomology weak equivalence if and only if the map Map pro(f, cK(R, n)) is a weak equivalence for every n 0, where K(R, n) is a fibrant Eilenb* *erg- Mac Lane space. (2) A map f is an R-homology weak equivalence if and only if the map Map pro(f, cK(M, n)) is a weak equivalence for every n 0 and every R-module M, where K(M, n) is a fibrant Eilenberg-Mac Lane space. Proof.The first claim follows immediately from Definition 5.2 and Proposition 5* *.6. The second claim follows immediately from Propositions 5.5 and 5.6. 6.The R-Cohomological and R-Homological Model Structures We next establish model structures whose weak equivalences are the R-cohom- ology weak equivalences and the R-homology weak equivalences. Definition 6.1. A cofibration of pro-spaces is an essentially levelwise cofibra* *tion. This is the same notion of cofibration as in the strict model structure [I2] * *or the ß*-model structure [I3]. In fact, the notion of cofibration is the same for* * every model structure on pro-spaces in the entire paper. Definition 6.2. An R-cohomology fibration is a map having the right lifting property with respect to all maps that are both cofibrations and R-cohomology weak equivalences. Theorem 6.3. The cofibrations, R-cohomology weak equivalences, and R-cohom- ology fibrations give a left proper simplicial model structure on the category * *of pro- spaces. Proof.The proof is an application of Theorem 2.2 with K equal to the set of Eilenberg-Mac Lane objects of the form K(R, n). By Corollary 5.7(1), the K-colo* *cal weak equivalences are the same as the R-cohomology weak equivalences. If ~ is any infinite cardinal, then an R-module M is ~-generated if M has a generating set of cardinality at most ~. Note that there is a set of isomorphi* *sm types of ~-generated R-modules. Also note that a ~-generated R-module might have more than ~ elements because R might be large. A pro-map is a ~-generated R-cohomology weak equivalence if it induces an isomorphism on cohomology with coefficients in all ~-generated R-modules. Si* *m- ilarly, a pro-map is a ~-generated R-cohomology fibration if it has the right lifting property with respect to all pro-maps that are both cofibrations and ~- generated R-cohomology weak equivalences. Theorem 6.4. For any infinite cardinal ~, the cofibrations, ~-generated R-co- homology weak equivalences, and ~-generated R-cohomology fibrations give a left proper simplicial model structure on the category of pro-spaces. Proof.The proof is an application of Theorem 2.2 with K equal to the set of Eilenberg-Mac Lane objects of the form K(M, n) with M a ~-generated R-module. COMPLETIONS OF PRO-SPACES 13 Actually, the ~-generated model structures are not so interesting. What we really want is a model structure in which the weak equivalences are detected by R-homology, or equivalently, according to Proposition 5.5, by cohomology with coefficients in all R-modules. This is the main purpose of the rest of this sec* *tion. Definition 6.5. An R-homology fibration is any map that is a ~-generated R-cohomology fibration for some ~. In other words, the class of R-homology fibrations is the union of the classe* *s of ~-generated R-cohomology fibrations as ~ ranges over all cardinals. By Proposit* *ion 5.5, the class of R-homology weak equivalences is the intersection of the class* *es of ~-generated R-cohomology weak equivalences as ~ ranges over all cardinals. Lemma 6.6. For any ~, the acyclic ~-generated R-cohomology fibrations are the same as the acyclic R-homology fibrations. Proof.For every ~, the acyclic ~-generated R-cohomology fibrations are detected* * by the same class of cofibrations, so these classes of acyclic ~-generated R-cohom* *ology fibrations are all equal. Therefore, if p is an acyclic ~-generated R-cohomolo* *gy fibration, then it is a ~-generated R-cohomology weak equivalence for all ~. Th* *is implies that p is an R-homology weak equivalence by Proposition 5.5. For the other direction, suppose that p is an acyclic R-homology fibration. T* *hen it is a ~-generated R-cohomology weak equivalence for every ~ and a ~-generated* * R- cohomology fibration for some ~. Thus, p is an acyclic ~-generated R-cohomology fibration for some ~. As in the first paragraph, this implies that p is an acy* *clic ~-generated R-cohomology fibration for all ~. Theorem 6.7. The cofibrations, R-homology weak equivalences, and R-homology fibrations are a left proper simplicial model structure on the category of pro-* *spaces. Proof.Most of the proof follows formally from the existence of the ~-generated R-cohomology model structures. Given any collection of classes that satisfy the two-out-of-three axiom, their intersection also satisfies the two-out-of-three * *axiom. Similarly, arbitrary unions and intersections of classes preserve the retract a* *xiom. Since Lemma 6.6 tells us what the acyclic R-homology fibrations are, one of the lifting axioms and one of the factoring axioms follows immediately from The- orem 6.4. For the other lifting axiom, suppose that i is an acyclic R-homology cofibration and p is an R-homology fibration. Then there exists some ~ such that p is a ~-generated R-cohomology fibration, and i is also an acyclic ~-generated R-cohomology cofibration. Thus i has the left lifting property with respect to* * p because of Theorem 6.4. Factorizations into acyclic R-homology cofibrations followed by R-homology fi- brations are significantly more difficult. We construct these below in Proposit* *ion 6.10. The simplicial structure also follows formally from the ~-generated model str* *uc- tures of Theorem 6.4. Namely, let i : A ! B be a cofibration and let p : X ! Y * *be an R-homology fibration. Then p is a ~-generated R-homology fibration for some ~, so the map Map(i, p) : Map (B, X) ! Map (A, X) xMap(A,Y )Map(B, Y ) is a fibration of simplicial sets because the ~-generated model structure is si* *mplicial. If i or p is acyclic, then it is a ~-generated R-cohomology weak equivalence, s* *o the 14 DANIEL C. ISAKSEN above map Map (i, p) is also a weak equivalence because of the ~-generated model structure of Theorem 6.4. Left properness follows immediately from the fact that every object is cofibr* *ant. Functorially in M, choose a fibrant Eilenberg-Mac Lane space K(M, n) for each finitely generated R-module M. Let pk(M, n) be the fibration K(M, n) k ! K(M, n)@ k. Now for any R-module M, define K(M, n) to be colimK(N, n), where the colimit ranges over all finitely generated submodules N of M. This construction is a sp* *ecific fibrant model for the Eilenberg-Mac Lane space of type (M, n). We also get maps pk(M, n), which are again fibrations because the colimit is filtered. Note also* * that K(M, n) k is equal to colimN M (K(N, n) k) because k is a finite simplicial set (and similarly for K(M, n)@ k). The constant map cpk(M, n) : cK(M, n) k ! cK(M, n)@ k is a ~-generated R-cohomology fibration if M is ~-generated. This (or rather its dual) is proved* * in [CI, Lem. 2.3(b)]; it can also be deduced from Proposition 5.6 using adjointnes* *s. Therefore, cpk(M, n) is an R-homology fibration for every M. A pro-space X is ~-bounded if each space Xs has at most ~ elements. Lemma 6.8. Let f : X ! Y be any map between ~-bounded pro-spaces. Then f has the left lifting property with respect to all maps of the form cpk(M, n) wi* *th M any R-module, if and only if f has the left lifting property with respect to al* *l maps of the form cpk(M, n) with M any ~-generated R-module. Proof.One direction is tautological. For the other direction, let f have the l* *eft lifting property with respect to all maps of the form cpk(M, n) with M any ~- generated R-module. Suppose there is a square X ______//cK(M, n) k | | | | | fflffl| fflffl|k Y _____//cK(M, n)@ with M any R-module; we want to produce a lift. This square can be represented as a square g k Xs _____//_K(M, n) | | | | | fflffl| fflffl|k Yt__h__//K(M, n)@ of ordinary spaces. For each x in Xs, f(x) belongs to K(Px, n) k for some finit* *ely generated submodule Px of M. Similarly, for each y in Yt, g(y) lies in K(Py, n)* *@ k for some finitely generated submodule Py of M. Since there are at most ~ choices for x and y, we can choose a ~-generated submodule N containing each Px and COMPLETIONS OF PRO-SPACES 15 each Py. Now we have a diagram X _____//_cK(N, n)_k____//cK(M, n) k | | | | | | | fflffl| fflffl|k fflffl|k Y _____//cK(N, n)@ ____//_cK(M, n)@ for some ~-generated submodule N of M. A lift exists in the left square by as- sumption, and this gives us the desired lift. The importance of Lemma 6.8 is that the second condition involves only a set * *of maps of the form cpk(M, n), while the first condition does not. Lemma 6.9. A cofibration i : A ! B induces an isomorphism in cohomology with coefficients in M if and only if it has the right lifting property with respect* * to the maps cpk(M, n). Proof.By Proposition 5.6, instead of considering whether Hn(i; M) is an isomor- phism for all n, we shall consider whether Mappro(i, cK(M, n)) : Map pro(B, cK(M, n)) ! Map pro(A, cK(M, n)) is a weak equivalence for all n. Since i is a cofibration, this map is a fibra* *tion of simplicial sets. It is an acyclic fibration if and only if it has the left * *lifting property with respect to the maps @ k ! k. By adjointness, this lifting proper* *ty is equivalent to the desired lifting property. Proposition 6.10. Any map f : X ! Y of pro-spaces factors into i : X ! Z followed by q : Z ! Y , where i is an acyclic R-homology cofibration and q is an R-homology fibration. Proof.Choose a cardinal ~0 such that X and Y are both ~0-bounded, and let f0 = f. Factor the map f0 into an acyclic ~0-generated R-cohomology cofibration f1 : X ! Z1 followed by a ~0-generated R-cohomology fibration Z1 ! Y . Here we are using Theorem 6.4. Proceeding inductively, choose a cardinal ~n such that X and Zn are both ~n- bounded. Factor the map fn into an acyclic ~n-generated cofibration fn+1 : X ! Zn+1 followed by a ~n-generated fibration Zn+1 ! Zn. This process yields a diagram X ! . .!.Z2 ! Z1 ! Y, and we let Z = limpronZn, where the limit is computed within the category of pro-spaces. Choose a cardinal ~1 such that ~1 ~n for each n. Note that X, Y , and each Zn are ~1 -bounded. Using the explicit construction of cofiltered limits * *in pro-categories given in Section 2.1, it follows that Z is also ~1 -bounded. If ~ ~, then the ~-generated R-cohomology weak equivalences are contained in the ~-generated R-cohomology weak equivalences. Therefore, the acyclic ~- generated R-cohomology cofibrations are contained in the ~-generated R-cohom- ology cofibrations. From the nature of lifting properties, it follows that the* * ~- generated R-cohomology fibrations are contained in the ~-generated R-cohomology fibrations. 16 DANIEL C. ISAKSEN Since ~n ~1 , each map Zn ! Zn-1 (and also Z1 ! Y ) is a ~1 -generated R-cohomology fibration. Thus, the map q : Z ! Y is a composition of a countable tower of ~1 -generated R-cohomology fibrations. Formal arguments with lifting properties imply that q is also a ~1 -generated R-cohomology fibration. This me* *ans that q is an R-homology fibration. Each map X ! Zn is a cofibration. Since cofibrations are closed under cofilte* *red limits [I1, Cor. 5.3], we conclude that i : X ! Z is a cofibration. By Lemma 6.* *9, it remains only to show that i has the left lifting property with respect to the m* *aps cpk(M, n) for all R-modules M. Suppose given a square X ______//cK(M, n) k | | | | | fflffl| fflffl|k Z _____//cK(M, n)@ with M a ~1 -generated R-module. Using the explicit construction of filtered li* *mits of pro-objects given in Section 2.1, this diagram factors as X _____________//cK(M, n) k | | | | | fflffl| fflffl|k Z _____//Zj____//cK(M, n)@ for some j. Because X and Zj are both ~j-bounded, this diagram further factors into X _____//_cK(N, n) k____//_cK(M, n) k | | | | | | | fflffl| fflffl| fflffl| Zj_____//cK(N, n)@ k____//cK(M, n)@ k as in the proof of Lemma 6.8, where N is a ~j-generated submodule of M. Lemma 6.9 tells us that there is a lift in the diagram X ______________//_cK(N, n) k____//_cK(M, n) k | ___55______ | ___________ | | | __________ | | fflffl|__________ fflffl| fflffl| Zj+1_____////_Zj_//cK(N, n)@ k____//cK(M, n)@ k because X ! Zj+1 is an acyclic ~j-generated cofibration and N is a ~j-generated R-module. This gives us the desired lift. 7. R-Completions The model structures of Theorems 6.3 and 6.7 allow us to define the R-complet* *ion of any pro-space. Definition 7.1. Let X be a pro-space. The cohomological R-completion X^R-c of X is a fibrant replacement for X in the cohomological model structure of The- orem 6.3. The homological R-completion X^R-hof X is a fibrant replacement for X in the homological model structure of Theorem 6.7. COMPLETIONS OF PRO-SPACES 17 Philosophically, the cohomological R-completion of a pro-space X should pre- serve the cohomology of X with coefficients in R but forget all other informati* *on. Similarly, the homological R-completion of X should preserve the cohomology of X with coefficients in any R-module but forget all other information. Thus, homol* *ogi- cal R-completion contains more information than the cohomological R-completion. See Example 5.3 for a pro-space X such that X^R-hand X^R-care distinct. These ideas are made precise in the following theorem. Theorem 7.2. A map f : X ! Y of pro-spaces (or of ordinary spaces) induces an isomorphism in cohomology with coefficients in R if and only if X^R-cand YR^* *-c are simplicially homotopy equivalent pro-spaces. Similarly, the map f induces an isomorphism in cohomology with coefficients in all R-modules if and only if X^R* *-h and YR^-hare simplicially homotopy equivalent pro-spaces. Of course, if two pro-spaces are simplicially homotopy equivalent, then they * *are strictly weakly equivalent or equivalent with respect to any reasonable notion * *of homotopy theory for pro-spaces. Proof.The maps X ! X^R-cand Y ! YR^-care R-cohomological weak equivalences. Thus H*(f; R) is an isomorphism if and only if X^R-cand YR^-care R-cohomologica* *lly weakly equivalent. Since every pro-space is cofibrant, X^R-cand YR^-care both c* *ofi- brant and fibrant with respect to the R-cohomological model structure of Theorem 6.3. Thus, X^R-cand YR^-care R-cohomologically weakly equivalent if and only if they are simplicially homotopy equivalent. The argument for homological R-completions is identical except that it uses t* *he R-homological model structure of Theorem 6.7. For any pro-space X such that each Xs is pointed and connected, we will now show how to construct X^R-hin terms of the Bousfield-Kan R-towers [BK ] of the spaces Xs. The moral is that at least for pointed connected spaces, the Bousfie* *ld- Kan R-tower (with a minor modification) is the same thing as fibrant replacement in the R-homological model structure. Proposition 7.3. Let X be a pro-object in the category of pointed connected spa* *ces, and let I be the indexing category of X. Construct a new pro-space Y with index* *ing category I x N by defining Ys,nto be the nth Postnikov section PnRnXs of the nth stage of the Bousfield-Kan R-tower for Xs. Then the strict fibrant replacement * *^Y of Y is a fibrant replacement for X in the R-homological model structure. Remark 7.4. In the previous proposition, it is also possible to define Ys,nto be just RnXs, not its nth Postnikov section. However, then ^Ymust be a ß*-fibrant replacement, i.e., a fibrant replacement in the model structure on pro-spaces in which weak equivalences are detected by pro-homotopy groups [I2]. Proof.Let K be the collection of all fibrant spaces of the form K(M, n), where * *n 0 and M is any R-module. From [BK , Cor. III.5.6] and [BK , Prop. III.5.3], we kn* *ow that the Postnikov tower of RnXs can be refined to a sequence of principal fibr* *ations whose fibers belong to K. Therefore, the Postnikov tower of PnRnXs consists of a finite sequence of principal fibrations whose fibers belong to K. Proposition* * 3.5 tells us that each PnRnXs is K-nilpotent, so Y is essentially levelwise K-nilpo* *tent. It follows from [CI, Prop. 3.7] that ^Yis also essentially levelwise K-nilpoten* *t, so Theorem 3.3 tells us that ^Yis fibrant in the R-homological model structure. 18 DANIEL C. ISAKSEN It remains to show that the map X ! ^Yis an R-homology weak equivalence. Since Y ! ^Yis a strict weak equivalence, it suffices to show that X ! Y is an R-homology weak equivalence. We know from [BK , Prop. III.6.5] (or [D ]) that cHk(Xs; R) ! Hk(P*R*Xs; R) is a pro-isomorphism for each s; here P*R*Xs is the pro-space . .!.P2R2Xs ! P1R1Xs ! P0R0Xs, and cHk(Xs; R) is the constant pro-group with value Hk(Xs; R). Now Hk(X; R) is isomorphic to limproscHk(Xs; R), where the limit is computed within the category of pro-groups. Similarly, Hk(Y ; R) is isomorphic to limprosHk(P*R*Xs; R) (see the construction of limits in pro-categories given in Section 2.1). Thus, the * *map Hk(X; R) ! Hk(Y ; R) is a pro-isomorphism. 8. Z=p-Cohomology In this section, fix a prime p, and let R be the finite ring Z=p. We will com* *pare the Z=p-cohomological model structure of Theorem 6.3 to the Z=p-cohomological model structure of [Mo ] and show that they yield the same homotopy categories. Throughout this section, whenever we discuss the category of pro-simplicial set* *s, we are always thinking of it equipped with the Z=p-cohomological model structur* *e. We first recall some ideas from [Mo ]. Let F be the category of finite sets. * *The category pro-F of pro-finite sets is equivalent to the category of totally disc* *onnected compact Hausdorff topological spaces. The main object of study in [Mo ] is the category spro-F of simplicial pro-fi* *nite sets (or, equivalently, simplicial totally disconnected compact Hausdorff topol* *ogi- cal spaces). The weak equivalences in this category are the continuous cohomolo* *gy isomorphisms with Z=p-coefficients, the cofibrations are the degreewise monomor- phisms, and the fibrations are defined by a lifting property. The main purpose of this model structure on spro-F is to describe a Z=p- completion functor. Given any set X, let X^be the pro-finite set of all finite * *quo- tients of X. Applying this construction degreewise gives a functor from simplic* *ial sets to simplicial pro-finite sets. The Z=p-completion of a simplicial set X is* * defined to be a fibrant replacement for ^X. In order to compare the category spro-F to the category of pro-simplicial set* *s, we need the intermediate category pro-sF of pro-simplicial finite sets. Despite cl* *aims in [Mo ], [R ], and elsewhere, this category is not equivalent to spro-F. See [I1,* * Ex. 3.7] for a counterexample. Beware that a simplicial finite set is not the same as a * *finite simplicial set. A finite simplicial set can only have finitely many non-degene* *rate simplices, while a simplicial finite set need only be finite degreewise. There is an inclusion functor i : pro-sF ! pro-sSetfrom pro-simplicial finite* * sets to pro-simplicial sets. We next define its adjoint. Definition 8.1. If cX is any constant pro-space, then FincX is the system of all simplicial finite quotients of X. If X is an arbitrary pro-space, then FinX* * is limprosFincXs, where the limit is calculated within the category of pro-simplic* *ial finite sets. Lemma 8.2. The functor Finis the left adjoint of the inclusion i. Proof.We begin by showing that FincX is a cofiltered system of spaces. Let X1 and X2 be two simplicial finite quotients of X. We need to find another simplic* *ial COMPLETIONS OF PRO-SPACES 19 finite quotient X3 of X that refines both X1 and X2. Note that X1 x X2 is a simplicial finite set but not necessarily a quotient of X because the canonical* * map X ! X1xX2 is not surjective. Define X3 to be the image of the map X ! X1xX2. Now X3 is a simplicial finite set because it is a subobject of X1 x X2. The map X ! X3 is surjective by construction, so X3 is a simplicial finite quotient of * *X. The two maps X ! X1 and X ! X2 factor through X3. This shows that FincX is a cofiltered system. Next we will show that FincX has the correct adjoint property. We want to show that Hom pro(FincX, Y ) is isomorphic to Hom pro(cX, Y ) for every pro-simplici* *al finite set Y . This follows from the fact that every map from X into a simplici* *al finite set factors through a simplicial finite quotient of X. Now let X be an arbitrary pro-simplicial set. The construction of limits in pro-categories given in Section 2.1 implies that Hom pro(limproaZa, cY ) = colimaHompro(Za, cY ) for any cofiltered system a 7! Za of pro-objects and any constant pro-object cY* * . The desired adjointness property for FinX now follows formally. The adjoint functors Finand i connect the categories of pro-simplicial sets a* *nd pro-simplicial finite sets. Now we have to connect the categories of pro-simpli* *cial finite sets and simplicial pro-finite sets. As described explicitly in [I1, x 3* *], there are functors F : pro-sF ! spro-F and G : spro-F ! pro-sF such that G is the left adjoint of F and such that the composition F G is naturally isomorphic to the identity on spro-F. This uses the fact that the category of simplicial fin* *ite sets is small and that the simplicial indexing category op has finite morphism sets. In other words, the category spro-F is a retract of the category pro-sF. * *As observed above, F and G are not inverse equivalences of categories because GF is not naturally isomorphic to the identity functor. The construction of G is complicated; fortunately we will not need the details here. For later reference, we describe the functor F . Let X be a pro-simplic* *ial finite set. For each n 0, Xn is a pro-finite set. Thus, [n] 7! Xn is a simpli* *cial pro-finite set, and this is F X. In order to pass between pro-simplicial sets and simplicial pro-finite sets, * *we use the compositions F O Fin and i O G. Unfortunately, these functors are the composition of a left adjoint and a right adjoint. Thus, they do not have nice adjointness properties. This means that we will not be able to produce a Quillen equivalence [Hi, Defn. 8.5.20] between pro-sSetand spro-F. One might hope that there is a Z=p-cohomology model structure on the inter- mediate category pro-sF. Then there would be a zig-zag of Quillen equivalences pro-sSet____//pro-sFoo_//_spro-F.oo_ However, since the category of simplicial finite sets is not a model category, * *the techniques used in this paper do not seem to apply. Possibly there is another approach altogether. In the absence of a Quillen equivalence, we have to show directly that the ho- motopy categories Ho(pro-sSet) and Ho(spro-F) are equivalent. Lemma 8.3. A map f of pro-simplicial finite sets is a Z=p-cohomology isomor- phism if and only if F f is a Z=p-cohomology isomorphism of simplicial pro-fini* *te 20 DANIEL C. ISAKSEN sets. A map g of simplicial pro-finite sets is a Z=p-cohomology isomorphism if * *and only if Gg is a Z=p-cohomology isomorphism of pro-simplicial finite sets. Proof.Let X be a pro-simplicial finite set. The cochain complex C*X used to compute H*(X; Z=p) is given by CnX = colimsHom ((Xs)n, Z=p). Using the de- scription of the functor F above, we see that this is equal to the cochain comp* *lex used to compute H*(F X; Z=p). This proves the first claim. For the second claim, let Y be a simplicial pro-finite set. We want to show t* *hat Y and GY have naturally isomorphic Z=p-cohomology. By the previous paragraph, it suffices to compare F Y and F GY . Now F GY is isomorphic to Y , so we just need to use the previous paragraph again. We have observed that the functor GF is not well-behaved categorically. Nev- ertheless, it does have good cohomological properties. Corollary 8.4. The counit natural transformation from the functor GF to the identity functor on pro-sF is a natural Z=p-cohomology isomorphism. Proof.If X is any pro-simplicial finite set, both parts of the proof of Lemma 8* *.3 imply that H*(X; Z=p) is isomorphic to H*(GF X; Z=p). Lemma 8.5. Let f : X ! Y be a map between pro-simplicial sets. Then f is a Z=p- cohomology isomorphism if and only if Fin(f) is a Z=p-cohomology isomorphism. Proof.We will show that for every pro-simplicial set X, the natural map X ! FinX is a Z=p-cohomology isomorphism. Because of Lemma 5.1 and the definition of Fin, it suffices to assume that X is a simplicial set. We must show that the natural* * map colimYHn(Y ; Z=p) ! Hn(X; Z=p) is an isomorphism, where Y ranges over all simplicial finite quotients of X. To* * do this, we consider reduced cochain complexes given in degree n by functions into Z=p from the non-degenerate part NXn of X in degree n. To show that the map of reduced cochain complexes is surjective, consider an arbitrary cochain ff, which is just a function NXn ! Z=p. We need to construct a simplicial finite quotient X0 of X and a cochain ff0 on X0 that pulls back to* * ff. Begin by defining an n-dimensional simplicial set Y whose (n-1)-skeleton is tri* *vial and whose non-degenerate n-simplices correspond to the elements of Z=p. There is an obvious map sknX ! Y induced by ff. Adjointness gives a map X ! cosknY . Since Y is a simplicial finite set, so is cosknY . Finally, take X0 to be the i* *mage in cosknY of X. To show that the map of reduced cochain complexes is injective, suppose that X0 and X00are two simplicial finite quotients of X, and let ff0and ff00be reduc* *ed cochains on X0 and X00respectively that pull back to the same reduced cochain ff on X. There exists a simplicial finite quotient Y of X refining both X0 and X00 COMPLETIONS OF PRO-SPACES 21 (see the proof of Lemma 8.2). We now have the diagram X1QQQB 1BBQQ 11BBQQQQB 11 BB QQQQ 11 !! ___QQQ((//_0 11 Y X 11 | | 0 11 | |ff 1,,fflffl|fflffl| X00 _ff00//_Z=p in which the outer quadrilateral and the two triangles are commutative. We want to show that the square is also commutative. This follows from the fact that the map X ! Y is surjective. Proposition 8.6. The functor F O Fininduces a functor Ho(pro-sSet) ! Ho(spro-F) on homotopy categories, and the functor i O G induces a functor Ho(spro-F) ! Ho(pro-sSet) on homotopy categories. Proof.By the universal property of localizations of categories, it suffices to * *show that the two functors take weak equivalences to weak equivalences. Let f be any Z=p-cohomology isomorphism of pro-simplicial sets. Lemmas 8.3 and 8.5 imply that F O Finf is a Z=p-cohomology isomorphism. Hence, F O Finpreserves weak equivalences. For i O G, this is the second part of Lemma 8.3. Theorem 8.7. The functors F O Finand i O G induce inverse equivalences between the homotopy categories Ho(pro-sSet) and Ho(spro-F). Proof.The composition (F O Fin) O (i O G) is isomorphic to the identity because FinO i is the identity by construction of Finand because F O G is isomorphic to* * the identity. On the other hand, Corollary 8.4 and Lemma 8.5 tell us that for every pro-simplicial set X, there are natural weak equivalences X __~__//FinXo~o_GF O FinX. Thus X and (i O G) O (F O Fin)X are naturally isomorphic in Ho(pro-sSet). 9. Free Groups The point of this section is to describe the both the cohomological and homo- logical Z=p-completions of the Eilenberg-Mac Lane space K(Fn, 1), where Fn is t* *he free group on n generators. This means that we need to find a fibrant replaceme* *nt for cK(Fn, 1) in the Z=p-cohomological model structure and the Z=p-homological model structure. As we will see below, these two completions turn out to be the same. Part of the definition of these fibrant replacements requires that the pro-sp* *ace be strictly fibrant. For the rest of this section, we will drop this requiremen* *t. This change preserves the homotopy type of each space in the cofiltered system becau* *se of the nature of strict weak equivalences. Thus, for calculational purposes we * *do not really need the strict fibrancy. 22 DANIEL C. ISAKSEN Theorem 9.1. Consider the system {K(Fn=H, 1)} as H ranges over all normal subgroups of Fn such that Fn=H is a finite p-group; the structure maps are in- duced by the canonical quotient maps. This pro-space is the fibrant replacement* * for K(Fn, 1) in either the Z=p-cohomological or Z=p-homological model structure. To be precise, we really should also produce a map cK(Fn, 1) ! X that induces an isomorphism in Z=p-cohomology. We will not worry about this because the map will be obvious and natural in everything that we do. Proof.For notational convenience, write XH for the space K(Fn=H, 1). We need to show that H*(X; M) is isomorphic to H*(Fn; M) for every Z=p-module M. This will show that X and K(Fn, 1) are weakly equivalent in both the Z=p-cohomologic* *al and Z=p-homological model structures. We also have to show that each space XH is nilpotent in the sense of Definition 3.1 with respect to the class of Eilenb* *erg-Mac Lane spaces of the form K(Z=p, n). This will show that X is fibrant in both mod* *el structures. First of all, the diagram X is a cofiltered system because Fn=(H \ K) is a fi* *nite p-group whenever Fn=H and Fn=K are finite p-groups. Since K(Fn, 1) is just a wedge of n circles, Hq(Fn; Z=p) is isomorphic to Z=p* * in dimension 0; to (Z=p)n in dimension 1; and to the zero group otherwise. This te* *lls us exactly what the cohomology of X should be. Let Z be any connected space whose homotopy groups are finite p-groups. Then Z is Z=p-nilpotent in the sense of Bousfield and Kan. This can be proved by showing that if G is a finite p-group acting on another finite p-group A, then G acts nilpotently. If in addition Z has only finitely many non-zero homotopy gro* *ups, then Z must be K-nilpotent in the sense of in the sense of Definition 3.1 becau* *se of [BK , Prop. III.5.3] and Proposition 3.5. Since each XH satisfies the hypotheses in the previous paragraph, we conclude that each XH is K-nilpotent. It remains to calculate the Z=p-cohomology of X. This is done below in Lemma 9.4. Lemma 9.2. There exists a normal subgroup H of Fn such that Fn=H is a finite p-group and such that the map H1(Fn; M) ! H1(H; M) is the zero map for all Z=p-modules M. Proof.By an argument similar to the one given after Definition 5.2, it suffices* * to consider the case M = Z=p. Let H be the kernel of the homomorphism Fn ! (Z=p)n that is the composition of abelianization with reduction modulo p. Now K(H, 1) ! K(Fn, 1) is a covering map of degree pn. More concretely, K(H, 1) is the Cayley graph of the group (Z=p)n relative to the standard basis. Thus, K(H,* * 1) has one vertex for each element of (Z=p)n. The edges of K(H, 1) are of the form* * v to v + ei, where v is any element of (Z=p)n and ei is any element of the standa* *rd basis. We use a wedge of n circles as our model for K(Fn, 1). Let ff be a 1-cocycle * *on K(Fn, 1) whose value on the ith circle of K(Fn, 1) is the element ffi of Z=p. L* *et fi be the 1-cocycle on K(H, 1) induced by ff. The value of fi on the edge from v to v + ei is ffi. We construct a 0-cocycle fl whose coboundary is fi. Let the value of fl on the vertex (v1, . .,.vn) of K(H, 1) equal ff1v1+. .+.ffnvn. Thus, fi is zero in coh* *omology, which means that the desired map is zero. COMPLETIONS OF PRO-SPACES 23 For each normal subgroup H of Fn such that Fn=H is a finite p-group, we have a short exact sequence H ! Fn ! Fn=H which gives rise to a fiber sequence K(H, 1) ! K(Fn, 1) ! K(Fn=H, 1). Each such sequence has an associated cohomological Serre spectral sequence Est2= Hs(Fn=H; Ht(H; M)) ) Hs+t(Fn; M), where Ht(H; M) is a local system on K(Fn=H, 1). Since the Serre spectral se- quence is natural and since filtered colimits respect filtrations, we can take * *colimits everywhere and get another spectral sequence Est2= colimHHs(Fn=H; Ht(H; M)) ) Hs+t(Fn; M). Recall how the structure maps of the colimit in the above formula are construct* *ed. If K is a subgroup of H, let ß be the projection Fn=K ! Fn=H. The map Hs(Fn=H; Ht(H; M)) ! Hs(Fn=K; Ht(K; M)) is the composition Hs(Fn=H; Ht(H; M)) ! Hs(Fn=K; ß*Ht(H; M)) ! Hs(Fn=K; Ht(K; M)), where ß*Ht(H; M) is a pullback of local systems. Lemma 9.3. The group colimHHs(Fn=H; Ht(H; M)) is zero unless t = 0. Proof.For t 2, each local system Ht(H; M) is zero because H is a free group. * *It only remains to consider the case t = 1. Because H is free, Lemma 9.2 implies t* *hat there exists a subgroup K such that the map ß*H1(H; M) ! H1(K; M) of local systems is zero. This gives the desired result for t = 1. Lemma 9.4. The map colimHHq(Fn=H; M) ! Hq(Fn; M) is an isomorphism, where the colimit ranges over all normal subgroups of Fn such that Fn=H is a fi* *nite p-group. Proof.By the previous lemma, the E2-term of the Serre spectral sequence describ* *ed above is concentrated on the line t = 0. This gives the desired isomorphism. 10.Questions The work in this paper leaves some obvious further questions unanswered. We mention a few of these here in the interest of encouraging future work on the s* *ubject. Question 10.1. Are the model structures of Theorems 1.1 and 1.2 are right prope* *r? The general machinery of localizations does not automatically produce right proper model structures. Presumably the Serre spectral sequence is the way to approach this problem, but one has to deal with twisted coefficients. Question 10.2. If X is a space considered as a constant pro-space, how do its t* *wo fibrant replacements compare? 24 DANIEL C. ISAKSEN We know that the model structures of Theorems 1.1 and 1.2 are distinct. How- ever, in Section 9 we showed that the two fibrant replacements of K(Fn, 1) are * *the same. It is easy to imagine that this would generalize to any space X with some kind of finiteness hypothesis on the cohomology of X. Question 10.3. If the ground ring R is Z=p, what is the difference between the two fibrant replacements of a pro-space that is an 'etale topological type? Certain kinds of pro-spaces are more relevant to applications than others. The pro-spaces that arise as 'etale topological types of well-behaved schemes [AM ]* * [F ] are particularly interesting. Perhaps theorems in algebraic geometry about 'et* *ale cohomology with finite coefficients can be used to conclude that the two fibrant replacements are the same. Question 10.4. Let R be an infinite ring. 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