STRICT MODEL STRUCTURES FOR PRO-CATEGORIES DANIEL C. ISAKSEN Abstract.We show that if C is a proper model category, then the pro-cate* *gory pro-C has a strict model structure in which the weak equivalences are th* *e lev- elwise weak equivalences. This is related to a major result of [10]. The* * strict model structure is the starting point for many homotopy theories of pro-* *objects such as those described in [5], [17], and [19]. 1. Introduction If C is a category, then the category pro-C has as objects all cofiltered dia* *grams in C and has morphisms defined by Hom pro-C(X, Y ) = limtcolimsHomC(Xs, Yt). Pro-categories have found many uses over the years in fields such as algebraic * *ge- ometry [2], shape theory [20], geometric topology [6], and possibly even applied mathematics [7, Appendix]. When working with pro-categories, one would frequently like to have a homotopy theory of pro-objects. The first attempts at this appear in [2] and [24] in wh* *ich pro-objects in homotopy categories are considered. The difficulty with this app* *roach is that the diagrams commute only up to homotopy, and this makes it virtually impossible to make sense of most of the standard notions of homotopy theory in * *this context. Much better is to first consider actually commuting cofiltered diagrams (of s* *paces or simplicial sets or spectra or whatever) and then to define a notion of weak * *equiva- lence between such pro-objects. This approach was first taken by [12] in a rest* *ricted context. It was also applied much more generally in [10]. The idea is to start with a * *model structure (i.e., a homotopy theory) on a category C and then to construct a str* *ict model structure on pro-C in which the weak equivalences are more or less just t* *he levelwise weak equivalences. The resulting homotopy theory is precisely suited* * to study homotopy limits [4, Ch. XI] of cofiltered diagrams. The strict model stru* *cture is a starting point for several other model structures such as those described * *in [5], [17], and [19]. The strict model structure on pro-C does not seem to exist for a completely a* *rbi- trary model category C. A niceness hypothesis was required in [10, p. 45] (whic* *h was weakened in [13]). Unfortunately, many important examples of model categories, such as the usual models for spectra, do not satisfy this hypothesis. The main * *pur- pose of this paper is to prove that the strict structure on pro-C exists whenev* *er C ____________ Date: August 24, 2001. 1991 Mathematics Subject Classification. 18G55, 55U35. Key words and phrases. Closed model structures, pro-homotopy theory, pro-spa* *ces. 1 2 DANIEL C. ISAKSEN is a proper model category. Almost all of the most important examples of model categories are proper. Another problem with [10] is that a non-standard set of axioms for model stru* *c- tures are used. From a modern perspective, it is harder to comprehend the techn* *ical details of [10] than to simply work out new proofs from scratch. The secondary * *goal of this paper is to give these new more modern proofs. Oddly, the two-out-of-th* *ree axiom is the most difficult part of the proof; in most model structures, it is * *automatic from the definition of weak equivalence. The last goal of the paper is to consider whether strict model structures on * *pro- categories are fibrantly generated. It is already known that these model struct* *ures are not cofibrantly generated in general, even when C is [17, x19]. We produce reasonable collections of generating fibrations and generating acyclic fibratio* *ns that have cosmall codomains, but these collections are not sets. In fact, the strict* * model structure for pro-simplicial sets is not fibrantly generated. We show that if t* *his strict structure were fibrantly generated, then in the category of simplicial sets the* *re would exist a set of fibrations that detect acyclic cofibrations. The paper is organized as follows. First we introduce the language of pro-cat* *egories and give some background results. Then we define the strict weak equivalences a* *nd prove that they satisfy the two-out-of-three axiom when C is proper. Next we pr* *ove that the strict model structure exists when C is proper. Finally, we consider w* *hether the strict model structure is fibrantly generated. We assume familiarity with model categories. The original reference is [23], * *but we follow the notation and terminology of [14] as closely as possible. Other re* *ferences include [9] and [15]. 2.Preliminaries on Pro-Categories We begin with a review of the necessary background on pro-categories. This material can be found in [1], [2], [8], [10], and [18]. 2.1. 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). Composition is defined in the natural way. A category I is cofiltering if the following conditions hold: it is non-empty* * and small; for every pair of objects s and t in I, there exists an object u togethe* *r with maps u ! s and u ! t; and for every pair of morphisms f and g with the same source and target, there exists a morphism h such that fh equals gh. Recall tha* *t a category is small if it has only a set of objects and a set of morphisms. A dia* *gram is said to be cofiltering if its indexing category is so. Beware that some mate* *rial on pro-categories, such as [2] and [21], consider cofiltering categories that are * *not small. All of our pro-objects will be indexed by small categories. Objects of pro-C are functors from cofiltering categories to C. We use both * *set theoretic and categorical language to discuss indexing categories; hence "t s* *ä nd "t ! s" mean the same thing when the indexing category is actually a directed s* *et. The word pro-object refers to objects of pro-categories. A constant pro-obje* *ct is one indexed by the category with one object and one (identity) map. Let c : STRICT MODEL STRUCTURES FOR PRO-CATEGORIES 3 C ! pro-C be the functor taking an object X to the constant pro-object with val* *ue 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 limpro* *for limits computed within the category pro-C. Let Y : I ! C and X : J ! C be arbitrary pro-objects. We say that X is cofinal in Y if there is a cofinal functor F : J ! I such that X is equal to the compos* *ite Y F . This means that for every s in I, the overcategory F # s is cofiltered. I* *n the case when F is an inclusion of directed sets, F is cofinal if and only if for e* *very s in I there exists t in J such that t s. The importance of this definition is tha* *t X is isomorphic to Y in pro-C. A level representation of a map f : X ! Y is: a cofiltered index category I; pro-objects ~Xand ~Yindexed by I and pro-isomorphisms X ! ~Xand Y ! ~Y; and a collection of maps fs : ~Xs! ~Ysfor all s in I such that for all t ! s in I, * *there is a commutative diagram X~t_____//~Yt | | | | fflffl| fflffl| X~s_____//~Ys and such that the maps fs represent a map ~f: ~X! ~Ybelonging to a commutative square f X _____//Y | | | | fflffl|fflffl| ~X__~__//~Y f in pro-C. That is, a level representation is just a natural transformation such* * that the maps fs represent the element f of limscolimtHomC(Xt, Ys) ~=limscolimtHomC(X~t, ~Ys). Every map has a level representation [2, App. 3.2] [21]. More generally, suppose given any diagram A ! pro-C : a 7! Xa. A level representation of X is: a cofiltered index category I; a functor X~: A x I ! C : (a, s) 7! X~as; and pro-isomorphisms Xa ! X~asuch that for every map OE : a ! b in A, X~OEis a level representation for XOE. In other words, X~ is a uniform l* *evel representation for all the maps in the diagram X. Not every diagram of pro-objects has a level representation. However, finite * *di- agrams without loops do have level representations. This makes computations of limits and colimits of such diagrams in pro-C relatively straightforward. To co* *mpute this limit or colimit, just take the levelwise limit or colimit of the level re* *presentation [2, App. 4.2]. 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 repre* *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. The following surprisingly g* *eneral 4 DANIEL C. ISAKSEN and very useful proposition about retracts of essentially levelwise maps is pro* *ved in [18, Thm. 5.5]. Proposition 2.2. Let C be any class of maps in a category C. Then retracts pres* *erve the class of maps in pro-C that belong to C essentially levelwise. 2.2. Cofiniteness. A directed set (I, ) is cofinite if for every t, the set of* * elements s of I such that s t is finite. A pro-object or level representation is cofi* *nite directed if it is indexed by a cofinite directed set. For every cofiltered category I, there exists a cofinite directed set J and a* * cofinal functor J ! I [10, Th. 2.1.6] (or [1, Expos'e 1, 8.1.6]). Therefore, every pro-* *object is isomorphic to a cofinite directed pro-object. Similarly, every map has a cof* *inite directed level representation. Thus, it is possible to restrict the definition * *of a pro- object to only consider cofinite directed sets as index categories, but we find* * this unnatural for general definitions and constructions. On the other hand, we find it much easier to work with cofinite directed pro-objects in practice. Thus, m* *ost of our results start by assuming without loss of generality that a pro-object i* *s in- dexed by a cofinite directed set. Cofiniteness is critical because many argumen* *ts and constructions proceed inductively. Definition 2.3. Let f : X ! Y be a cofinite directed level representation of a * *map in a pro-category. For every index t, the relative matching map Mtf is the map Xt! lims s and a map hts: Yt ! Xs belonging to* * a commutative diagram Xt _____//_Yt | ____| | __ | fflffl|"fflffl|"__ Xs ____//_Ys. In effect, the maps htsrepresent the inverse of f. By restricting to cofinal su* *bsets, we may assume that such a diagram exists for every t > s. We choose a map hts: Yt! Xs for each pair t > s. Factor each map fs : Xs ! Ys into a cofibration Xs ! Zs followed by a fibrati* *on Zs ! Ys. We shall define structure maps making Z into a pro-object. Define a category J whose objects are the elements of I and whose morphisms t ! s are finite chains t = u0 > u1 > . .>.un = s. Composition is defined by concatenation of chains. Note that J is not a directed set; it is not even cofi* *ltered. For every morphism t = u0 > u1 > . .>.un = s in J, define a map Zt ! Zs by the composition hu0u1 Zt____//_Yt= Yu0__//_Xu1___//Xu2___//._._._//Xun = Xs___//Zs. A diagram chase shows that this makes Z into a diagram indexed by J. There may be more than one map from a given Zt to a given Zs, but another diagram chase shows that they become equal after composition with some map Zu ! Zt. Consider the category K defined to be a quotient of J as follows. The objec* *ts of K are the same as the objects of J, and two morphisms from t to s in J are iden* *tified in K if the corresponding maps from Zt to Zs are equal in C. Now K is a cofilte* *red category, and we may consider it as the indexing category of Z. The projection functor K ! I is cofinal, so we may reindex X and Y along this functor. More diagram chases show that the maps Xs ! Zs and Zs ! Ys assemble into level representations X ! Z and Z ! Y . It remains only to show that these maps are pro-isomorphisms. This follows from the commutative diagrams Xt _____//Zt Zt _____//Yt | ____| | """| | __ | | "" | |fflffl~|fflffl~__fflffl|fflffl|~~"" Xs _____//Zs Zs _____//Ys for every pair t > s. The diagonal maps in the above diagrams are the compositi* *ons Zt! Yt! Xs and Yt! Xs ! Zs respectively. Remark 3.4. There is a more obvious argument where a level representation X ! Y is functorially factored into a levelwise cofibration X ! Z followed by a level* *wise 6 DANIEL C. ISAKSEN fibration Z ! Y . This does not give the desired factorization; the structure m* *aps for Z are wrong. The following lemma appears in [17, Prop. 10.4], but the previous lemma makes the technical details of that proof clearer. Lemma 3.5. If C is a proper model category, then the strict weak equivalences of pro-C are closed under composition. Proof.It suffices to assume that there is a cofinite directed level representat* *ion for the diagram f h g X ____//_Yoo__Z _____//W in which f and g are levelwise weak equivalences while h is a pro-isomorphism (* *but not a levelwise isomorphism). We must construct a levelwise weak equivalence is* *o- morphic to the composition gh-1f. By Lemma 3.2, after reindexing we can factor h : Z ! Y into a levelwise cofib* *ra- tion Z ! A followed by a levelwise fibration A ! Y such that X ! Y A Z ! W is a level representation in which the first and fourth maps are levelwise weak* * equiv- alences, and the second and third are pro-isomorphisms. Let B be the pullback X xY A, and let C be the pushout AqZ W . Since pushouts and pullbacks can be constructed levelwise, the maps B ! A and A ! C are levelwise weak equivalences. Here we use that the model structure is left and r* *ight proper. Moreover, the maps B ! X and W ! C are pro-isomorphisms since base and cobase changes preserve isomorphisms. Hence the composition B ! C is the desired levelwise weak equivalence. Lemma 3.6. Suppose that C is a proper model category, and let f and g be two composable maps in pro-C. If g and gf are strict weak equivalences, then f is a strict weak equivalence. If f and gf are strict weak equivalences, then g is a * *strict weak equivalence. Proof.We first prove the first claim. We may consider a cofinite directed level representation f X ____//_Y | | | g| fflffl| fflffl| W _____//Z where the vertical maps are levelwise weak equivalences and the bottom horizont* *al map is a pro-isomorphism (but not a levelwise isomorphism). We want to show that the top horizontal map is an essentially levelwise weak equivalence. By Lemma 3.2, after reindexing there exists a level representation X _____//B____//_Y | | | | | | fflffl|fflffl| fflffl| W ____//_A___//_Z such that B is the levelwise pullback A xZ Y ; the first and third vertical map* *s are levelwise weak equivalences; the map W ! A is a levelwise acyclic cofibration a* *nd a pro-isomorphism; and the map A ! Z is a levelwise fibration and a pro-isomorphi* *sm. STRICT MODEL STRUCTURES FOR PRO-CATEGORIES 7 Because of right properness, the map B ! A is also a levelwise weak equivalence. By the two-out-of-three axiom in C, the induced map X ! B is a levelwise weak equivalence. On the other hand, the map B ! Y is an isomorphism because base changes preserve isomorphisms. Hence X ! B is isomorphic to f. The proof of the second claim is similar. We start with a cofinite directed l* *evel representation X _____//W f || || fflffl|fflffl| Y __g__//_Z where the vertical maps are levelwise weak equivalences and the top horizontal * *map is a pro-isomorphism. We want to show that the bottom horizontal map is an essentially levelwise weak equivalence. We produce a level representation X _____//A____//W | | | | | | fflffl|fflffl| fflffl| Y _____//B____//Z such that B is the levelwise pushout A qX Y ; the first and third vertical maps are levelwise weak equivalences; the map A ! W is a levelwise acyclic fibration and a pro-isomorphism; and the map X ! A is a levelwise cofibration and a pro- isomorphism. The map B ! Z is the desired level representation. The above lemmas imply that our definition of strict weak equivalences agrees with the definition of [10, x3.3]. Proposition 3.7. When C is a proper model category, the strict weak equivalences of Definition 3.1 agree with the weak equivalences of [10, x3.3]. Proof.The weak equivalences of [10, x3.3] are by definition compositions of ess* *en- tially levelwise weak equivalences. By Lemma 3.5, these compositions are again essentially levelwise weak equivalences. On the other hand, every levelwise we* *ak equivalence can be factored into a levelwise acyclic cofibration followed by a * *levelwise acyclic fibration. Therefore, every levelwise weak equivalence is a weak equiva* *lence in the sense of [10, x3.3]. The preceding proposition is closely related to the main result of [22]. Howe* *ver, we make a useful observation missed there. Namely, it is not necessary to satur* *ate the essentially levelwise weak equivalences; they are already saturated when C * *is proper. 4. Strict Model Structures Beginning with a proper model structure on a category C, we now establish a model structure on the category pro-C. Definition 4.1. The strict cofibrations of pro-C are the essentially levelwise * *cofi- brations. Definition 4.2. A map in pro-C is a special fibration if it has a cofinite dire* *cted level representation p for which every relative matching map Msp is a fibration* *. A map in pro-C is a fibration if it is a retract of a special fibration. 8 DANIEL C. ISAKSEN In order to help us understand these definitions, we need some auxiliary noti* *ons. Definition 4.3. A map in pro-C is a special acyclic fibration if it has a cofin* *ite directed level representation p for which every relative matching map Msp is an acyclic fibration. Remark 4.4. Every special acyclic fibration is a special fibration, so it is al* *so a strict fibration. We shall see below in Proposition 4.14 that the class of strict acy* *clic fibrations is equal to the class of retracts of special acyclic fibrations. Lemma 4.5. Special acyclic fibrations are essentially levelwise acyclic fibrati* *ons. In particular, they are strict acyclic fibrations. Proof.Special acyclic fibrations are special fibrations, so they are strict fib* *rations by definition. This means that the second statement follows from the first. Suppose given a cofinite directed level representation p : X ! Y for which ea* *ch Msp is an acyclic fibration. The map ps : Xs ! Ys factors as Xs_Msp_//Ysxlimt