A GENERALIZATION OF QUILLEN'S SMALL OBJECT ARGUMENT BORIS CHORNY Abstract.We generalize the small object argument in order to allow for i* *ts application to proper classes of maps (as opposed to sets of maps in Qui* *llen's small object argument). The necessity of such a generalization arose wi* *th the appearance of several important examples of model categories which w* *ere proven to be non-cofibrantly generated [2, 7, 8, 17]. Our current appro* *ach allows for the construction of functorial factorizations and localizatio* *ns in the equivariant model category on diagrams of spaces [10] and in two differe* *nt model structures on the category of pro-spaces [11, 17]. The examples above suggest a natural extension of the framework of cof* *i- brantly generated model categories. We introduce the concept of a class- cofibrantly generated model category, which is a model category generated by classes of cofibrations and trivial cofibrations satisfying some reas* *onable assumptions. 1.Introduction Quillen's definition of a model category has been slightly revised over the l* *ast decade. The changes applied to the first axiom MC1 requiring the existence of all finite limits and colimits, and to the last axiom MC5 requiring the existen* *ce of factorizations. The modern approaches to the subject [14, 16] demand the existe* *nce of all small limits and colimits in MC1. This gives some technical advantages w* *hile treating transfinite constructions, such as localizations, in model categories.* * The modern version of the axiom MC5 requires the factorizations to be functorial. Functoriality of the factorizations is a very important part of the structure* * of Quillen's model category. Most examples of model categories have functorial fac- torizations, and many works on abstract homotopy theory assume that condition. For example, there are two recent constructions of homotopy limits and colimits* * in abstract model categories equipped with functorial factorizations [5, 14]. The most widely known model category without functorial factorizations is the category of pro-spaces or, more generally, of pro-objects (in the sense of Grot* *hen- dieck) in a proper model category C [11, 19] and its Bousfield localization mod* *elling the 'etale homotopy theory [4, 17]. The construction of functorial factorizatio* *ns in these model categories was one of our main goals during the work on this paper. However, this task would not be accomplished without an observation that the we* *ll- known theorem of C.V. Meyer [21] implies immediately the existence of a functor* *ial replacement of a pro-map by a levelwise pro-map. Unfortunately this construction depends on the choice of a functor which is inverse to the equivalence of categ* *ories ____________ Date: January 28, 2004. 1991 Mathematics Subject Classification. Primary 55U35; Secondary 55P91, 18G* *55. Key words and phrases. model category, functorial factorization, pro-categor* *y. 1 2 BORIS CHORNY constructed by Meyer. It makes our construction less explicit and perhaps not applicable for concrete computations. We provide a partial compensation for this drawback by giving in Appendix A an explicit construction of functorial fibrant replacement in the pro-categories with the strict model structure or, more prec* *isely, by proving that Isaksen's construction of (non-functorial) factorizations becom* *es functorial when applied just to fibrant replacements. The purpose of the append* *ix is also to discuss whether the construction of factorizations in [19] can be ma* *de functorial, provided that now we have the functorial levelwise replacements. We* * do not arrive at a definite conclusion to this question, but we show the specific * *point in the proof which distinguishes between the simpler case of fibrant replacemen* *ts and the case of general factorizations. The main tool for the construction of functorial factorizations in model cate* *gories and localizations thereof is Quillen's small object argument [14, 16, 22]. Howe* *ver, in its original form, the argument is applicable neither to the category of dia* *grams with the equivariant model structure [10], nor to pro-categories, since it allo* *ws for the application in cofibrantly generated model categories only. We propose here a generalization which may be used in a wider class of model categories. The collections of generating cofibrations and generating trivial cofibrations may * *now form proper classes, satisfying the conditions of the following theorem: Theorem 1.1 (The generalized small object argument). Suppose C is a category containing all small colimits, and I is a class of maps in C satisfying the fol* *lowing conditions: (1) There exists a cardinal ~, such that each element A 2 dom (I) is ~-small relative to I-cof; (2) There exists a functor S :Map C ! Map C equipped with an augmenta- tion t: S ! IdMapC, such that S(f) 2 I-coffor every f 2 Map C and any morphism of maps i ! f with i 2 I factors through the natural map t(f): S(f) ! f. Then there is a functorial factorization (fl, ffi) on C such that, for all morp* *hisms f in C, the map fl(f) is in I-cof and the map ffi(f) is in I-inj. We say that a class I of maps in a category C permits the generalized small o* *bject argument if it satisfies conditions (1) and (2) of Theorem 1.1. This theorem is the second attempt by the author to generalize the small obje* *ct argument. The previous version appeared in the study of the equivariant localiz* *a- tions of diagrams of spaces [6]. The specific properties of the equivariant mo* *del category of D-shaped diagrams of spaces and also the non-functorial factorizati* *on technique developed by E. Dror Farjoun in [10] suggested the rather complicated technical notion of instrumentation. It is essentially a straightforward üf nct* *orial- izationö f Dror Farjoun's ideas. The classes of generating cofibrations and g* *en- erating trivial cofibrations of diagrams satisfy the conditions of instrumentat* *ion, but we were unable to apply the same version of the argument to the category of pro-spaces. The conditions of Theorem 1.1 on the class I of maps generalize tho* *se of instrumentation, as we explain in Section 3. Therefore, this paper shows that the two rather different homotopy theories of pro-spaces and of diagrams of spaces fit into a certain joint framework. In ord* *er to describe the similarity between the two cases let us give the following Definition 1.2. A model category C is called class-cofibrantly generated if GENERALIZED SMALL OBJECT ARGUMENT 3 (1) there exists a class I of maps in C (called a class of generating cofibr* *ations) that permits the generalized small object argument and such that a map is a trivial fibration if and only if it has the right lifting property wit* *h respect to every element of I, and (2) there exists a class J of maps in C (called a class of generating trivial cofibrations) that permits the generalized small object argument and such that a map is a fibration if and only if it has the right lifting proper* *ty with respect to every element of J. Obviously, the class-cofibrantly generated model categories are equipped with functorial factorizations. The categorical dual to a class-cofibrantly generat* *ed model category is called class-fibrantly generated. The purpose of this paper is to show that the equivariant model structure on the diagrams of spaces is class-cofibrantly generated and the both known model structures on the category of pro-spaces are class-fibrantly generated. The applications of Quillen's small object argument are not limited to abstra* *ct homotopy theory. A similar argument is used, for example, in the theory of cat- egories to construct reflections in a locally presentable category with respect* * to a small orthogonality class [3, 1.36]. Recently another generalization of the sm* *all object argument was considered by the category theorists J. Ad'amek , H. Herrli* *ch, J. Rosick'y and W. Tholen [1]. Their version of the argument applies to the "in- jective subcategory problem" in locally ranked categories - a generalization of* * the notion of a locally presentable category which includes topological spaces. We * *hope that our generalization of the small object argument will be applicable to the * *ö r- thogonal subcategory problemä nd "injective subcategory problem" with respect to some reasonable classes of morphisms. The rest of the paper is organized as follows: Section 2 is devoted to the proof of the generalized cosmall object argument. Next, we review some of our previous results about the diagrams of spaces in Section 3 and show how they fit into the newly established framework. After providing the necessary preliminaries on pro- categories in Section 4 we give our main applications of the generalized (co)sm* *all object argument in Section 5. Appendix A is devoted to an alternative, explicit construction of a functorial fibrant replacement in pro-C. This construction is* * based on the construction of factorizations given by D. Isaksen [19]. We also discuss* * the difficulty which arises while trying to check whether the construction of gener* *al factorizations in [19] is functorial. Acknowledgements. I would like to thank Dan Isaksen for many fruitful conversa- tions and suggestions for improving this paper. In particular I owe him the ide* *a of the proof of Theorem 5.3. 2.Proof of the generalized small object argument Proof of Theorem 1.1.Given a cardinal ~ such that every domain of I is ~-small relative to I-cof, we let ~ be a ~-filtered ordinal. To any map f :X ! Y we will associate a functor Zf :~ ! C such that Zf0= X, and a natural transformation æf: Zf ! Y factoring f, i.e., for each fi < ~ the 4 BORIS CHORNY triangle f X ?``````````//`???"Y ?? """ ?ØØ " æffi Zffi is commutative. Each map iffi:Zffi! Zffi+1will be a pushout of a map of the form S(f), i.e., iffi2 I-cof, since I-cof is closed under pushouts. We will define Zf and æf: Zf ! Y by transfinite induction, beginning with Zf0= X and æf0= f. If we have defined Zfffand æffffor all ff < fi for some limit ordinal fi, define Zffi= colimff> q | ` | >> q |fflO qq | | >> k| q | | >>> fflffl|qq | {fi}|| >>> ZkDD g2M| | >>> | || | >> |pk | fflffl| >>> fflfflfflffl| fflfflfflffl| {Yi}_____________>>__________________//_Yk_________//_BDD<< ?? >> x ?? >> x x ?? >> x ?? >ØØ x ?? x X x ?? i i x ?? | x ?? xiqi x ?? |||fflffl|||||||x|||| ?? ||| ||| x ?? |xiZi||||| x x ??u||||uu|||x| u??|x|p||||x|uuuu uuuuu??i|i|x|||||| uuu |?ØØfflfflfflffl|x||||||| S({fi}) |xiYi|||||||||||||||||||| In order to verify the factorization property, fix an arbitrary map {fi} ! g * *with g 2 M being a fibration between constant objects. It follows immediately from the definition of the morphism set between two pro-objects that any map into a constant pro-object factors through a map of the form 'ifor some i 2 I. Applying this to the category pro-Map (C) ~=Map (pro-C), we find an index k 2 I such that the fixed map {fi} ! g factors through Xk ! Yk. In the diagram above the maps xiZi! Zk and xiYi! Yk are projections. The dashed map xiYi! B is the composition of the projection with the map Yk ! B. Finally, the dashed map Zk ! A is a lifting in the commutative square, which exists in the model category C. Now we are able to apply the generalized cosmall object argument to produce for every map in pro-C its functorial factorization into a trivial cofibration * *followed by a fibration. To obtain the second factorization we repeat the construction above for the c* *lass N of trivial fibrations between constant objects and factorize all the maps Xi!* * Yi into cofibrations followed by trivial fibrations. Hence the cosmall object argument may be applied to provide every map f with a functorial factorization into a cofibration followed by a trivial fibrat* *ion in pro-C. GENERALIZED SMALL OBJECT ARGUMENT 11 Let S denote the category of simplicial sets with the standard model structur* *e. Our next goal is to construct functorial factorizations in the localized model * *struc- ture of [17]. From now on the words cofibration, fibration and weak equivalence refer to Isaksen's model structure. Since the procedure of (left Bousfield) localization preserves the class of c* *ofibra- tions and, hence, the class of trivial fibrations, the functorial factorization* * into a cofibration followed by a trivial fibration was constructed in the theorem abov* *e. We keep the class N of generating trivial fibrations the same as in the strict * *model structure. The class of generating fibrations L is defined to be the class of * *all co-n-fibrations (see [17, Def. 3.2]) between constant pro-objects for all n 2 N. Proposition 5.2. The class of trivial cofibrations equals L-proj. Proof.Every element of L-proj is a strong fibration [17, Def. 6.5]; therefore e* *very trivial cofibration has the left lifting property with respect to L by [17, Pro* *p. 14.5]. Conversely, if a map i has the left lifting property with respect to L, then i * *has the left lifting property with respect to the class L0of all retracts of L-cocell c* *omplexes. By [19, Prop.5.2] L0contains all strong fibrations. But then [17, Prop. 6.6] im* *plies that L0contains all fibrations. Therefore, i must be a trivial cofibration. Theorem 5.3. Isaksen's model structure on pro-S may be equipped with functorial factorizations. Proof.It suffices to construct a functorial factorization of every morphism of * *pro-S into a trivial cofibration followed by a fibration. We apply the same construct* *ion as in Theorem 5.1 to the class L, except for the factorizations of the levelwise representation {fi}. Apply first the Marde~si'c functor in order to guarantee that our pro-system * *is indexed by a cofinite strongly directed set. Since the Marde~si'c functor is na* *turally isomorphic to the identity, we abuse notation and keep calling the indexing cat* *egory I. We construct the factorizations of the maps fi by induction on the number n(* *i) of predecessors of i and factor fi into an n(i)-cofibration qi followed by a co* *-n(i)- fibration, which is possible by [17, Prop. 3.3]. This an induction on the set * *of natural numbers, since I is now cofinite. For any element g :A ! B of L, there is a number n 2 N such that g is a co-n-fibration. We may always enlarge k such that n(k) n and hence qk will be an n-cofibration by [17, Lemma 3.6]. Finally, the lift in the commutative squar* *e in the diagram in the proof of Theorem 5.1 exists by [17, Def. 3.2]. Appendix A. An explicit construction of a functorial fibrant replacement in pro-C The purpose of this appendix is to give an explicit construction of functorial fibrant replacements in the strict model category on pro-C. More precisely, the purpose is to prove that the construction of fibrant replacements in [19] is fu* *nctorial. We do not know whether the construction of arbitrary factorizations is functori* *al in [19], but in view of Remark A.2 it seems unlikely. Proposition A.1. If C is a proper model category with functorial factorizations, then there exists a functorial fibrant replacement X,~!^Xin the category pro-C * *with the strict model structure. 12 BORIS CHORNY Proof.Given a pro-object X, we apply first the Marde~si'c functor in order to r* *eplace it by an isomorphic pro-object M(X) indexed by a cofinite strongly directed set. Since M( . ) is naturally isomorphic to the identity functor, we suppress the n* *otation and quietly assume that all pro-objects are indexed by cofinite strongly direct* *ed sets. Our objective is to find for every pro-object X indexed by a cofinite strongly directed set S, a functorial factorization of the map f :X ! * into a trivial c* *ofi- bration i: X,~!^Xfollowed by a fibration p: ^Xi * in the strict model structure* * on pro-C. We apply the factorization algorithm of [19], with a mild alteration, on* * the simplest level representation of f: (idS, {Xs ! *}). We define X^ by induction. The only difference between the current construc- tion and [11, x4.3] [19] is our vision of the bonding maps of X^. In the origi* *nal construction they were induced by the fibrations pk below. For every s0 2 S such that there are no s 2 S with s s0, define ^Xs0by appl* *ying the functorial factorization of C on the map Xs0! *: "~ Xs0 Øis__//^Xs0`p////_lim{s l > m. We define X^kand ik by applying the functorial factorization of C on the map Xk ! lims