LOCALIZATION WITH RESPECT TO A CLASS OF MAPS I - EQUIVARIANT LOCALIZATION OF DIAGRAMS OF SPACES BORIS CHORNY Abstract.Homotopical localizations with respect to a set of maps are kno* *wn to exist in cofibrantly generated model categories (satisfying additiona* *l as- sumptions) [3, 12, 20, 29]. In this paper we expand the existing framewo* *rk, so that it will apply to not necessarily cofibrantly generated model cat* *egories and, more important, will allow for a localization with respect to a cla* *ss of maps (satisfying some restrictive conditions). We illustrate our technique by applying it to the equivariant model ca* *tegory of diagrams of spaces [11]. This model category is not cofibrantly gener* *ated [7]. We give conditions on a class of maps which ensure the existence of* * the localization functor; these conditions are satisfied by any set of maps * *and by the classes of maps which induce ordinary localizations on the generaliz* *ed fixed-points sets. Introduction Homotopy idempotent constructions, or homotopical localizations, play an im- portant role in algebraic topology and algebraic geometry. Homotopical localiza* *tion is a functor L in a model category that carries weak equivalences into weak equ* *iva- lences and is equipped with the natural transformation Id ! L, that induces weak equivalences LX ' LLX for all X. The idea behind such construction is to öf rget information" in a consistent, functorial way. The amount of the information öf rgotten" by L is measured pre- cisely by the class of maps SL which are turned into weak equivalences by L. In practice, we usually know what kind of information we would like to discar* *d, i.e. SL, and we are looking for the functor L. E.g., SL = {homological equivale* *nces of spaces}; and L is Bousfield's localization functor [2]. If we are able to encode the "informative contentö f SL by just a set of maps S SL, and if the underlying model category is cofibrantly generated (plus some other conditions), then the construction of the localization functor L is given* * by the classical framework established by A.K Bousfield, E. Dror Farjoun, P.S. Hirschh* *orn, J. Smith [3, 12, 20, 29]. But if we need to invert a proper class of maps or if we happen to work in a non-cofibrantly generated model category, then the standard framework does not guarantee the existence of the localization functor. Such situations are n* *ot rare. Recently several important model categories were shown to be non-cofibran* *tly ____________ Date: December 10, 2003. 1991 Mathematics Subject Classification. Primary 55U35; Secondary 55P91, 18G* *55. Key words and phrases. model category, localization, equivariant homotopy. During the preparation of this paper the author was a fellow of Marie Curie * *Training Site hosted by Centre de Recerca Matem`atica (Barcelona), grant no. HPMT-CT-2000-000* *75 of the European Commission. 1 2 BORIS CHORNY generated [1, 7, 9, 23]. Also the existence of the localization functors with r* *espect to the class of cohomological equivalences of spaces is a long-standing open probl* *em. In this work we develop a new approach that allows for construction of homo- topical localizations with respect to a class of maps (satisfying some conditio* *ns) in a not necessarily cofibrantly generated model category. As an application, we g* *ive an example of a non-cofibrantly generated model category (taken from [7]) and a class of maps which fits into our framework but is not covered by the classical* * lo- calization frameworks. In our example the model structure is generated by class* *es of cofibrations and trivial cofibrations which carry an additional structure su* *fficient to guarantee the applicability of a version of Quillen's small object argument.* * The set of conditions on a class of maps (in an abstract category) that allows one * *to apply the generalized version of the argument is given the name instrumentation. The main example of this paper involves homotopical localizations in the equi- variant homotopy theory. Classical Bredon's homotopy theory of spaces with a group action was generalized to the equivariant homotopy theory over an arbitra* *ry small category D by E. Dror Farjoun and A. Zabrodsky in the mid-80's [13, 10, 1* *1]. The question of the existence of equivariant localization functors was consid* *ered previously by V. Halperin [19]. Only strong equivariant localizations were con- structed in that work, which are not localizations in any familiar model catego* *ry on diagrams of spaces. Organization of the paper. In the preliminary section we recall basic definition and results about the homotopy theory of diagrams of spaces. We also make expli* *cit the connection between two notions of orbits: Dwyer-Kan [17] and Dror Farjoun- Zabrodsky [13, 11]. Namely the orbits in the later works are also orbits from t* *he first work. In Section 2 we develop a formalism necessary to describe the additional stru* *cture carried by a class of maps in order to satisfy the conditions of Quillen's smal* *l object argument. We introduce here the central notion of instrumentation of a class of maps. In Section 3 we illustrate this notion by constructing instrumentations o* *n the classes of generating cofibrations and trivial cofibrations in the category of * *diagrams of spaces equipped with the equivariant model structure of [11]. Section 4 is devoted to the proof of the generalized Quillen small object arg* *ument, which also finishes the proof that the factorizations in the model category of * *[11] are functorial. Before turning to the construction of localizations in the equivariant model * *cate- gory of D-shaped diagrams of spaces we prove in Section 5 that this model categ* *ory is proper. We also answer affirmatively the question posted by E. Dror Farjoun * *in [11, 2.3] and show that in the category of diagrams of simplicial sets every ob* *ject is cofibrant. In Section 6 we list the properties required from a class of maps in the equi* *variant category of diagrams in order to ensure the existence of the localization funct* *or with respect to that class of maps. Afterwards we construct these localization funct* *ors and prove their basic properties. Finally, in Section 7 we apply the technique of Section 6 in order to constru* *ct the fixed-pointwise localization functor with respect to a map of spaces (for the c* *ase of G-spaces, where G is a compact Lie group, such localizations were constructed by J. P. May, J. McClure and G. Triantafillou [26] with respect to the ordinary non-equivariant homology theory). EQUIVARIANT LOCALIZATION OF DIAGRAMS 3 In Appendix A we discuss the notion of contractible objects in a general model category. Their properties were useful in the proof of the properties of locali* *zation functors in Section 6. Acknowledgements. I would like to thank Emmanuel Dror Farjoun for his support and many helpful ideas. I am grateful to Carles Casacuberta who has patiently read and helped me to improve the early version of this paper. This paper was significantly influenced by the fundamental treatise of P. S. * *Hir- schhorn [20]. I am obliged to the author who made available through the internet the early versions of his manuscript. 1. Preliminaries on the diagrams of spaces In this section we review the equivariant homotopy theory of diagrams of spac* *es and establish basic notation. The only novelty introduced here is that an orbit diagram in the sense of Dror Farjoun-Zabrodsky [10, 11, 13] is also an orbit in* * the sense of Dwyer-Kan [17]. The topological (versus simplicial) version of the main proposition appeared previously in [5]. In this paper the category of spaces S is the category of simplicial sets or * *com- pactly generated topological spaces with the standard model structure. Most of * *the results are true in the category of pointed spaces. If D is a small category en* *riched over S, then the category of (D-shaped) diagrams of spaces SD is the category * *of continuous functors from D to S with natural transformations as morphisms. SD is a simplicial category and we denote by hom(., .) the simplicial function com* *plex; homD (., .) will denote the set of morphisms in SD . The two functors are relat* *ed as follows: hom (Xe, Ye)n = homD (Xe n, Ye). There are several well-known model structures on the category of diagrams of spaces. One of the most widely used is the Bousfield-Kan model category, in whi* *ch weak equivalences and fibrations are objectwise and cofibrations are obtained by the left lifting property with respect to trivial fibrations. Another example * *(not used in this paper) is A. Heller's model category, in which weak equivalences a* *nd cofibrations are objectwise and fibrations are obtained by the right lifting pr* *operty with respect to trivial cofibrations. These model categories are cofibrantly ge* *ner- ated. In this article we work mostly with the equivariant model structure on the category of diagrams of spaces constructed by E. Dror Farjoun [11] and described in Definition 1.2. Recall from [17] that a set {Oe}e2E of objects of a category M, enriched over simplicial sets, is said to be a set of orbits for M if the following axioms ho* *ld: Q0: M is closed under arbitrary direct limits. op Q1: For every e 2 E, the functor hom (Oe, .): M ! Sets commutes, up to homotopy, with the pushouts of the following form: Oe0 "K`______//Xa | | | | fflffl| fflffl| Oe0 L _____//Xa+1 where K ,! L is an inclusion of finite simplicial sets. 4 BORIS CHORNY Q2: For every e 2 E, the functor hom (Oe, .): M ! Sets op commutes, up to homotopy, with transfinite compositions of maps Xa ,! Xa+1 as in Q1. Q3: There is a limit ordinal ~ such that, for every e 2 E, the functor hom(O* *e, .) strictly commutes with ~-transfinite compositions of maps Xa ,! Xa+1 as in Q1. According to [10, 11, 13], a D-diagram Teis an orbit if colimDTe= *; OD denot* *es the full subcategory of orbits. Often OD is a large subcategory of SD . The obj* *ective of this section is to show that the orbits in this sense satisfy the axioms Q0-* *Q3. Nevertheless, the collection of orbits is not a set of orbits for SD , since it* * may be a proper class rather than a set. Any subset of obj(OD ) is a set of orbits for S* *D and defines a model category structure; see [17]. Axiom Q0 is obvious in the category of diagrams of spaces. Axiom Q2 was verified in [11]. Axioms Q1 and Q3 were left to the reader in [11]. We decided * *to include them into current exposition. Axiom Q3 will be proved in Proposition 3.1 below. The following proposition verifies Axiom Q1. Proposition 1.1. If Teis an orbit, K ,! L is an inclusion of (finite) simplicial sets and the square (1) Te "K`_____//_Xea | | | | fflffl| fflffl| Te L_____//Xea+1 is a pushout diagram in SD , then for any other orbit Te0the commutative square hom(Te0,"Te` K)____//_hom(Te0, Xea) | | | | fflffl| fflffl| hom (Te0, Te _L)__//hom(Te0, Xea+1) is a pushout diagram, up to`homotopy, in the category of simplicial sets, i.e.,* * the natural map hom(Te0, Xea) hom(T0,T K)hom(T 0, T L) ! hom(T 0, Xa+1) is a weak equivalence of simplicial sets.e e e e e e Proof.The argument for S = T opdiffers from the case S = Sets op. We will treat first the case of the simplicial sets. Let us prove first the following private case: for any orbit Te0, the functor hom(Te0, .) strictly commutes with the pushouts of the form (2) Te K"_`____//Xea" ` | | | | fflffl|" fflffl| Te LØ____//Xea+1, where ,! are cofibration in the sense of Definition 1.2 bellow, or transfinite * *com- positions of the maps as in Q1 above. Note that the restriction applies only on* * the lower horizontal map; the right vertical map is a cofibration also in (1). EQUIVARIANT LOCALIZATION OF DIAGRAMS 5 It will suffice to show that in each dimension n 0 the commutative square hom D(Te0 n, Te K)_____//_homD(Te0 n, Xea) | | | | fflffl| fflffl| hom D(Te0 n, Te L)____//homD(Te0 n, Xea+1) is a pushout in the category of sets. The functor of tensoring with a simplici* *al set W is equal to the product, in the category of diagrams, with the constant diagram containing W in each entry. Hence, it commutes with homD (Te0 n, . ). Additionally, wenrecall that the tensor . n is the left adjoint of the coten* *sor functor ( . ) , therefore the commutative square above becomes hom D(Te0 n, Te) x homD (Te0, K_n_)_//homD(Te0 n, Xea) | | | | fflffl| n fflffl| homD (Te0 n, Te) x homD (Te0,_L___)//homD(Te0 n, Xea+1) where the simplicial sets K n and L n are thought of as constant diagrams. Let C be the set of connected components of the nerve of D. Since Te0is an orbit, any map from Te0to a constant diagram with simplicial`set W in each entry is determined by the image of colimDTe0= 0 in colimDW = CW , but 0 can hit only one component, determined by Te0, i.e., hom D(Te0, W ) = W0 - the set * *of 0-simplices of W . We conclude that n n 0 n n homD (Te0, K ) = (K )0 = Kn, hom D (Te, L ) = (L )0 = Ln. ` Additionally, note that colimDXea+1= colimDXea` K L, and, on the`level of the n-simplices, (colimDXea+1)n = (colimDXea)n Kn Ln = (colimDXea)n (Ln\Kn). Then the set of maps from Te0 n to Xea+1may be decomposed into the disjoint union of sets parameterized by the (colimDXea+1)n as follows: hom D(Te0 n, Xea+1) = 0 1 a a homD (Te0 n, colimDXeaxcolimDXfa+1Xea+1) @ hom xD(T 0 n, Px)A, x2Ln\Kn e e where Pexis the pullback of the map Xea+1 ! colimDXea+1 over the n-simplex x 2 Ln \ Kn, i.e., the map x: n ! colimDXea+1; and hom xDis the set of all equivariant maps which induce the map x on the colimits. The restriction imposed on the commutative square (2) and [11, Lemma 2.1] imply that the following commutative squares are pullbacks: " Ø " Te LØ_______//_Xea+1 Xea___________//_Xea+1 | | | | | | | | fflffl|Ø " fflffl| fflffl|Ø " fflffl| L _______//colimDXea+1, colimD Xea_____//colimDXea+1. 6 BORIS CHORNY Therefore, Pex= Tex n and colimDXeaxcolimDXfa+1Xea+1= Xea. Finally we can conclude: a homD (Te0 n, Xea+1) = homD (Te0 n, Xea) (hom D(Te0 n, T ) x (Ln \ Kn)) Assembling all the information obtained so far, we rewrite the initial commut* *a- tive square as homD (Te0 "n,`Te) x Kn____________//homD(Te0 n, Xea) | | | | fflffl| fflffl|` hom D(Te0 n, Te) x Ln___//homD(Te0 n,Xfa) homD(Te0 n,T)x(Ln\Kn). We need to show that this square is a pushout of sets. It is implied by the fol* *lowing decomposition of this square into the disjoint union of two pushout squares: homD (Te0 n, Te) x Kn___//homD(Te0 n, Xea) || || || || || || homD (Te0 n, Te) x Kn___//homD(Te0 n, Xea) and ;________________________________; | | | | fflffl| fflffl| hom D(Te0 n, Te) x (Ln \ Kn)___homD(Te0 n, T ) x (Ln \ Kn). Therefore, the functor hom (Te, . ) commutes with pushouts of the form (2). For the general case, factor the map Te L ! Xea+1into a cofibration followed by a trivial fibration (this is legitimate from the point of view of [11] since* * Q1 is only required for the construction of the second factorization; see [17, 2.2]) Te "K`______//Xe0a~___////_Xea" ` | | | | | | fflffl|" fflffl|~ fflffl| Te LØ_____//Xe0a+1__////_Xea+1, and`take Xe0a= Xe0a+1xXfa+1Xea. Then the left square is a pushout: Xe0a+1= Xe0a T K T L by the topos theory result [24, IV.7.2] applied to the topoi SD # X 0 e eD op eba+1andySth#eXea+1(recallbthataSs=eSetschanisgaetopos),fsinceritomaymbetobtain* *edhe outer square along the map X 0 ! X . Then the left square is of the form (2) and in the diagram * * e a+1 e a+1 hom (Te0,"Te` K)____//hom(Te0, Xe0a)~_////_hom(Te0, Xea) | | | | | | fflffl| fflffl| ~ fflffl| hom (Te0, Te L)___//hom(Te0, Xe0a+1)////_hom(Te0, Xea+1), the left square is a pushout and the outer square is a pushout up to homotopy, * *as required. EQUIVARIANT LOCALIZATION OF DIAGRAMS 7 In the case S = T op, there is the topological function space homT op:(T opD)* *opx T opD ! T op. In order to pass from the topological to simplicial function spac* *e one applies the singular functor. It was proven in [5, Proposition 3.1] that the fu* *nctor homT op(Te0, . ) commutes with the pushouts of the form (1). This means that the commutative square homT op(Te0,"Te` K)___//_homT(opTe0, Xea) i|| || fflffl| fflffl| hom T op(Te0, Te__L)_//homT op(Te0, Xea+1) is a pushout in the category of compactly generated topological spaces. We need to show that the application of the singular functor preserves this pushout up to homotopy, though it is not true in general. It follows from [20, 13.5.5], since the map i is a cofibration (i is equal to hom T op(Te0, Te) x |K* *| ,! homT op(Te0, Te) x |L|). Although the collection of orbits is not, in general, a set of orbits for SD * *, it determines a model structure given by the following Definition 1.2. We say that a model structure on the category of the D-shaped diagrams of spaces is generated by a collection O of orbits if a map f :Xe! Yeis o a weak equivalence if and only if the induced map hom (Te, f): hom (Te, Xe) ! hom(Te, Ye) is a weak equivalence of simplicial sets for any orbit Te2 O; o a fibration if and only if the induced map hom (Te, f): hom (Te, Xe) ! hom(Te, Ye) is a fibration of simplicial sets for any orbit Te2 O; o a cofibration if and only if it has the left lifting property with respe* *ct to trivial fibrations. We say that the model structure is equivariant if O = obj(OD ). Example 1.3. If D = G is a group, then a G-orbit is just a homogeneous space G=H for some subgroup H < G (hence the name - orbit). In this case the collecti* *on of orbits forms a set {G=H | H < G}. Weak equivalences from Definition 1.2 coin- cide with the classical G-equivariant homotopy equivalences introduced by G. Br* *e- don in [4]. The equivariant model category generated by the set of orbits was constructed in [17]. Example 1.4. If D = (o ! o) is the category with two objects and one non-identi* *ty morphism, then for any space X 2 S there corresponds the orbit TX = (X ! *). The full subcategory of orbits in this case is equivalent to the category of sp* *aces S. Although the full homotopical information in the equivariant model category is encoded, usually, by the class of orbits, for a fixed diagram there exists a se* *t of orbits, which captures its homotopical information. Definition 1.5. The category of orbits of Xe is a full small subcategory of OD generated by the set of orbits {Tex= * xcolimDXfXe| x: * ! colimDXe}, for S = T op; 8 BORIS CHORNY or n 0 n op {Tex= 0 xcolimDX nX | x: ! colimX , n 0}, for S = Sets . f e D e The category of orbits of Xeis denoted by OXf. Let E OD be a small full subcategory. Then obj(E) forms a set of orbits for SD . For any diagram Xe, we can form a diagram XeEof spaces over the category E* *op with XeE(Ee) = hom(Ee, Xe), Ee2 obj(E). We will call it the diagram of orbit-po* *ints of Xe, since it generalizes the notion of the diagram of fixed-points of a spac* *e with a group action. According to [17], there exists a model category on SD with a map f being a weak equivalence or fibration if and onlyoifpfE is a weak equival* *ence or fibration in the Bousfield-Kan model category on SE . Moreover, the functor ( . )E has a left adjoint and this is a Quillen equivalence of the model catego* *ries. Fix a diagram Xe, and assume that Xe is of orbit type E, i.e., OXf E. We will use the result [10, Lemma 3.7], which shows that XeEis Eop-free. This mean* *s, essentially, that the orbit-points functor preserves cofibrant diagrams of orbi* *t type E, which is not typical for a right adjoint. 2.Instrumented classes of maps In this section we introduce several notions from Category Theory. Some of them are well-known, others are new. The main new concept presented here (Defi- nition 2.11) is the instrumented class of maps. This notion will allow us to ge* *neralize Quillen's small object argument in Section 4. Any set of morphisms with small d* *o- mains (see below) in any category may be thought of as an instrumented class. A non-trivial example of an instrumented (proper) class of maps in the category of diagrams of spaces is given in Section 3. Throughout this section, let C be a category and O a (not necessarily small) * *full subcategory of C. Map C will denote the category of maps of C with commutative squares as morphisms. Definition 2.1. For any category C, power category Pow C is the category of all subsets of obj(C) and enriched functions as morphisms, i.e., for any A, B obj* *(C) the set of morphisms from A to B is the set of all functions F :A ! mor(C), such that for any a 2 A, dom (F (a)) = a and codom (F (a)) 2 B. Suppose A, B, C obj(C). If F1: A ! mor(C) is a morphism from A to B and F2: B ! mor(C) is a morphism from B to C in Pow C, then their composition F3 = F2 O F1 is defined by F3(a) = F2(codom (F1(a))) OC F1(a). For any set A, the identity morphism idA is a function that corresponds to every a 2 A, the identity morphism ida. Definition 2.2. Let O be a class of objects in C. For any object c of C, a set of arrows _i:oi ! c has the factorization property with respect to O if for eve* *ry o 2 O and _ :o ! c there is a commutative triangle in C o`?``//?oi " _ ØØc"_i"" for some i. The next definition has appeared in [11, 1.1]. EQUIVARIANT LOCALIZATION OF DIAGRAMS 9 Definition 2.3. A class O of objects in a category C is locally small if for any object c 2 C there exists a set of arrows _i:oi ! c with oi 2 O, that has the factorization property with respect to O. A full subcategory O of a category C * *is locally small if its class of objects is locally small. Example 2.4. In the category of sets any class S of pairwise non-isomorphic objects is locally small, since S contains sets of cardinality bigger than any * *fixed cardinality ~, i.e. for any set X we can find a set Y in S which admits a surje* *ctive map f :Y ! X. This map has the factorization property. Remark 2.5. The condition for a class of maps to be locally small is dual to the classical solution-set condition [25]. The previous definition admits a functorial version: Definition 2.6. Let O be a locally small subcategory of C with a class of ob- jects O = obj(O). A factorization setup for the pair (C, O) is a functor F :C ! Pow (Map C) such that, for every c 2 obj(C) and f 2 morC(c, c0), we have (1) F (c) is a set of maps of C of the form o ! c, o 2 O; (2) F (c) has the factorization property with respect to O; (3) F (f) is a function that corresponds to any element o ! c of F (c) a com- mutative square (a morphism in Map C) of the form o ----! o0 ?? ? y ?y c --f--!c0 where o0! c0is an element F (c0). Example 2.7. In the category of groups the class of free groups has a factoriza* *tion setup F : for any group G, F (G) is the set with one element {"G :FG ! G}, where "G is the canonical map of the free group generated by the elements of G onto G (counit of the adjunction of the free functor with the forgetful functor); fo* *r a homomorphism ': G1 ! G2 of groups, F (') is defined to be a function which assigns to "G1 the following commutative square: FG1 --F'--!FG2 ?? ? y ?y G1 --'--! G2 where F' is the result of applying the free functor on the map of underlying se* *ts ': G1 ! G2. The factorization property of F (G) follows from the universal prop- erty of free groups. We accept the following convention for the notation: if a class of objects in* * a category C is denoted by a capital letter, e.g. I, then the full subcategory of* * C generated by the class I is denoted by the script letter I and the factorizatio* *n setup for the pair (C, I) is denoted by the calligraphic letter I. Example 2.8. If a full subcategory M of C is small, i.e., M = obj(M) is a set, then there exists a factorization setup M: C ! Pow (Map C), such that for every 10 BORIS CHORNY object c in C, a M(c) = morC (m, c) m2M and for any morphism f :c ! c0 in C, M(f): M(c) ! M(c0) is a function which corresponds to any element ': m ! c of M(c) the commutative square m -idm---!m ? ? '?y ?yf' c ----!fc0. A simple property of factorization setups is given by the following Proposition 2.9. Let J and K be two disjoint classes of objects in a category C, supplied with factorization setups J and K for the pairs (C, J) and (C, K), res* *pec- tively. Then the class of maps L = J t K is supplied with a factorization setup. Proof.For any c 2 C define L(c) = J (c) t K(c), and for any morphism f :c ! c0in C define L(f) = J (f)tK(f) obj(Map C). It is routine to check Definition 2.6. Corollary 2.10. If J is a factorization setup for the pair (C, J) and K is a se* *t of objects in C, then the pair (C, J [ K) may be supplied with a factorization set* *up. Proof.This follows from Proposition 2.9 applied for the disjoint classes J and * *K0= K \ (K \ J). A factorization setup for K0 is provided by Example 2.8. Let C be a category, and let I be a class of maps in C. Recall [20, 22] that* * a map in C is I-injective if it has the right lifting property with respect to ev* *ery map in I; a map in C is I-projective if it has the left lifting property with respe* *ct to every map in I; a map in C is I-cofibration if it has the left lifting property* * with respect to every I-injective map; a map in C is I-fibration if it has the right* * lifting property with respect to every I-projective map. The classes of I-injective map* *s, I-projective maps, I-cofibrations and I-fibrations are denoted I-inj, I-proj, I* *-cof and I-fib, respectively. Suppose that C contains all small colimits. A relative I-cell complex is a tr* *ansfi- nite composition of pushouts of elements of I. We denote the collection of rela* *tive I-cell complexes by I-cell. Let D be a collection of morphisms of C, A an object of C and ~ a cardinal. We say that A is ~-small relative to D if, for all ~-filtered ordinals ~ and all ~* *-sequences X0 ! X1 ! . .!.Xfi! . .,. such that each map Xfi! Xfi+1is in D for fi + 1 < ~, the natural map of sets colimfi<~C(A, Xfi) ! C(A, colimfi<~Xfi) is an isomorphism. We say that A is small relative to D if it is ~-small relati* *ve to D for some ~. We say that A is small if it is small relative to mor(C). Finally, we introduce the main notion of the current section: instrumented cl* *ass of maps. Definition 2.11. Let C be a category. A locally small class I of maps in C is c* *alled instrumented (or equipped with an instrumentation) if (1) there exists a cardinal ~ s.t. any A 2 dom(I) is ~-small relative to I-c* *ell, EQUIVARIANT LOCALIZATION OF DIAGRAMS 11 (2) there exists a factorization setup I for the pair (Map C, I) (see Defini- tion 2.6). 3.Example: Diagrams of spaces Let D be a small category. In this section we consider the equivariant model category on the D-shaped diagrams of spaces (see Definition 1.2) and show that the classes of generating cofibrations and generating trivial cofibrations (see* * below) may be equipped with an instrumentation. 3.1. Generating cofibrations and generating trivial cofibrations. Let O be the category of D-orbits. We define the class of generating cofibrations to be I = {Te @ n ,! Te n}Te2O,n 0and the class of generating trivial cofibrations to be J = {Te nk~,!Te n}Te2O,n k 0. Let I and J denote the full subcategori* *es of Map SD with classes of objects I and J, respectively; I and J generate the m* *odel structure of Definition 1.2 in the sense that the class of trivial fibrations e* *quals I-inj and the class of fibrations equals J-inj. The following proposition verifies fo* *r classes I and J the first part of the definition of the instrumented class. 3.2. Smallness of domains. Proposition 3.1. Any Te K 2 dom(I) [ dom(J), where Teis an orbit and K is a finite simplicial set, is @0-small with respect to both the I-cell and J-cell. Proof.Denote by H the class of maps {Te K ! Te L}, where Te2 O and K ,! L is a cofibration (inclusion) between finite simplicial sets. It will suffice to* * discuss colimits of I-cell maps along the first infinite ordinal ! only. Consider an !-* *sequence of H-cellular spaces ; = Ze0! . .!.Zeff! . .!.Ze!, where Ze!= colimff> | "" | | " ~| fflffl|" fflfflfflffl| XeE_____//ZE e By Yoneda's lemma for any d 2 D and an arbitrary diagram W , W (d) ~=W E(F d). Hence, the lift in the above square naturally `reduces' ifn tfhe obvifous sense* * to the lift in the original square. Remark 5.2. The last proposition settles affirmatively the conjecture stated in* * [11, 2.3]. Corollary 5.3. The equivariant model category on SD is left proper. Proof.This follows from Proposition 5.1 and C. L. Reedy's theorem [28] (see [20, 13.1.2] for a modern exposition) which asserts that every pushout of a weak equ* *iv- alence between cofibrant objects along a cofibration is a weak equivalence. Proposition 5.4. The equivariant model category on the diagrams of topological spaces is left proper. Proof.Suppose we are given a pushout in the category T opD of diagrams of topo- logical spaces: Ae__f~_//"B`e g|| || fflffl|fflffl| Xe_____//Ye We have to show that the map Xe! Yeis a weak equivalence. By Corollary 4.2 the cofibration g :Ae,! Xe is a retract of an I-cellular map g0:Ae,! Xe0. Hence, the pushout of f along g is a retract of the pushout of f a* *long g0 and it will suffice to show that the last map is a weak equivalence. The I-cellular cofibration g0 has a decomposition into a (transfinite) sequen* *ce Ae= Ae0,! Ae1,! . .,.! Aen,! . .,.! Xe0, such that Aen+1is a pushout of Aenalong a map from I. Therefore, the pushout of f along g is the colimit of the consecu* *tive pushouts of f along the maps gn :Aen,! Aen+1for all n. Hence, by Proposition 3.* *1, it is enough to show that the pushout of a weak equivalence along gn is a weak equivalence. Any map in I has a form i: Te K ,! Te L, where K ,! L is an inclusion of finite simplicial sets and Teis an orbit. Consider the following commutative di* *agram EQUIVARIANT LOCALIZATION OF DIAGRAMS 19 in which the left and right squares are pushouts: Te K" _____//_`Aenfn~//_"B`en i|| gn|| || fflffl| fflffl| fflffl| Te L_____//Aen+1___//_Ben+1 hence the outer square is a pushout. By the axiom Q1, which was verified in Proposition 1.1, for any orbit Te0, in* * the induced diagram of simplicial sets hom(T0,fn) hom (Te0,"Te` K)____________//hom(T 0, An)____e~______//hom(T 0, Bn) | rrkK| e e qq | e e | rrr | qqq | | rrr | qqq | | xxrrr | xxqqq | hom(Te0,i)|| qP18L``8` ` ` `||``e`e`e`e`//P2M22eeee || | qqqqq L LeLeeeee|eeeee MM M | | qqq eeeeeeee L | M | fflffl|qqeeeee L&&fflffl| M&&fflffl| hom (Te0, Te L)___________//hom(Te0, Aen+1)_________//hom(Te0, Ben+1) the left and outer squares are, up to homotopy (i.e., up to a weak equivalence), pushout diagrams. Consider the pushouts P1 and P2 of the left and outer square respectively, i.e., a P1 = hom(Te0, Te L) hom(T 0, An) hom(Te0,Te K) e e and a P2 = hom(Te0, Te L) hom(T 0, Bn), hom(Te0,Te K) e e then the natural maps P1 ! hom (Te0, Aen+1) and P2 ! hom (Te0, Ben+1) are weak equivalences of simplicial sets. The map i in T opD is a monomorphism, therefore the induced map hom (Te0, i) is a monomorphism (or a cofibration) of simplicial sets, hence its cobase change hom(Te0, Aen) ! P1 is also a cofibration. ` Finally, [20, 7.2.14] implies that P2 = P1 hom(T0,A )hom(T 0, Bn) and the map P1 ! P2 is a weak equivalence by the left properneess enof siempleicial sets. T* *herefore, we can conclude from the `2 out of 3' property that the map hom (Te0, Aen+1) ! hom(Te0, Ben+1) is a weak equivalence of simplicial sets and, hence, the origin* *al map Aen+1! Ben+1is a weak equivalence of diagrams of topological spaces. 6. Construction of the localization functor Let S be a class of maps in SD . Without loss of generality, we may assume that the elements of S satisfy the conditions listed in 6.1 below. In this sec* *tion we construct for such class S a coaugmented functor LS :SD ! SD such that for each Xe2 SD , LS(Xe) is S-local and the natural map jXf:Xe ! LS(Xe) is an S-equivalence (see below). We prove also that the natural map jXf:Xe ! LSXe is initial (up to homotopy) among all the maps of Xeinto S-local spaces, thus LSXe is characterized up to weak equivalence. Our construction is an extension of the classical constructions of localizati* *on functors with respect to a set of maps in a cofibrantly generated model category 20 BORIS CHORNY satisfying additional conditions [3], [20]. A useful summary of the classical * *con- struction is available at [18, Appendix]. 6.1. Preliminaries on S-local spaces and S-equivalences. Throughout this section let us suppose that S is a class of maps such that every f 2 S, f :Ae,!* * Beis a cofibration between cofibrant diagrams. Assume, for simplicity, that all cofibr* *ations in S are non-trivial. We are going to construct the localization functor with r* *espect to S in this section, provided that S satisfies the following conditions: (1) The class of horns of S a Hor(S) = { n Ae @ n B ! n B | S 3 f :A ! B, n 0} @ n Ae e e e e may be equipped with an instrumentation with a factorization setup H. (2) All the elements of dom(Hor(S)) are ~-small with respect to cofibrations* * of of diagrams for some fixed cardinal ~. These conditions are satisfied for example by any set of non-trivial cofibratio* *ns S. An example of a proper class of maps satisfying the conditions above will be gi* *ven in Section 7 below. Definition 6.1. Let S be a class of maps such that every f 2 S, f :Ae,! Beis a cofibration between cofibrant diagrams. o A diagram Xe is called S-local if Xe is fibrant and for every f 2 S, the induced map hom(f, Xe): hom (Be, Xe) ! hom(Ae, Xe) is a weak equivalence of simplicial sets. If S consists of the single m* *ap f :A ! B, then an S-local diagram will also be called f-local. o A map g :Ce! Deis an S-local equivalence (or just an S-equivalence) if f* *or any cofibrant replacement ~gof g and every S-local diagram Pethe induced map hom(~g, Pe): hom (D~e, Pe) ! hom(C~e, Pe) is a weak equivalence of simplicial sets. If S consists of the single m* *ap f :A ! B, then an S-local equivalence will also be called an f-local equ* *iv- alence (or an f-equivalence). Remark 6.2. Of course one needs to check that the notion of S-equivalence is we* *ll- defined, i.e., it does not depend on the choice of the cofibrant replacement. * *It follows from [20, 9.7.2]. We shall use also an S-local version of the Whitehead theorem (see [20, 3.2.13] for the proof). Proposition 6.3 (S-local Whitehead theorem). A map g :Q1 ! Q 2is a weak equivalence of S-local spaces if and only if g is an Se-loceal equivalence. Proposition 6.4. A cofibration g :Ce,! De of cofibrant diagrams is an S-local equivalence if and only if g has the homotopy left lifting property (see [20, 9* *.4.2] for the definition) with respect to all the maps of the form Ze! *, where Zeis * *an S-local diagram. Proof.Follows immediately form the definitions and [20, 9.3.1]. Proposition 6.5. A diagram Xe is S-local if and only if the map Xe! * has the right lifting property with respect to the following families of maps: EQUIVARIANT LOCALIZATION OF DIAGRAMS 21 o generating trivial cofibrations J; o Hor(S). Proof.The right lifting property for the members of J is satisfied since Xeis f* *ibrant by definition. For every element f 2 Hor(S) it follows, by adjunction, from the* * right lifting property of the map hom (f, Xe): hom (Be, Xe) i hom (Ae, Xe) with respe* *ct to the generating cofibrations of simplicial sets {@ n ,! n | n 0}. Proposition 6.6. The class of maps K = J [ Hor(S) may be equipped with an instrumentation. Proof.The existence of a factorization setup K for the pair (Map SD , K) follows from Proposition 2.9, since we have chosen only non-trivial cofibrations in S, * *there- fore the classes J and Hor(S) are disjoint. The existence of the cardinal ~ suc* *h that the elements of dom(K) are ~-small relative to K-cell follows from the assumpti* *ons on S. 6.2. Construction of the functor LS. We construct the coaugmented functor LS by applying the generalized small object argument, with respect to the instrume* *nted (by Proposition 6.6) class of maps K, to factorize the map Xe! * into a K-cellu* *lar map, followed by a K-injective map. The obtained functorial factorization jX Xe- f!LS(Xe) -! * provides us with the coaugmented functor LS, such that for any diagram Xe, the diagram LS(Xe) is S-local by Proposition 6.5. It remains to show that the natural coaugmentation map jXfis an S-equivalence. Lemma 6.7. Every map in K is an S-equivalence. Proof.Every map g 2 J is a trivial cofibration between cofibrant diagrams, i.e., for any S-local (in particular fibrant) diagram Ze, hom (g, Ze) is a trivial fi* *bration of simplicial sets. Hence, g is an S-equivalence. It remains to show that every map in Hor(S) is an S-equivalence. Every map in Hor(S) is a cofibration between cofibrant objects, hence, by Proposition 6.4, i* *t is enough to show that every map in Hor(S) has the homotopy left lifting property with respect to any map of the form Ze! *, where Zeis S-local. This last proper* *ty is implied by [20, 9.4.8(1)]. Lemma 6.8. A pushout of an S-equivalence g that is also a cofibration between cofibrant objects Ae"_`__//Xe g|| || fflffl||fflffl Be_____//Ye in the equivariant model category on SD is an S-equivalence again. Proof.The following proof is a straightforward generalization of [20, 1.2.21]. * *This argument significantly relies on the left properness. Factor the map Ae! Xe as Ae!u Ce!v Xe, where u`is a cofibration and v is a trivial fibration. If we let D be the pushout Be A C , then we have the commuta* *tive ee 22 BORIS CHORNY diagram ü v Ae"Ø`__//Ce"~`////_Xe g|| |k| |h| fflffl|fflffl|sfflffl|t Be_____//De____//Ye in which u and s are cofibrations,`and so Ceand Deare cofibrant. [20, 7.2.14] i* *mplies that Yeis a pushout Xe CD . Since k is a cofibration and we are working in a (* *left) proper model category (beyeCorollary 5.3 and Proposition 5.4), the map t is a w* *eak equivalence. Thus, k is a cofibrant approximation to h, and so it is sufficien* *t to show that k induces a weak equivalence of mapping spaces to every S-local diagr* *am. In any simplicial model category, a class of maps with the homotopy left lift* *ing property with respect to a map p is closed under pushouts [20, 9.4.9]. Consider the collection of maps P = {pZe:Ze! * | Zeis S-local}. Then, by Proposition 6.4, a cofibration between cofibrant objects is an S-local equivalence if and only i* *f it has the homotopy left lifting property with respect to any element of P . But t* *he last property is preserved under pushouts, hence k in the diagram above is an S-equivalence and so is h. Lemma 6.9. The class of S-equivalences between cofibrant diagrams in the sim- plicial model category SD is closed under coproducts (in the category of maps), and the class of S-equivalences which are cofibrations is closed under transfin* *ite compositions in the left proper model category SD . Proof.Let gffbe an S-equivalence for each ff 2 A; then for any S-local diagram * *Ze a Y hom ( gff, Z) = hom (gff, Z), ff2A e ff2A e where hom (gff, Ze) is a weak equivalence of simplicial sets, therefore their p* *roduct is also a weak equivalence. Let Ee0,! Ee1,! . .,.! Eefi,! . .b.e a ~-sequence of cofibrations which are also S-equivalences. By [20, 17.9.4] we may suppose, without loss of generalit* *y, that all the diagrams Eeiare cofibrant. Then for each S-local diagram Zethere i* *s a ~-sequence of trivial fibrations of simplicial sets hom (Ee0, Ze)j~hom (Ee1, Ze)j~. .j.~hom(Eefi, Ze)j~. ... The inverse limit of the last sequence is a homotopy inverse limit, in particul* *ar, the natural map hom (Ee0, Ze) limfi<~hom(Eefi, Ze) = hom(colimfi<~Eefi, Ze) is a * *weak equivalence (compare to the inverse limit of the constant tower). Thus, the nat* *ural map Ee0,! colimfi<~Eefiis an S-equivalence. Proposition 6.10. Every map in K-cell is an S-equivalence. Proof.This follows from Lemmas 6.7, 6.8 and 6.9. Corollary 6.11. The coaugmented functor LS is the S-localization functor. Proof.By the construction of LS, the natural map jXf:Xe ! LSXe is in K-cell, so, by the proposition above, jXf is an S-equivalence. We postpone the proof of universality of LS until the next section. EQUIVARIANT LOCALIZATION OF DIAGRAMS 23 Lemma 6.12. For any diagram Xe, either cofibrant or not, the coaugmentation morphism jXf:Xe! LSXe satisfies: for any S-local diagram Pe, hom(jXf, Pe): hom (LSXe, Pe) '!hom (Xe, Pe) is a weak equivalence of simplicial sets. Proof.Corollary 6.11 implies that jXfis an S-equivalence, hence the result foll* *ows from Definition 6.1 and [20, 13.2.2(1)]. 6.3. Universality and other properties of LS. Proposition 6.13 (LS is initial). For any map g :Xe! Peinto an S-local dia- gram there exists a factorization Xe! LSXe ! Pewhich is unique up to simplicial homotopy. Proof.By the construction of LS, the natural map jXf:Xe! LSXe is in K-cell. By Proposition 6.5 the map Pe! * is in K-inj. Then in the diagram Xe___g__//"P`e== _ K-cell3jXf||__ |2K-inj fflffl|_fflfflfflffl|| LSXe_____//* the lift exists and provides the required factorization. To show the uniqueness up to homotopy of the factorization above, consider the map hom (jXf, Pe): hom (LSXe, Pe) -'! hom(Xe, Pe), which is a weak equivale* *nce by Lemma 6.12, since Peis S-local. Then, by [20, 9.5.10], jXfinduces a bijection of the sets of simplicial homotopy classes of maps j*X:[LSX , P] ~= [X , P]. T* *he uniqueness, up to simplicial homotopy, of the liftinfg folleowes from ethee inj* *ectivity of the map j*X. f Remark 6.14. By [27, II.2.5], if two maps are simplicially homotopic, then they* * are both left and right homotopic. Corollary 6.15. The natural maps jLSXf, LS(jXf): LSXe ' LSLSXe are simpli- cially homotopic weak equivalences. Proof.The naturality of j implies that the square jX Xe - -f--! LSXe ? ? jXf?y ?yjLSXf LSXe -----! LSLSX LS(jXf) e is strictly commutative. Two possible paths provide two different factorizatio* *ns of the map Xe ! LSLSXe. By Proposition 6.13 the natural maps jLSXf, LS(jXf) are simplicially homotopic. Proposition 6.3 implies that the map jLSXfis a weak equivalence; so does LS(jXf). Proposition 6.16 (Divisibility). For any two maps g, h: LSXe ' Peinto any S- local diagram Pe, one has h s~g if and only if h O jXf~sg O jXf. 24 BORIS CHORNY Proof.Suppose h s~g. The simplicial homotopy between the maps h and g is a 1-simplex in hom (LSXe, Pe), so its image under j*Xprovides a simplicial homoto* *py between h O jX and g O j . f f Xfs Conversely, if h O jXf~ g O jXf, then the injectivity of the map j*X:[LSX , * *P] ~= s f e e [Xe, Pe] implies that h ~ g. Proposition 6.17 (No zero divisors). Suppose W is a retract of LSX for some X . If the composition X ! L X ! W is null fhomotopic (see Appendeix A for the edefinition), then W ' *. S e f f Proof.First notice that the diagram W is S-local (in particular, fibrant) as a * *retract of the S-local diagram LSX . Supposefthat the composition X ! L X ! W is null homotopic; then there exisets, by Proposition A.3, a fibrant coSnetrafctible di* *agram U such that the following solid arrow diagram commutes: * * e jX xx_i________ XeC__f_//LSX_r___//ØW== CCC Ø e ----f CCCpØ --q- C!!Cfflffl-- Ue' *e. The dashed arrow p exists by the universal property of Proposition 6.13 and mak* *es the left triangle commutative. By the divisibility property of Proposition 6.1* *6, r s~q O p, since, upon precomposing with jXf, these two maps are equal. Recall that W is a retract of LSX , hence idW = r O i s~q O (p O i). Therefo* *re, W is a homotopy fretract of U. e f * * f Now apply the general maechinery for detection of weak equivalences [20]: the map q :Ue! W between two fibrant spaces is a weak equivalence if and only if f* *or any cofibrafnt A the induced map on simplicial homotopy classes q : [A , U] ! [* *A , W ] is a (natural) ebijection. But [A , U] = *, as U is fibrant and c*ontreacetible* *,eanfd [A , W ] is a retract of [A , U], i.e., [Ae, eW] = *. Heence, W ' U ' *. * * e f e e e f f e Corollary 6.18. LS(Ue) ' *efor any contractible diagram Ueand LS(nullmap) is a null map. Proof.LS(Ue) is a retract of LS(Ue) by the identity morphism. Moreover, idLS(Ue* *)O jUe:Ue! LS(Ue) is null, since Ueitself is contractible. Hence, by the proposit* *ion above LS(Ue) ' *e. The second property follows immediately from the first one. Proposition 6.19. A map g :Xe ! Ye is an S-local equivalence if and only if LS(g): LSXe ! LSYe is a weak equivalence. Proof.In the commutative diagram Xe --g--! Ye ? ? jXf?y ?yjYe LSXe ----! LSY LS(g) e the vertical arrows are S-local equivalences by construction. Hence, the map g is an S-local equivalence if and only if LS(g) is an S-local equivalence by the* * `2 EQUIVARIANT LOCALIZATION OF DIAGRAMS 25 out of 3' property of S-equivalences [20, 3.2.3]. But LS(g) is a map between two S-local spaces, hence LS(g) is an S-local equivalence if and only if LS(g) is a* * weak equivalence by Proposition 6.3. The coaugmentation map jXfis a cofibration for any diagram Xe, hence the sub- category of cofibrant diagrams is stable under localizations. Consider the rest* *riction of LS to the subcategory of cofibrant objects (do nothing for the diagrams of s* *im- plicial sets) and denote the new functor by LrS. Then LrSis terminal with respe* *ct to S-local equivalences. In more detail, we have the following Proposition 6.20 (LrSis terminal). On the subcategory of cofibrant objects the coaugmentation map jXf:Xe ! LSXe = LrSXeis terminal, up to homotopy, among all S-local equivalences, i.e., for any S-equivalence of cofibrant diagrams g :* *Xe! Ye, there exists an extension Xe! Ye!lLrSXethat is unique up to (simplicial) homoto* *py with l O g s~jXf:Xe! LrSXe. Proof.By Proposition 6.19, LrS(g) = LS(g) is a weak equivalence. Moreover, this is a weak equivalence between two objects which are both fibrant and cofibrant,* * so in the following commutative diagram of solid arrows Xe___g___//_Ye jXf|| jY || fflffl|~e fflffl| LSXebLSg_//_LSYeb L ` r q the map LS(g) has a simplicial homotopy inverse q (all notions of homotopy of maps between objects which are fibrant and cofibrant coincide). Define l = qjYe; then, using the commutativity of the diagram above and [20, 9.5.4], we obtain lg = qjYeg = (qLS(g))jXf~sidLS(Xf)jXf= jXf. To show the uniqueness up to simplicial homotopy of l, suppose there exists another map l0:Ye! LSXe such that l0g s~jXf. Then l0factors through LSYe(since LS is initial), i.e. there exists an arrow q0:LSYe ! LSXe such that l0 = q0jYe.* * It will suffice to show that q0~sq since [20, 9.5.4] implies that l0~sl. By assump* *tion, l0g s~jXf, so q0jYeg s~jXfor, equivalently, q0LS(g)jXf~s idLSXfjXf. By divisib* *ility, q0LS(g) s~idLSXf. But LS(g) is a weak equivalence of cofibrant objects, therefo* *re, by [20, 9.5.12], LS(g)*: [LSYe, LSXe] ! [LSXe, LSXe] is a bijective map of simp* *licial homotopy classes which satisfies LS(g)*([q]) = LS(g)*([q0]), hence [q] = [q0] or q s~q0. 7.Fixed-pointwise localization _ Localization with respect to a class of maps In the previous section we developed the localization theory of diagrams of s* *paces with respect to a class of maps of diagrams subject to certain conditions 6.1. * *But a large part of the equivariant homotopy theory [4, 10, 13] uses the `fixed-point* *wise' approach and its generalizations, so it is natural to ask whether for any map f :A ,! B of simplicial sets there exists a localization functor L, which induc* *es f-equivalences of fixed-point sets hom (Te, Xe) ! hom(Te, LXe) for each orbit T* *e. We 26 BORIS CHORNY do not know in general whether it is possible to find a set of maps of diagrams such that the localization with respect to it gives the required functor L. But* * in one simple case f :; ,! *, discussed in the companion paper [6], we know that t* *his is impossible. In this section we show how to apply the generalized small object argument to the localizations with respect to certain classes of maps. There is no point in considering fixed-pointwise localizations with respect t* *o a set of maps since, in the category of spaces, the localization with respect to * *any set of maps is equivalent to the localization with respect to a single map: tak* *e this single map to be the coproduct of the set of maps, in case that there is no map of the form ; ,! X, X 6= ;; otherwise, this map ; ,! X will induce the same localization functor as the whole set. Fixed-pointwise localization with respect to homology in the category of spac* *es with a compact Lie group action was constructed in [26]. Our construction is new only for the diagram shapes which lead to non-cofibrantly generated model structures on SD [7], otherwise it is covered by the classical localization fra* *mework. Given a non-trivial cofibration f :A ,! B of simplicial sets, consider the fo* *llowing class of maps F = {f Te:A Te,! B Te| Te2 O}. Then a diagram Zeis F -local if and only if it is fibrant and for each orbit Tethe space of `Te-fixed points', * *hom(Te, Ze), is f-local: o Zeis fibrant , hom(Te, Ze) is fibrant for each Te2 O, by definition; o hom (f Te, Ze): hom (B Te, Ze) ! hom (A Te, Ze) is a weak equivale* *nce , hom (B, hom(Te, Ze)) ! hom (A, hom(Te, Ze)) is a weak equivalence for each Te2 O, by adjunction. Example 7.1. Let f :; ,! * be a map in S; then F = {f Te:; ,! Te| Te2 O}. The f-localization functor on the category of spaces assigns to any space X a contractible space Lf(X), for a space is f-local iff it is fibrant and contract* *ible. By the considerations above, a diagram Xeis F -local iff its Te-fixed-point spa* *ce is f-local for any orbit Te, i.e., fibrant and contractible, hence Xe is F -local * *iff it is fibrant and contractible. Any map of diagrams is an F -equivalence, likewise any map of spaces is an f- equivalence. In other words, a map of diagrams g :Xe! Yeis an F -local equivale* *nce if and only if the induced map of fixed-point spaces hom (Te, g): hom (Te, Xe) ! hom(Te, Ye) is an f-equivalence for each orbit Te2 O. The example above has the following generalization: Proposition 7.2. A map g :Xe! Yeis an F -local equivalence if and only if for each orbit Te2 O the map hom (Te, g): hom (Te, Xe) ! hom(Te, Ye) is an f-equiva* *lence of simplicial sets. Proof.Let E O be a full small subcategory of the category of orbits such that Xeand Yeare of orbit type E. The set of orbits obj(E) is a set of orbits in the sense of [17] (see [11] for the proof), and induces a simplicial model structur* *e on the category of D-diagrams which is Quillen equivalent to the Bousfield-Kan mod* *el category on Eop-shaped diagrams of spaces [17]. Moreover, these model categories are cofibrantly generated, therefore they have functorial factorizations, and i* *t was shown in [15] that their simplicial homotopy categories are homotopy equivalent. A map g :Xe! Yeis an F -local equivalence if for any cofibrant replacement (in the model category generated by all_orbits) ~g:~Xe! ~Yeand any F -local diagram EQUIVARIANT LOCALIZATION OF DIAGRAMS 27 Zethe induced map on function complexes hom (~g, Ze): hom (Y~e, Ze) ! hom (X~e,* * Ze) is a weak equivalence of simplicial sets. By the construction (in Section 3) of* * the cofibrant replacement (which is a D-CW -complex of the same orbit type as the original space), X~eand ~Yeare cofibrant in the model category generated by the objects of E and Zeis fibrant in both model categories, hence the function com- plexes hom (X~e, Ze) and hom (Y~e, Ze) are weakly equivalent to the homotopy fu* *nction complexes of the simplicial homotopy category [15]. But (Xe)E and (Ye)E are cof* *i- brant in the Bousfield-Kan model category as free diagrams over Eop, and (Ze)E * *is fibrant, hence the function complexes hom ((X~e)E, (Ze)E) and hom ((Y~e)E, (Ze)* *E) are also weakly equivalent to the homotopy function complexes of the corresponding simplicial homotopy category. The homotopy equivalence between simplicial homotopy categories implies that the map hom (~g, Ze): hom (Y~e, Ze) ! hom(X~e, Ze) is a weak equivalence if and* * only if the map hom ((~g)E, (Ze)E): hom ((Y~e)E, (Ze)E) ! hom ((X~e)E, (Ze)E) is a weak* * equiv- alence. But X~eand ~Yeare weakly equivalent to Xe and Ye, hence (X~e)E ~= (Xe)E and (Y~e)E ~=(Ye)E are weak equivalences of cofibrant (free) objects in the Bou* *sfield- Kan model category. Therefore, the maps hom(~g, Ze) and hom((~g)E, (Ze)E) are w* *eak equivalences if and only if hom ((g)E, (Ze)E): hom ((Ye)E, (Ze)E) ! hom((Xe)E, * *(Ze)E) is a weak equivalence. Compare [10, 5.13]. Now we can prove the proposition. Suppose that all maps induced by the map g on fixed-point sets are f-equivalences. Let Ze be any F -local diagram. The equivariant function complex hom ((Xe)E, (Ze)E) may be represented as a homotopy inverse limit over the twisted arrow category aEop (objects of aEop are morphis* *ms of Eop and arrows of aEop are commutative squares e0 ---- e00 ?? ? y ?y e1 ----! e01 in Eop) [16, 3.3]. Since (Xe)E is cofibrant (because it is free) and (Ze)E is f* *ibrant, the following map is a weak equivalence: hom((Xe)E, (Ze)E) -~!holimaEophoma((Xe)E, (Ze)E), where hom a((Xe)E, (Ze)E) is an aEop-diagram of simplicial sets in which for ea* *ch (e0 ! e1) 2 aEop there is assigned the simplicial set hom ((Xe)E(e0), (Ze)E(e1)* *). By assumption, the induced map between aEop-diagrams hom a(gE, (Ze)E): hom a((Ye)E, (Ze)E) ! homa((Xe)E, (Ze)E) is an objectwise weak equivalence, since each entry of the diagram (Ze)E is an * *f-local space. Hence, the induced map on the homotopy inverse limits is a weak equivale* *nce hom((g)E, (Ze)E): hom ((Ye)E, (Ze)E) -~!hom ((Xe)E, (Ze)E). Then, by the discus* *sion above, the map hom(~g, Ze): hom (Y~e, Ze) ! hom(X~e, Ze) is a weak equivalence * *for any F -local diagram Ze, i.e., the map g is an F -local equivalence. Alternatively, one can think of an equivariant function complex as an end. Th* *en this is a homotopy end, since (Xe)E is free, and the same conclusion follows fr* *om the results in [14]. Suppose now that g is an F -local equivalence. We need to show that for each * *orbit Tethe induced map on the Te-fixed-point space is an f-equivalence. Let VTe(Te0)* * = 28 BORIS CHORNY homEop(Te0, Te) be a diagram of simplicial sets over E; then by the dual of Yon* *eda's lemma we obtain the natural isomorphism (Xe)E EopVTe~=(Xe)E(Te) ~=hom(Te, Xe). Take W to be any f-local simplicial set. Then in the commutative diagram ~= E ~= E hom((Ye)E(Te), W_)__//hom((Ye) EopVTe, W_)__//_hom((Ye) , hom(VTe, W )) hom(gE|Te,W)|| || hom(gE,hom(VTe,W))|| fflffl| ~= fflffl| ~= fflffl| hom((Xe)E(Te), W_)__//_hom((Xe)E EopVTe,_W_)_//hom((Xe)E, hom(VTe, W )), where hom (VTe, W )) is an Eop-diagram of f-local spaces [12, A.8(e.2)], the le* *ft ver- tical arrow is a weak equivalence if and only if the right vertical arrow is a * *weak equivalence. But the diagram hom(VTe, W ) may be replaced, up to objectwise weak equivalence, by a diagram (Ze)E, where Zeis a fibrant approximation of the real* *iza- tion of hom(VTe, W ) as a D-diagram, i.e., Zeis an F -local diagram and therefo* *re the map hom (gE, hom(VTe, W )) is a weak equivalence. Proposition 7.3. The class of maps a Hor(F ) = { n A Te @ n B T ! n B T | n 0, T 2 O} @ n A Te e e e is instrumented with a factorization setup H and a cardinal ~ > |A| + |B|. Proof.The factorization setup H is constructed as follows. Any map u = (u1, u2) ` u1 n A Te @ n A T @ n B T _____//X e e e | | | | fflffl| fflffl| n B Te____u2_________//_Ye is uniquely given by a map of Teinto the nodes of the diagram: hom (@ n B, Xe) m | mmmm | mmmm | vvmmmm | hom( n B, Y)___________________//hom(@ n B, Y) | | e | e | | | | | | | | | fflffl| | hom( n A, X)__________|_________//_hom(@ n A, X) | nn e | mm e | nnnn | mmmm | nnnn | mmmm fflffl|vvnn fflffl|vvmmm hom( n A, Ye)__________________//hom(@ n A, Ye). Take the `3-dimensional pullback' (the inverse limit) in the diagram above W g,* *n. Then, similarly to the proof of Proposition 3.4, define Hg,nto be the set off a* *ll maps u above which correspond bijectively, by adjointness, to the set of maps O(W g,* *n); assign H(g) = S f n 0 Hg,n. The factorization property readily follows. It is straightforward to check that the domains of maps in Hor(F ) are ~-small relative to Hor(F )-cell. EQUIVARIANT LOCALIZATION OF DIAGRAMS 29 Proposition 7.3 verifies for the class F the conditions listed in 6.1. Hence* * the results of Section 6 apply and we obtain an example LF of the localization with respect to the class_of maps F . Appendix A. Contractible objects and null homotopies in a model category The purpose of this appendix is to discuss the fundamental notions of contrac* *tible objects and null-homotopic maps in a model category. Using these notions, we prove Proposition 6.17 in a manner that allows for immediate generalization to * *an arbitrary simplicial model category (satisfying some assumptions), as do the re* *st of the proofs in Section 6. Some authors use these notions in pointed model categories [8], where the def* *i- nitions are clear: contractible objects are weakly equivalent to the zero objec* *t and a null map is homotopic to a map which factors through the zero object. Our aim here is to discuss contractible objects and null maps in any model category, wh* *ile generalizing the pointed model category case. Definition A.1. A category with contractible objects is a pair (M, pt), where M* * is a model category and pt is a distinguished object in M, which satisfies that ev* *ery retract of pt is naturally isomorphic to pt. This distinguished object is calle* *d the one-point object or singleton. An object U in M is called contractible if U is * *weakly equivalent to the one-point object pt. A map f :A ! X is called null homotopic (nullmap or just null) if it factors up to homotopy (both left and right) throu* *gh the one-point object. Example A.2. In any model category M the initial and terminal objects may be chosen to be one-point objects. They lead, however, to different notions of contractibility and null maps. If M is a pointed model category (; = * = 0), th* *en the only object which may be taken as a singleton is the zero object pt= 0. The notions of a singleton, contractible object and nullmap are self-dual. In the present paper we always take pt = *, the terminal object in the category of diagrams of spaces. Proposition A.3. Let M be a model category with a singleton pt and let f :A ! X be null homotopic. (1) If A is a cofibrant object in M, then f factors through a cofibrant cont* *ractible object. (2) If X is a fibrant object in M, then f factors through a fibrant contract* *ible object. Proof.We will prove part (1); the proof of part (2) is dual. Since f is null homotopic, there exists a map g = '', where A -'! pt- '!X homotopic (both left and right) to f. Choose a good cylinder object A ^ I and a left homotopy H :A ^ I ! X between f and g. Then the following solid arrow 30 BORIS CHORNY diagram is commutative, ' A fi_l_______A____//"p`t"_` __i0_____________________________________________________* *____________________||________________________________ ____"b___________________________________________________* *_____________________________________fflOi1|fflO|_____________@ ____&&__________________________________________________* *_______________________________fflffl|fflffl|'________________@ _________________________________________________A/^/I* *_CA _____________________________________________________* *____________________C ____________________________________________________* *_________________________________________C ___________________________________________________* *________H_____________________________________________C _____________________________________~~__________* *______________________________________________________________@ f ______________22_______________________________* *______________________________________________________________@ ` where CA = pt A A ^ I. 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