Class-combinatorial model categories

CLASS-COMBINA TORIAL MODEL CA TEGORIES B. CHORNY AND J. R OSICK ´ Y ∗ Abstract. W e extend the framew ork of com binatorial model cat- egories, so that the category of small presheaves o ver large indexing categorie s and ind-categor ies would b e em bra ced b y the new ma- chinery called class-c o mbinatorial mo del ca tegories. The definition of the new cla ss o f model categor ies is based on the correspo nding extension of the theory of locally presentable and a ccessible ca tegories develop ed in the compa nion pap er [10], where we intro duced the concepts of lo ca lly c lass-pres entable and class-ac cessible catego ries. In this work we prov e that the category o f w eak equiv alences of a nice class -combinatorial mo del categ ory is class-a ccessible. Our extension of J. Smith lo c alization theorem depends on the verifi- cation of a cosolution-set condition. The deep est result is that the (left Bousfield) lo ca lization of a cla ss-combinatorial model cate- gory with resp ect to a str ongly class -accessible lo caliza tion functor is class-co m binato r ial again. 1. Intro duction The theory of com binatorial mo del categories pioneered by J. Smith in the end of ’90- s has b ecome a standard framework for abstract ho- motop y theory . The foundatio ns of the sub ject may b e found in [2] and [13, 14]; a concise exp osition has app eared in [24, A.2.6]. A mo del category M is combin ato r ia l if it satisfies t w o conditions. The first condition requires that the underlying category M is lo cally presen table (see, e.g., [1] for the definition and an introduction to the sub je ct). The second condition demands that the mo del structure will b e cofibrantly generated (see, e.g., [19] for the definition and discus- sion). Date : O ctob er 1 8, 2 011. Key wor ds and phr ases. class -combinatorial mo del category , Bousfield lo calization. ∗ Suppo rted by MSM 00216 2240 9 and GA ˇ CR 2 01/1 1/052 8. The ho spitality of the Austr alian National Universit y is gratefully acknowledged. 1 2 B. CHORNY AND J. R OSI CK ´ Y Sev eral in teresting examples of non-com binatorial mo del categories app eared ov er the past decade. F or example the categories of pro- spaces and ind-spaces w ere applied in new con texts in homo t o p y the- ory [11, 20] resulting in non- cofibra n tly generated mo del structures constructed on non-lo cally presen table categories. The maturation o f the calculus of homoto py functors [16] stim ulated the dev elopmen t of the abstract homotopy t heory of small functors ov er large categories [9] resulting in f orm ulation of the basic ideas of Go odwillie calculus in the language of mo del categories [3]. The mo de l categories used for this purp ose are also not cofibrantly generated and the underlying category of small functors from spaces to spaces (or sp ectra) is not a lo c ally presen table catego ry . Ho w ev er, all the mo del categories from the examples ab ov e are class- c ofibr antly generated (except for the pro- categories, whic h are class- fibr antly generated). This extension of the classical definition was in- tro duced in [7], whic h in turn dev elop e d the ideas by E. D ror F a rjoun originated in the equiv ariant homotopy theory [15]. The purp ose of the current pap er is to dev elop a fr a mew o r k exte nding J. Smith’s com binato rial mo del categories, so tha t the mo del categor ies of small preshea v es o v er large categories, ind-catego r ies of mo del cate- gories (the opp os ite catego ries of pro- categories) w ould b e come the ex- amples of the newly defined class-c ombinatorial mo del categories. The definition of the class-com binatorial mo del cat ego ry consists, similarly to the combinatorial mo del category , of tw o conditions: the underly- ing category is required to b e lo c al ly class - pr esentable and the mo del structure m ust b e class-cofibran tly generated. As w e men tioned ab o ve , the second condition was studied in the earlier work [7], while the first condition relies on a concept of the lo cally class-presen table category , whic h w as intro duced and studied in the companion pro ject [10], which is a prerequisite for reading this pap er. The main results of our pap er are generalizing the corresp onding results ab out the combinatorial mo del categories. In Theorem 2.10 w e prov e that the lev elwise w eak equiv alences in the category of small preshea ves form a class-accessible category (see [10] for the definition). In Remark 2.12 w e form ulate the mild conditions, whic h g uaran tee that the class-combin ato rial mo del category has the class-accessible sub cat- egory of w eak equiv alences. Such class-combin ato rial mo del cat ego ries are called nice in this pap er. The cen tral result of J. Smith’s theory is the lo c alization theorem, stating the existence of the (left Bousfield) lo calization of any com- binatorial mo del category with resp ect to an y set of maps. After a brief discussion of construction of lo calization functors with respect CLASS-COMBINA TORI AL MODEL CA TEGORIES 3 to cone-coreflectiv e cla sses of cofibrations with bo unded presen tability ranks of do mains and co domains , we pro v e in Theorem 3.10 a v ari- an t of a lo calization t heorem for nice class-com binatoria l mo de l cate- gories with r esp ect to strongly class-accessible ho mo t op y lo calization functors (i.e., lo calization functors preserving λ -filtered colimits a nd λ -presen table o b jects for some cardinal λ ) . Although a n application of our lo calization theorem dep ends on the v erification of a cosolution- set condition for the class of intende d generating trivial cofibrations, w e are able to c hec k t his condition in many interes ting situations. In the last Theorem 3.14 we prov e that in the cases where the lo c alization with resp ect to a strongly class-accessible functor exists, the lo calized mo del category is class-com binatorial ag a in. W e conclude the pap er b y sev eral examples of lo calized mo del categories. Using Theorem 3.14 w e sho w that the n -p o lynomial mo del category constructed in [3] is class- com binatorial (Example 3.16). On the other hand, there is a mo del category constructed in [8] as a lo calization of a class-com binatorial mo del category with resp ect to an inaccessible lo calization functor that happ ens to b e non- cofibran tly generated ( Example 3.17). 2. Class-combina torial model c a tegories Recall that a w eak f a ctorization system ( L , R ) in a lo cally class- λ - presen table category K w as called c ofibr an tly class - λ - gener ate d in [10] 4.7 if L = cof ( C ) for a cone-coreflectiv e class C of mor phisms suc h that (1) morphisms fro m C ha v e λ -presen table domains a nd co domains and (2) an y morphism b etw een λ - presen ta ble ob jects has a weak fac- torization with the middle ob ject λ - presen ta ble. T o b e c one-c or efle ctive means for eac h f there is a subset C f of C suc h that eac h morphism g → f in K → with g ∈ C factorizes as g → h → f with h ∈ C f . If the w eak factorization is functorial, a cofibrantly class- λ -generated w eak factor ization system is cofibran tly class- µ -generated f or eac h reg- ular cardinal µ D λ . Without functoriality , condition ( 2 ) do es not need to go up to µ and th us w e will make it a part of the follo wing definition. Definition 2.1. Let K b e a class lo cally- λ -presen table mo del category . W e say that K is class - λ - c ombinatorial if b oth (cofibrations, trivial fibrations) a nd (trivial cofibrations, fibrations) are cofibran tly class- µ - generated w eak fa cto r ization systems for eve ry regular cardinal µ D λ . 4 B. CHORNY AND J. R OSI CK ´ Y It is called class-c ombinatorial if it is class- λ -combinatorial for some regular cardinal λ . An y combinatorial mo del category is class-com binatorial. The rea- son is that weak factorizations are functorial and, moreov er, the result- ing functors are accessible. Thus they are strongly accessible and this prop erty go es up fo r λ ⊳ µ (cf. [26]). Example 2.2. Let SSet denote the category of simplicial sets. Given a simplicial category A , b y abuse of no tation, P ( A ) will denote the category o f smal l simplicia l pr esh e aves o n A . The ob jects are functors A op → SSet whic h are small w eigh ted colimits o f simplicial repre- sen ta ble functors (see [12]). In [10], we used this notation for small preshea ves on a category A but it will cause any misunderstanding. The simplicial category P ( A ) is complete prov ided that A is complete (see [12]); completeness is mean t in the enric hed sense (see, e.g., [4] o r [22]). The category P ( A ) is alw a ys class-finitely-accessible , b ecause each small simplicial presheaf is a conical colimit of preshea v es from G = { hom A ( − , A ) ⊗ K | A ∈ A , K ∈ SSet } . Therefore eac h small presheaf is a filtered colimit of finite colimits of elemen ts of G . The elemen ts of G are, in turn, filtered colimits of the elemen t s of G fin = { hom( − , A ) ⊗ L | A ∈ A , L ∈ SSet fin ) } where SSet fin denotes the full sub cat ego ry of SSet consisting of finitely presen table simplicial sets. Therefore, ev ery small presheaf is a filtered colimit of finite colimits of the elemen ts of G fin . W e are going to sho w that, for a complete simplicial category A , t he category P ( A ) equipp ed with the pro jectiv e mo de l structure is class- com binatorial. W e will need the f o llo wing result. Lemma 2.3. L et A b e a c ompl e te s i m plicial c a te gory and µ a n un- c ountable r e gular c ar din a l. Then µ -pr esentable obje cts in P ( A ) ar e close d under fi n ite weighte d limits. Pr o of. A w eight W : D → SSet is finite if D has finitely many ob jects and all its ho m-ob jects D ( c, d ) and all v alues W ( d ) are finitely pre- sen ta ble simplicial sets (see [23 ] 4.1.) F ollowing [23] 4.3, finite w eigh ted limits can b e constructed from finite conical limits and cotensors with finitely presen table simplicial sets. Th us w e ha v e to show that µ - presen table ob jects in P ( A ) are closed under these limits. F ollo wing [4] 6.6.1 3 and 6.6.16, finite conical limits in P ( A ) coincide with finite limits in the underlying category P ( A ) 0 . This underlying catego r y is a sub category of the catego r y P ( A 0 ) of small functors from [10] 2.2 CLASS-COMBINA TORI AL MODEL CA TEGORIES 5 (2) whic h is closed under limits and colimits. Thus µ - presen ta ble ob- jects in P ( A ) are µ -presen table in P ( A 0 ). Since the la tter ob jects are closed under finite limits (see [21] 4.9), µ - presen ta ble ob jects in P ( A ) are closed under finite conical limits. It r emains to show that they are closed under cotensors with finitely presen table sim plicial sets. The later a r e finite conical colimits of cotensors with ∆ n , n = 1 , 2 , . . . (see the pro o f of [12 ], 5.2). Th us we hav e to sho w that µ -presen table ob jects in P ( A ) a re closed under cotensors with ∆ n ’s. Let H b e µ -presen table in P ( A ). Since H is a µ -small colimit of tensors H i of finitely presen t a ble simplicial sets with represe ntables and b oth colimits and cotensors in P ( A ) are p oint wise, w e ha v e H ∆ n ( A ) = (colim H i ) ∆ n ( A ) = (colim H i ( A )) ∆ n = hom (∆ n , colim H i ( A )) ∼ = colim hom(∆ n , H i ( A )) = colim H i ( A ) ∆ n = (colim H ∆ n i )( A ) for eac h A in A . Hence H ∆ n ∼ = colim H ∆ n i and th us it suffices to sho w that each H ∆ n i is µ -presen table. Since each H i is equal to V ⊗ hom( − , B ) for some finitely presen table simplical set V and B in A , w e get fo r the same reasons as ab ov e hom(∆ n , V ⊗ ho m( A, B )) = hom(∆ n , V × hom( A, B ) ) ∼ = hom(∆ n , V ) × hom(∆ n , hom( A, B )) ∼ = V ∆ n × hom( A, B ∆ n ) = ( V ∆ n ⊗ hom( − , B ∆ n ))( A ) and th us ( V ⊗ hom( − , B )) ∆ n ∼ = V ∆ n ⊗ hom( − , B ∆ n ) . The latter ob jects are µ - presen table.  Prop osition 2.4. L et A b e a c omplete simplicial c ate gory. Then P ( A ) is clas s - λ -c ombin a torial with r esp e ct to the pr oje ctive mo d el structur e for e ach unc ountable r e gular c ar dina l λ . Pr o of. F ollow ing [9], P ( A ) is a mo d el category where the generat- ing classes I and J of cofibrations and trivial cofibrations are cone- coreflectiv e and satisfy condition [10] 4.7 (1) for an y regular cardinal λ . In fact, I consists of morphisms ∂ ∆ n ⊗ hom( − , A ) → ∆ n ⊗ hom( − , A ) and J of mor phisms Λ k n ⊗ hom( − , A ) → ∆ n ⊗ hom( − , A ) , 6 B. CHORNY AND J. R OSI CK ´ Y and all inv olv ed domains a nd co doma ins a r e finitely presen table. W e ha v e to sho w that they satisfy [10] 4.7 (2) as we ll, i.e., that they are b ounded. Let λ be uncountable and f : G → H be a morphism b et w een λ -presen t able ob jects a nd consider a morphism g → f where g ∈ I . F ollo wing the pro of of 3.7 in [9], this morphism corresp onds to a morphism hom( − , A ) → P where P is t he pullbac k G ∂ ∆ n × H ∂ ∆ n H ∆ n . Since P is λ - presen ta ble (see 2.3), there is a c hoice of a set T f from [10] 4.8 (2) whose car dina lity do es not exceed λ . Since a ll morphisms from I hav e finitely presen table domains and co domains, the factorizat io n of f stops at ω . Th us the cardinality of T ∗ f is smaller than λ . F ollowing [10] 4.8 (2) , condition [10] 3.7 (2) is satisfied. The argumen t fo r J is the same.  Remark 2.5. A ve ry useful prop erty of the comb inator ial mo del cat- egories is that the class o f w eak equiv alences is an accessible and a c- cessibly embedded sub category of the catego ry of morphisms K → (see [27] 4 .1 or [24] A2.6.6). T ogether with Smith’s theorem [2 ] it consti- tutes the lo calization theorem for com binatorial mo de l categories with resp ect to sets o f maps. It would b e natural to exp ect tha t a similar prop erty holds in the class-com binatorial situation. Unfortunately w e w ere unable to prov e it in this generality . But in man y in teresting sit- uations w e are able to pro v e that t he class o f w eak equiv alences is a class-access ible subcatego r y of the catego ry of morphisms. Lemma 2.6. L et A b e a c omplete si m plicial c ate gory. Then P ( A ) admits a str ongly class-ac c essible fibr ant r ep l a c ement functor. Pr o of. The functor Ex ∞ : SSet → SSet is the finitely accessible fibrant replacemen t simplicial functor (see [17]). F or a small simplicial f unctor F : A op → SSet , let Fib( F ) b e the comp os ition A op F − − − − → SSet E x ∞ − − − − − − → SSet . W e will sho w that this comp osition is small. The category of finitely accessible simplicial functors SSet → SSet is equiv alent to the category of simplicial functors SSet SSet fin . This equiv a lence sends a finitely a ccessible functor SSet → SSet to its restriction on SSet fin . Thu s hom-functors hom( S, − ) : SSet → SSet with S finitely presen ta ble corresp ond to hom-functors hom( S, − ) : SSet fin → SSet . Since ev ery simplical functor SSet fin → SSet is a w eigh ted colimit of hom-functors, ev ery finitely accessible simplicial functor SSet → SSet is a weigh ted colimit of hom-functors hom( S, − ) with S finitely presen table. Th us the comp o sition Ex ∞ F is a w eigh ted CLASS-COMBINA TORI AL MODEL CA TEGORIES 7 colimit of functors hom ( S, − ) F with S finitely presen table. But the functor hom( S, − ) F is small b ecause it is isomorphic to the cotensor F S . The reason is that natural transformations A ( − , A ) → ho m ( S, − ) F = hom( S, F − ) corresp ond to morphisms S → F A , i.e., t o morphisms S → P ( A )( A ( − , A ) , F ) whic h, by the definition of the cotensor, corresp ond to morphisms A ( − , A ) → F S . Consequen tly , Ex ∞ F is small as a weigh ted colimit of small functors. W e hav e obtained the functor Fib : P ( A ) → P ( A ) whic h clearly has fibran t v alues. Moreo v er, the p oin twis e tr ivial cofibration Id SSet → Ex ∞ yields a w eak equiv alence Id P ( A ) → Fib. Th us Fib is a fibrant re- placemen t functor on P ( A ). Since Ex ∞ is finitely accessible , so is Fib. W e kno w that Ex ∞ is a w eigh ted colimit of hom-functors ho m( S, − ) with S finitely presen table. The corresp onding w eight is λ -small for an uncoun table regula r cardinal λ . Let F b e λ -presen table in P ( A ). Then Fib( F ) is a λ -small weigh ted colimit o f hom( S, − ) F ∼ = F S and the latter functors ar e λ -presen table follo wing 2 .3. Hence Fib( F ) is λ -presen table (the argumen t is analogous to [23], 4.14). Th us Fib is strongly class- λ -accessible.  Definition 2.7. Let A b e a complete simplicial category and f : A → B b e a morphism in P ( A ). The Serr e c o n struction o n f is the ob ject S ( f ) of P ( A ) defined as a pullbac k S ( f ) r / / q   B ∆ 1 B j   B f / / B where j : ∆ 0 → ∆ 1 sends 0 t o 0. Remark 2.8. The Serre construction w as used in the PhD t hesis of J.P . Serre in order to replace a n arbitrary map of top o logical spaces b y a fibration. W e are going to use it prett y m uch for the same purp ose in P ( A ). The adv antage ov er the mo dern metho ds of factorization is the functoriality of S ( − ). 8 B. CHORNY AND J. R OSI CK ´ Y Lemma 2.9. L e t A b e a c omplete simp l i c ial c ate go ry and f : A → B a morphism o f fibr ant obj e cts in P ( A ) . Then ther e exists a factorization A i − − − − → S ( f ) p − − − − → B of f wher e i is a we ak e quivalenc e a nd p is a fibr ation. Pr o of. The pullbac k in 2 .7 may b e split in t w o pulbac ks S ( f ) r / / q 1   B ∆ 1 B u   A × B f × id B / / q 2   B × B B v   A f / / B where ∆ 0 v − − − − → ∆ 0 + ∆ 0 u − − − − → ∆ 1 is the factorization of j . Since b o th u and v a re cofibrations and B is fibran t, the v ertical morphisms B u and B v are fibrations (see [18] 9.3.9 (2a)). Moreov er, since j is a trivial cofibration, B j is a trivial fibra t ion. Th us q 1 and q 2 are fibrations and q = q 2 g 1 is a t r ivial fibratio n. Let t denote the unique morphism ∆ 1 → ∆ 0 . Since, B j f ∆ 1 A t = f A j A t = f , there is a unique morphism i : A → S ( f ) suc h that q i = id A and r i = f ∆ 1 A t . Since q is a trivial fibration, i is a w eak equiv alence. Since B v : B × B → B is the first pro jection of t he pro duct, q 2 : A × B → A is the first pro jection as w ell. Let p 2 : A × B → B , p 2 : B × B → B b e the second pro jections and v ′ : ∆ 0 → ∆ 0 + ∆ 0 b e t he second injection of the copro duct. Then p 2 = B v ′ and p 2 q 1 i = p 2 ( f × id B ) q 1 i = p 2 B u r i = p 2 B u f ∆ 1 A t = B v ′ B u f ∆ 1 A t = B uv ′ f ∆ 1 A t = f A uv ′ A t = f . Since B is fibran t, p 2 is a fibration and th us p = p 2 q 1 is a fibration. W e ha v e f = pi .  Theorem 2.10. L et A b e a c om plete simplicial c ate gory and denote by W the class of we ak e quivalenc es in the pr oje ctive mo del structur e on P ( A ) . Then W is a class-ac c essib le c ate gory s tr ongly ac c essi b ly emb e dde d in P ( A ) → . CLASS-COMBINA TORI AL MODEL CA TEGORIES 9 Pr o of. Let Fib : P ( A ) → P ( A ) b e the strongly class-accessible fibran t replacemen t functor constructed in 2.6. Consider the functor R : P ( A ) → → P ( A ) → assigning to a morphism f : A → B the fibration p : S (Fib( f )) → Fib( B ) f r om 2.9. Since the construction of S ( f ) uses only finite limits, the functor S ( − ) : P ( A ) → → P ( A ) is strongly class-accessible b y 2.3. Therefore the functor R ( − ) is a lso strongly class-accessible . A morphism α : F → G in P ( A ) is a w eak equiv a lence if and only if Fib( f ) is a w eak equiv alence, i.e., if and only if R ( f ) is a trivial fibration. Let F 0 denote the full sub category of P ( A ) → consisting of trivial fibrations. F ollow ing 2.4 and [10] 4.9, F 0 is class- λ -accessible and strongly class- λ -accessibly em b edded in P ( A ) → for ev ery uncoun table regular cardinal λ . Since W is giv en b y the pullback K → R / / K → W O O / / F 0 O O whose vertical leg on the right is transp ortable, W is equiv alen t to the pseudopullbac k (see [10] 3 .2). Thus [10] 3 .1 implies that W is a class-access ible subcatego r y of P ( A ) → .  Definition 2.11. A class-com binatorial mo del category K is nic e if the class of w eak equiv alences W is a class-accessible, strongly class- accessibly embedded sub category of K → . Remark 2.12. W e hav e just prov ed that P ( A ) equipped with the pro jectiv e mo del structure is a nice mo del category for any complete simplicial category A . The same argumen t applies to ev ery simplicial class-com bina t o rial mo del category whic h is equipp ed with a strongly class-access ible fibran t replacemen t functor and whose µ -presen table ob jects are closed under finite w eighted limits for eac h µ ≥ λ (where λ is a cardinal). W e are not a ware of any example of a class-com binatoria l mo del cat ego ry , whic h w ould fail to b e nice. Theorem 2.13. L et K b e a lo c al ly class- λ -pr esentable c ate gory, I a λ -b ounde d cla ss of morphisms and W a class of morphism of K such that (1) W is a class- λ - ac c e s sible and str ongly class - λ -ac c essibly emb e d- de d sub c ate g ory of K → with the 2-out-of-3 pr op erty, 10 B. CHORNY AND J. R OSI CK ´ Y (2) I  ⊆ W , and (3) cof ( I ) ∩ W is close d under p usho ut and tr ansfinite c omp osition and c one- c or efle ctive in K → . Then, taking cof ( I ) for c ofibr ations and W for we ak e quivalen c es, we get a mo del c ate gory structur e on K . Pr o of. Since I is λ -b ounded, (cof ( I ) , I  ) is a cofibrantly class- λ -gene- rated w eak factorization system. F or ev ery λ -presen table w ∈ W , we construct a factorization in K in to a cofibration j follow ed b y a trivial fibration. By 2-o ut-of-3 prop ert y for W a nd (2), j is in W . Let J b e the class o f these morphisms j for all λ -presen table w ∈ W . W e will che c k no w the conditions of Lemma [2, 1.8]. W e hav e to sho w that f or eve ry morphism i → w in K → with i ∈ I and w ∈ W there exists j ∈ J that f a ctors it i → j → w . First note tha t there exists a λ -presen table w ′ ∈ W , whic h factors the or ig inal morphism, since eve ry w is a λ -filtered colimit of λ -presen t a ble ob jects W and ev ery i ∈ I is λ -presen table in K → ; w e used here t hat the inclusion of W to K → preserv es λ -presen table ob jects. Next, decomp ose that morphism w ′ in to a cofibration j ∈ J follo w ed by a trivial fibration. The lifting axiom in K finishes t his argumen t. Lemma [2, 1 .8 ] implies that cof J = cof I ∩ W . The requ iremen t that cof I ∩ W is cone-coreflectiv e in K → ensures that J is cone-coreflectiv e as well (by the same argumen t as ab ov e). By construction, the do- mains of all the eleme nts in J are λ -presen table. Hence J satisfies the assumptions of [10 ] 4.3 and thus (cof ( J ) , J  ) is a weak f a ctorization system. Since W is closed under retracts in K → (cf. [1] 2.4 and 2.5), w e get a mo del category structure on K .  Remark 2.14. Let K b e a lo cally presen table category , I a set of morphisms and W a class of morphism of K suc h tha t (1) W ha s the 2-out - of-3 prop erty and is closed under retracts in K → , (2) I  ⊆ W , and (3) cof ( I ) ∩ W is closed under pushout and transfinite comp osition. Then, taking cof ( I ) for cofibrations and W fo r w eak equiv alences, w e get a com binatoria l mo del category if a nd only if the inclusion of W in K → is accessible. This is the con tent of the Smith’s theorem (see [2 ] for sufficiency and [24] or [27] for necessit y). W e do no t kno w whether this can b e generalized to class-accessible setting and 2.13 is what w e are able to do. The question is whether cone-coreflectivit y of cof ( I ) ∩ W fo llo ws fro m the other assumptions. CLASS-COMBINA TORI AL MODEL CA TEGORIES 11 W e also do not kno w whether the mo del category in 2.13 is class - com binatorial. Indeed, we only know that the class J satisfies [1 0] 4.7 (1). Homotop y equiv alences can b e defined in an y catego ry K with finite copro ducts whic h is equipp ed with a w eak f actorization system ( L , R ) (see [27]). Recall that a cylinder obje ct C ( K ) of an ob ject K is giv en b y an ( L , R ) factorization o f the co diagonal ∇ : K + K γ K − − − − − → C ( K ) σ K − − − − − → K W e denote b y γ 1 K , γ 2 K : K → C ( K ) the comp ositions of γ K with the copro duct injections. Then, as usual, w e say tha t morphisms f , g : K → L are homotopic , and write f ∼ g , if there is a morphism h : C ( K ) → L suc h that the follo wing diag r am commutes K + K ( f ,g ) / / γ K $ $ H H H H H H H H H H H H H L C ( K ) h = = { { { { { { { { { { { { Here, ( f , g ) is induced by f and g . The homotop y relation ∼ is clearly reflexiv e, symmetric, compat ible with the comp osition and do es not dep end on the c hoice of a cylinder ob ject. But, it is not tra nsitive in general and w e will denote its transitiv e h ull by ≈ . W e get the quotient functor Q : K → K / ≈ . A morphism f : K → L is called a homotopy e quivalenc e if Qf is the isomorphism, i.e., if there exists g : L → K suc h that b oth f g ≈ id L and g f ≈ id K . The full sub categor y of K → consisting of homo t o p y equiv a lences w.r.t. a weak factorization system ( L , R ) will b e denoted b y H L . The fo llowing result generalizes [27], 3.8. Prop osition 2.15. L e t K b e a lo c al ly class-pr esentable c ate gory a nd ( L , R ) b e a we ak fac toriza tion system with a str ongly cl a ss-ac c essible cylinder functor. Then H L is a ful l im age of a str ongly class-ac c essible functor into K → . Pr o of. Giv en n < ω , let M n b e the category wh ose ob j ects are (4 n + 2 )- tuples τ = ( f , g , a 1 , . . . , a n , b 1 , . . . , b n , h 1 , . . . , h n , k 1 , . . . , k n ) 12 B. CHORNY AND J. R OSI CK ´ Y of morphisms f : A → B , g : B → A , a 1 , . . . , a n : A → A , b 1 , . . . , b n : B → B , h 1 , . . . , h n : C ( A ) → A and k 1 , . . . , k n : C ( B ) → B . Mor- phisms are pairs ( u, v ) of morphisms u : A → A ′ and v : B → B ′ suc h that f ′ u = v f , g ′ v = u g , uh i = h ′ i C ( u ) and v k i = k ′ i C ( v ) for i = 1 , . . . , n . This category is obtained b y an inserter construction inserting o ur n + 2 morphisms among Id and C . Since C is strongly class-access ible, the pro cedure of the pro of o f [10] 3.9 yields that M n is a class-accessible category . Let M n b e the full sub category of M n suc h that h 1 γ A = ( g f , a 1 ), h i γ A = ( a i , a i +1 ), h n γ n = ( a n , id A ), k 1 γ A = ( f g , b 1 ), k i γ A = ( b i , b i +1 ) and k n γ n = ( b n , id B ) where 1 < i < n . This category is obtained f rom M n b y an equifier construction and, by the same reason as ab ov e, the pro cedure of the pro of o f [10] 3.7 yields that M n is class-accessible. Moreo v er, its inclusion into M n is strongly accessible. W e hav e full em b eddings M m,n : M m → M n , for m < n , whic h tak e the missing a i , b i , h i , k i as the identities. The union M of all M n ’s is a class-accessible category . Since all M n ’s are strongly access ibly em b edded to M n , M is strongly accessible em b ed- ded by to M . Let F : M → K → sends each (4 n + 2)-tuple ab o ve to f . This is a strongly class-accessible functor whose image is H L .  3. Left Bousfield localiza tions Recall that ˜ h is a c ofibr a n t appr ox imation o f h if there is a commu - tativ e square A v / / h   ˜ A ˜ h   B w / / ˜ B where v and w are weak equiv alences. Definition 3.1. Let K b e a class-com binato r ia l simplicial mo del cat- egory and F a class of morphisms of K . Assume t hat F contains only cofibrations b etw een cofibran t ob jects. An ob ject K in K is called F - lo c al if it is fibran t and hom( f , K ) : hom( B , K ) → hom( A, K ) CLASS-COMBINA TORI AL MODEL CA TEGORIES 13 is a weak equiv alence of simplicial sets for eac h f : A → B in F . A morphism h of K is called an F - lo c al e quivalenc e if ho m( ˜ h, K ) is a we ak equiv alence f o r each F -lo cal ob ject K ; here, ˜ h is a cofibrant appro ximation of h . The full sub category of K consisting of F -lo cal ob j ects is denoted Lo c( F ) and t he full sub category of K → consisting of F -lo cal equiv a- lences is denoted LEq( F ). W e sa y that there exists a left Bousfield lo c aliza tion of F if cofibra- tions in K and F -lo cal equiv alences form a mo del category structure on K . Remark 3.2. (1) It is easy to see that the definition of a lo cal F - equiv a lence do es not dep end on the c hoice o f a cofibran t approximation. (2) F ollowing [18], 9.3.3 (2), any w eak equiv alence in K is an F -lo cal equiv a lence. On the other hand, ev ery F -lo cal equiv alence b etw een F -lo cal ob jects is a w eak equiv alence in K (cf. [17] X.2.1. 2)). (3) If K is left prop er then t he in tersection of cofibrations and F - lo cal equiv alences is closed under pushout and transfinite comp o sition (see [18], 13.3.10 , 17.9.4 for a non-trivial par t of the pro of ); the trivial part is t ha t hom( − , K ) sends colimits to limits and cofibrations t o fibrations. It is also closed under r etracts in K → of course. Giv en a morphism f , { f } - lo cal ob jects are called f -lo cal and analo- gously for f -lo cal equiv alences. The corresp onding categories are called Lo c( f ) and LEq( f ). Prop osition 3.3. L et K b e a class-c ombi n atorial simplicial mo del c at- e gory a n d F a set of c ofibr ation s b etwe e n c ofibr an t obje c ts of K . Then Lo c( F ) is a class-ac c ess i b le c ate g o ry str ongly ac c es s i b ly em b e d d e d in K . Pr o of. K hom( f , − ) / / SSet → Lo c( f ) O O / / W O O is a pullbac k where W denotes w eak equiv alences in SSet . Since the v ertical leg on the right is tra nsp ortable, Lo c( f ) is a pseudopullbac k and thus it is a class-accessible and its inclusion to K is strongly class- accessible (see [10] 3.1 , 3.2 a nd 2.10). Since Lo c( F ) = \ f ∈F Lo c( f ) , 14 B. CHORNY AND J. R OSI CK ´ Y the result follo ws fr o m [10] 3.3.  Definition 3.4. Let K b e a simplicial mo del categor y and f : A → B a cofibration of cofibrant ob jects. Consider a pushout ∂ ∆ n ⊗ A id ⊗ f / / i n ⊗ id   ∂ ∆ n ⊗ B p n 1   ∆ n ⊗ A p n 2 / / P f ,n where i n : ∂ ∆ n → ∆ n is the inclusion of the b oundary into a simplex. Let h f ,n : P f ,n → ∆ n ⊗ B b e the canonical morphism, whic h is a cofibration since K is simplicial. Cofibrations h f ,n , n = 0 , 1 , . . . are called f - horns . If F is a class o f cofibrations, then we denote b y Hor( F ) the collection of a ll f -horns, for all f ∈ F . Remark 3.5. Every h f ,n ∈ Hor( F ) is an F -lo cal equiv alence b ecause the morphism hom( h f ,n , K ) : hom(∆ n ⊗ B , K ) → hom( P f ,n , K ) is a w eak equiv alence for ev ery F - lo cal ob j ect K . In fact, the morphism hom(id ⊗ f , K ) : hom(∆ n ⊗ B , K ) → hom(∆ n ⊗ A, K ) is a w eak equiv alence b ecause K is F -lo cal and hom( p n, 2 , K ) is a trivial fibration as a pullbac k of the trivial fibration hom(id ⊗ f , K ) : hom( ∂ ∆ n ⊗ B , K ) → hom( ∂ ∆ n ⊗ A, K ) . Th us it suffices t o use t he 2-out- of-3 prop erty . W e used the fa ct t ha t f - horns are cofibrations b etw een cofibran t ob- jects and that the definition o f an F -lo cal equiv alence do es not dep end on the c ho ice of a cofibran t approximation. Lemma 3.6. L et K b e a class-c ombin atorial simplicial mo del c ate gory and F a class of c ofibr ations b etwe en c o fi br ant obje cts of K . Then a fibr ant obje ct K of K is F -lo c al if and on l y if it is inj e ctive to al l f -h o rns fo r f ∈ F . Pr o of. Since eac h f ∈ F is a cofibrations, hom( f , K ) is a fibration for each fibra n t ob ject K . Th us a fibran t ob ject K is F -lo cal if and only if ho m( f , K ) is a trivial fibration for each f ∈ F . This is t he same as ha ving t he righ t lifting prop ert y with resp ect to eac h inclusion i n : ∂ ∆ n → ∆ n . The latter is clearly equiv alent to b eing injectiv e with resp ect to f -horns h f ,n for all f ∈ F .  CLASS-COMBINA TORI AL MODEL CA TEGORIES 15 Lemma 3.7. L et F b e a c one -c or efle ctive class of c o fi br ations b etwe en λ -pr esentable c ofibr a nt obje cts. Th en Hor( F ) is c one-c or efle ctive class of morphisms b etwe en λ - p r esentable obje c ts. Pr o of. Since ∂ ∆ n ⊗ B and P f ,n are λ -presen table pro vided that A a nd B are λ -pr esen table, w e ha ve to prov e that Hor( F ) is cone-coreflectiv e. Let f : A → B b e a n elemen t of F . Given a commu tative square P f ,n / / h f ,n   X g   ∆ n ⊗ B / / Y with h f ,n ∈ Hor( F ) and g arbitrary , w e form, by adjunction, the fol- lo wing comm uta tiv e square: A f   / / X ∆ n g ′   B / / Q g ,n where Q g ,n = X ∂ ∆ n × Y ∂ ∆ n Y ∆ n . Since F is cone-coreflectiv e, there exists a set of morphisms F g ′ = { f ′ : A ′ → B ′ } ⊂ F suc h that an y morphism f → g ′ in K → factors through some elemen t f ′ ∈ F g ′ . Unrolling bac k the adjunction, w e obtain the set of horns Hor( F g ′ ) = { h f ′ ,n : P f ′ ,n → ∆ n ⊗ B ′ | n ≥ 0 } whic h dep ends en tirely on g . Th us Hor( F ) is cone-coreflectiv e.  Remark 3.8. (1) Let F b e a cone-coreflec tive class of cofibrations b et w een λ -presen table cofibra n t ob jects in a class λ -com binatorial sim- plicial mo del category K . Then Lo c( F ) is weak ly reflectiv e and closed under λ - filt ered colimits in K (fo llo wing 3.7, 3.6 and [10] 4.4). Recall that a weak reflection r K : K → K ∗ is obtained as a factorization K r K − − − − − → K ∗ − − − → 1 . in (cof (Hor( F ) ∪ C ) , (Hor( F ) ∪ C )  ) where C is a b ounded class suc h that cof ( C ) are cofibra t ions in K . Thus r K b elongs to cof (Hor( F ) ∪C ). If K is left prop er then, follo wing 3.2 (3), cof (Ho r ( F ) ∪ C ) ⊆ cof ( C ) ∩ LEq( F ). Hence r K is b o t h a cofibrat io n and an F -lo cal equiv alence. But this do es not mean that weak reflections are functorial, i.e., that there exists a functor L : K → Lo c( F ) and a natural transformation 16 B. CHORNY AND J. R OSI CK ´ Y η : Id → L suc h that η K = r K for eac h K in K . Suc h a functor L is called an F - lo c al i z ation func tor . (2) Giv en a mo del category K and a functor L : K → K , then LEq( L ) will denote the class of morphisms sen t by L to w eak equiv alences. If b oth the left Bousfield lo calization and a lo calization functor L exist for F , then LEq( F ) = LEq( L ). In fact h is an F -lo cal equiv alence iff its cofibrant appro ximation ˜ h is an F - lo cal equiv alence. Since η K is an F - lo cal equiv alence f o r eac h K , ˜ h is an F - lo cal equiv alence iff L ( ˜ h ) is an F -lo cal equiv alence, i.e., a w eak equiv alence in K (see 3.2 (2)). Prop osition 3.9. L et K b e a nic e class-c ombinatorial mo de l c ate gory and L : K → K b e a str ongly class-ac c essible functor. Then LEq( L ) i s a class-ac c essible c ate gory str on gly class-ac c ess ibly emb e dde d in K → . Pr o of. By a ssumption, the class W of w eak equiv alences is class-acce- ssible and strongly class-access ibly em b edded in K → . Since LEq ( L ) is giv en by the pullback LEq( L ) / /   W   K → L → / / K → ha ving the v ertical leg on the righ t transp ortable, LEq( L ) is a pseu- dopullbac k and th us class-accessible a nd strong ly class-accessibly em- b edded in K → (see [10] 3.1 and 3 .2).  Theorem 3.10. L e t K b e a nic e, class-c omb inatorial, left pr op er, sim- plicial mo del c ate gory and let F b e a c lass of morph i s m s in K . Supp ose ther e exists a str ongly class-ac c essible F -lo c alization functor L : K → K . Then the left Bousfield lo c alization of K w ith r esp e ct to F exists if and only if the interse ction of LE q ( F ) with the c ofib r ations of K is a c one-c or efle c tive class o f morphisms. Pr o of. Necess ity immediately fo llows from t he existence of the (trivial cofibration, fibration) factorizations in the lo calized mo del cat ego ry cf. [10] 4.2 (2)). In order to establish sufficiency , w e will v erify the conditions of 2.1 3. By 3.8 (2) and 3.9 the sub catego r y LEq( F ) is class-accessible . There is a regular car dina l λ suc h that LEq( F ) is class- λ -accessible and K is class- λ -com binatorial. In fa ct, it LEq( F ) is class- µ - accessible and K is CLASS-COMBINA TORI AL MODEL CA TEGORIES 17 class- ν -combinatorial, it suffices to t ak e µ, ν ⊳ λ . Let I b e the gener- ating class of cofibratio ns in K . Then I  ⊆ LEq( F ) b ecause LEq( F ) con tains all w eak equiv alences. F ollowing 3.2 (3), cof I ∩ LEq( F ) is closed under pushouts a nd transfinite comp ositions.  Example 3.11. Let A b e a complete simplicial category . Then P ( A ) equipped with the pro jectiv e mo del structure is a class-com binatorial mo del category (see 2.4). Let f : V → W b e a cofibration of simplicial sets. Then the class F = { f ⊗ hom( − , A ) | A ∈ A} is b ounded. The argumen t is the same as in the pro of of 2.4. The lo calization o f P ( A ) with resp ect to F is equiv alen t to the lev elwise lo calizatio n with respect to f . Let L f : SSet → SSet b e the f -lo calization functor, i.e., a fibrant replacemen t functor in the f -lo calized mo del category structure on SSet . Then L f is finitely accessible prov ided t ha t V and W a re finitely presen table. Moreo v er, L f is alw ay s simplicial (see [28], 24.2) . Similarly to L emma 2.6, w e get a strongly class-accessible simplicial functor L : P ( A ) → P ( A ) assigning to F the comp osition L f F . Since LEq( L ) = LEq( F ), LEq( F ) is a class-access ible sub category of P ( A ) → b y 3.9 . In a general case, L f is accessible and w e would need an extension of 2.3 to λ -small w eighted limits. This is v alid but w e ha v e not burdened our pap er with a pro of. Remark 3.12. Let F b e a set of cofibrations b et w een cofibrant ob jects in a nice class-com bina t o rial left pro p er mo del category K suc h that K admits a strongly class-accessible fibran t replacemen t functor a nd Hor( F ) is b ounded. Since Ho r( F ) is a set, there is a strongly class- accesible w eak reflection on Hor( F )- injectiv e ob jects (see [10] 4.8 (1)). W e can assume that the b ot h functors are strongly class- λ -a ccessible (see [10] 2.8). Th us they are strongly class- λ + -accessible ([10] 2.8 again) and, following [1 0 ] 4.8 (5 ), there is a strongly class- λ + -accessible F - lo calization functor L . Since LEq( F ) = LEq ( L ), LEq( F ) is stro ng ly class-access ible and strongly class-accessibly em b edded in K → . F ollow- ing 2.13, the left Bousfield lo calization of F exists pro vided that the in tersection of cofibration with LEq( F ) is cone-coreflectiv e. Let K b e a mo del category . A functor L : K → K a equipp ed with natural transformatio n η : Id K → L is called homotopy idemp otent if Lη K and η LK are w eak equiv alences for eac h K in K . Definition 3.13. Let K b e a mo del category equipp ed with a homo- top y idemp oten t functor L : K → K preserving w eak equiv alences. A left Bousfield lo calization of K with resp ect to L , or just L - lo c aliz ation of K is a new mo del structure on K suc h that the class of cofibrations 18 B. CHORNY AND J. R OSI CK ´ Y coincides with the or ig inal class of cofibratio ns in K and the class of w eak equiv alences is LEq( L ). New fibra tions are called L - fib r ations . Theorem 3.14. L et K a nic e, pr op er, simplicial c l a ss-c ombi n atorial mo del c ate gory and L : K → K a str ong l y clas s-ac c essible homo topy idemp otent functor pr es e rving we ak e quivale nc es. Supp ose additional ly, that pul lb acks of L -e quiva l e nc es al o ng L -fibr ations ar e L -e quivalenc es . Then the L -lo c aliza tion exists and is cla ss-c ombi n atorial. Pr o of. It w as sho wn in [6, App endix A], that the pair (cof ( I ) ∩ LEq ( L ) , (cof ( I ) ∩ LEq( L ))  ) is a weak factorization system. They argue as f ollo ws. T a k e i ∈ cof ( I ) ∩ L Eq( L ) and f : X → Y . F or an y morphism i → f in K → w e p erform the follo wing construction: A / /  _ i   X / / f    o @ @ @ @ @ @ @ LX  _ O   Z _     @ @ @ @ @ @ @ P / / w w w w o o o o o o o o o o o o o o o W     B / / > > ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Y / / LY . After applying t he functor L on the morphism f we factor Lf as a trivial cofibration follo w ed b y a fibration in K , obtaining the L -fibra tion W → LY , since this is a fibration of L -lo cal ob j ects. Then constructing P = W × LY Y w e obtain an L -fibration P → Y a s a pullbac k of an L -fibration and an L -equiv alence P → W due t o the additional assumption. The induced morphism X → P is an L -equiv alence b y the 2-o ut-of-3 prop erty . Now w e factor the morphism X → P in to a cofibration follow ed b y a trivial fibration in K . As the comp osition of t w o L -fibrations, the morphism Z → Y is an L -fibration, hence there exists a lift B → Z , sho wing that cof ( I ) ∩ LEq( F ) is cone-coreflectiv e. Lik e in the pro of of 3 .10, there exists a regular cardinal λ suc h that K is λ -comb inato r ia l and L strongly class- λ -accessible. Assume that X and Y are λ -presen table. Then LX, LY and W are λ -presen table. Since K is lo cally λ -presen table simplicial category , the functor E : K → P ( A ) from t he pro of of [10] 2 .6 tak es v a lues in simplicial presheav es and th us P ( A ) can b e ta k en in the sense of 2.2. The functor E sends λ -presen table ob j ects to finitely presen table ones and thu s it is strongly class- λ -acccessible . Since E preserv es limits (see [10] 2.6), 2.3 implies CLASS-COMBINA TORI AL MODEL CA TEGORIES 19 that E P is λ -presen table. Since K is closed under λ -filtered colimits in P ( A ) , P is λ -presen table. Th us Z is λ - presen ta ble. Consequen tly , the L -lo calized mo del catego ry is class- λ -com binatorial.  Example 3.15. T ake f : V → 1 in Example 3.11. F o r suc h maps f - lo calization functor is called also V -n ullification. Then the resulting class of f -equiv alences satisfies the conditions of 3.14, since the nullifi- cation o f spaces (i.e., the lo calization with resp ect to f for f as ab o v e) is a right pro p er mo del cat ego ry (see, e.g., [5]). Hence the mo del cat - egory resulting from the lev elwise nullific ation o f the pro jectiv e mo del structure on the category of small functors is class-com binatorial a gain. Example 3.16. Consider the category P ( SSet op ) of small simpli- cial functors from simplicial sets to simplicial sets equipped with the pro jectiv e mo del structure (see 2.4). Consider the lo calization func- tor L : P ( SSet op ) → P ( SSet op ), L = P n ◦ Fib constructed in [3], where P n is Go o dwillie’s n -th p olynomial a ppro ximation [16] a nd Fib : P ( SSet op ) → P ( SSet op ) is the strongly class-accessible fibran t replace- men t functor f r om 2.6 . Since P n is a countable colimit of finite homo- top y limits of cubical diagrams applied on homo t op y pushouts (joins with finite sets used to construct P n in [16] may b e expresses a s ho- motop y pushouts), it is strongly class-accessible. Th us L is strongly class-access ible, hence the p olynomial mo del structure constructed in [3] is class-combinatorial. The condition on the lo calization functor to b e strongly class-acce- ssible may not b e omitted in 3.14 as the follo wing example show s. Example 3.17. The follow ing lo calizatio n of t he class-com binatorial mo del category P ( SSet ) was constructed in [8]. The lo calization func- tor L : P ( SSet ) → P ( SSet ) is the comp osition of the ev a lua tion func- tor at the one po in t space ev ∗ ( F ) = F (1) with the fibrant replacemen t d ( − ) in simplicial sets and the Y oneda em b edding Y : SSet → P ( SSet ), i.e., L ( F ) = hom( − , [ F (1)). This lo calization functor satisfies the con- ditions of A 6 in [6] (pullback o f an L -equiv alence along an L -fibra tion is a n L -equiv alence a gain), a nd hence there exists the L - lo cal mo del structure o n P ( SSet ). The fibran t ob jects in the lo calized mo del cat- egory ar e the lev elwise fibrant functors w eakly equiv alen t to the repre- sen ta ble functors, but they are no t closed under filtered colimits, since filtered colimit of represen table functors need not b e represen table, no matter ho w lar g e the filtered colimit is. On the other hand, in a class- cofibran tly generated mo del category suffic iently large filtered colimits of fibrant ob jects are fibrant ag ain. In other words, w e obtained the a lo calization of P ( SSet ) , whic h is not class-cofibran tly generated. The 20 B. CHORNY AND J. R OSI CK ´ Y reason is that the lo calization functor L is not class-accessible. See [8] for more details on t his mo del structure. Reference s [1] J. Ad´ amek and J . Rosick´ y, L o c al ly Pr esent able and A c c essible Cate gories , Cambridge Univ ersity Pr e ss 199 4 . [2] T. Beke, S he afifiable ho motopy mo del c ate gories , Math. Pro c. Cam br. Phil. So c. 129 (20 00), 447 –475 . [3] G. Biederma nn, B. Chor ny and O. R¨ ondigs, Calculus of funct ors and mo del c ate gories , Adv ances in Mathematics 214 (2007 ), 92–11 5. [4] F. Borceux, Handb o ok of Cate goric al Algebr a 2 , Cambridge Univ. P r ess 1994. [5] A. K . Bo usfield, On the telesc opic homotopy the ory of sp ac es , T rans. Amer . Math. So c. 35 3 (2001 ), 23 9 1–24 26. [6] A. K. Bousfield and E. M. F riedlander , Homotopy t he ory of Γ -sp ac es, sp e c- tr a, and bisimplicial set s , In Geo metric Applications of Homo to py Theory II, Le c t. Notes in Math. 658, Spr inger-V erlag 197 8. [7] B. Chorny , A gener alization of Quil len ’s smal l obje ct ar gument , J. Pure Appl. Alg. 2 0 4, Issue 3 (2 006), 568–5 83 . [8] B. Cho rny , Br own r epr esentability for sp ac e-value d fun ct ors , Israel J . of Math., to app ear . [9] B. Cho rny and W. G. Dwy er, Homotopy the ory of smal l diagr ams over lar ge c ate gories , F or um Ma th. 21 (2009 ), 167– 1 79. [10] B. Chorny and J. Rosick´ y, L o c al ly class-pr esentable and class-ac c essible c ate gories , preprint 2011 . [11] D. Christens e n and D. Isakse n, Duality and pr o-sp e ctr a , Algebr. Geom. T op ol. 4 (2004), 781– 812. [12] B. Day and S. Lack, Limits of sm al l functors , J. Pure Appl. Alg. 2 10 (2007), 6 5 1–66 3. [13] D. Dugg e r, Universal homotopy the ories , Adv. Math. 164 (200 1), 14 4 –176 . [14] D. Dugger, Combinatorial mo del c ate gories have pr esent ations , Adv. Math. 164 (20 01), 177 –201 . [15] E. Dror F arjoun, Homotopy the ories for diagr ams of sp ac es , Pro c. Amer. Math. So c. 10 1 (1987 ), 18 1 –189 . [16] T. G. Go o dwillie , Calculus III: T aylor Series , Geometry & T op o logy , 7 (2003), 6 4 5–71 1. [17] P .G. Go er ss and J. F. Jar dine, Simplicia l Homotopy the ory , Birkh¨ auser 1999. [18] P . S. Hirschhorn, Mo del Cate gories and their L o c alizations , Amer. Math. So c. 2003. [19] M. Hov ey , Mo del c ate gories , Amer. Math. So c. 1 999. [20] D. Isa ksen, A mo del structur e on the c ate gory of pr o-simplicial sets , T rans. Amer. Math. So c. 35 3 (2001 ), 2805 – 2841 . CLASS-COMBINA TORI AL MODEL CA TEGORIES 21 [21] P . Kara zeris, J. Rosick´ y and J . V elebil, Completeness of c o c ompletions , J. Pure Appl. Alg. 1 9 6 (200 5 ), 229-2 50. [22] G M. Kelly , Basic c onc epts of enriche d c ate gory the ory , Cambridge Uni- versit y Pre s s 1982 . [23] G. M. Kelly , St ructur es define d by finite limits in the enriche d c ontext , I. , Cahiers T op. G´ eom. Diff. XXI I I (198 2 ), 3-42. [24] J. Lurie, Higher T op os the ory , Princeton Univ. Press 2 009. [25] M. Makk ai and R. Par´ e, A c c essible Cate gories: The F oundation of Cate- goric al Mo del The ory , Cont. Math. 104, AMS 19 89. [26] J. Ro sick´ y, Gener alize d Br own r epr esentability in homotopy c ate gories , Th. Appl. Ca t. 1 4 (2005 ), 45 1 -479 . [27] J. Rosick´ y, On c ombinatorial mo del c ate gories , Appl. Ca t. Str. 17 (200 9 ), 303-3 16. [28] M. Shulman. Homotopy limits and c olimits and enrich e d homotopy the ory , arXiv:math.A T/06101 94 . B. Chorny Dep a r tment of Ma thema tics University of Haif a a t Oranim Tiv on, I srael chorny@ma th.haif a . ac.il J. R osick ´ y Dep a r tment of Ma thema tics and St a tistics Masar yk University, F acul ty of Sciences Brno, Czech Republic rosicky@ma th.muni. cz