Locally class-presentable and class-accessible categories

CLASS-LOCALL Y PRESENT ABLE AND CLASS-A CCESSIBLE CA TEGORI ES B. CHORNY AND J. R OSICK ´ Y ∗ Abstract. W e generalize the conce pts of lo ca lly presentable and accessible categorie s. Our framework includes s uc h ca tegories as small presheav es o ver large categor ie s and ind-categories . This gen- eralization is int ended for a pplications in the abstr act homotopy theory . 1. Intr oduction The concept o f a lo cally presen table category w as in tro duced by Gabriel and Ulmer [19]. It w as further generalized by Makk ai and P ar ´ e who introduced accessible categories in the monograph [21] whic h con vincingly demonstrated the imp ortance of this notion. Since then lo cally presen table and accessible categories found n umerous applica- tions in algebra and, most prominently , in homotop y theory , where the concept of a lo cally presen table category w as adapted by J. Smith as a foundation for his theory of com binatorial mo del categories [5, 15, 1 6]. In the pa ssed decade there w ere sev eral interesting examples of non- com binatorial mo del structures constructed on non-lo cally presen table categories [6, 11, 13, 20]. The goal of our work is to extend the not io ns of the lo cally presen table and accessible categories, so that it could serv e as a catego rical foundation fo r an appropriate g eneralization of J. Smith’s theory . Suc h generalization will app ear in the companion article [12]. The definition of an a ccessible cat ego ry consists of a com bination of a completeness condition and a smallness condition. The smallness condition demands the existence of a set A of λ -presen table ob jects suc h that each ob ject is a λ -filtered colimit of ob jects of this set. In our pap er, w e are dropping the assumption that A is a set and we call the resulting concept a class-accessible category . This is no t a new Date : December 27 , 2011 . Key wor ds and phr ases. cla ss-lo cally presentable categor y , class-a ccessible cate- gory , weak factor ization s ystem. ∗ Suppo rted by MSM 00216 22409 and GA ˇ CR 201/1 1/052 8. The hospitality o f the Austr alian National Universit y is gratefully acknowledged. 1 2 B. CHORNY AND J. ROSICK ´ Y idea. Long b efore the app earance o f [21], this concept w as intro duced in [4] under the name of a λ -algebroidal category . The main disadv a n- tage of A b eing a class is that images of its ob jects by a functor can ha v e arbitrarily large presen tation ranks. Nev ertheless, w e will sho w that surprisingly many results ab o ut accessible cat ego ries can b e gen- eralized to the class-accessible setting. In particular, class-accessible categories are closed under lax limits. F urthermore, there is a satisfac- tory theory of injectivit y and w eak factorizatio n systems in class-lo cally presen table categories which is a starting p oin t for systematic applica- tions in homot o p y theory (see [12]) leaning on sev eral existing results in this direction (see [6], [9], [10] and [11]). The main example of a class-acce ssible category whic h is not a c- cessible is the category of small preshea v es on a large category . The omnipresence of this categor y was the main motiv a tion for our work. Since o rthogonality a nd factorization systems b eha v e ev en b etter than injectivit y and we ak factorization systems, a w a y is op en for dealing with small shea v es. An early work in this direction is [25]. W e hav e to distinguish b et w een sets a nd classes. This could b e formalized b y sa ying that w e are working in the G¨ odel-Berna ys set theory . Eac h set is a class, class whic h is not a set is called prop er. A category consists of a class o f o b jects but hom( A, B ) are sets fo r eve ry pair of ob jects A, B . Sometimes, suc h categor ies a re called lo cally small (whic h forces us to c hange the terminology in tro duced b y top ologists [14] for classes of morphisms satisfying the cosolution set condition — w e call them cone-cor eflectiv e in t his pap er). A category is small if it has a set of ob jects. 2. Class-accessible ca tegorie s Let us recall that a category K is called λ - ac c essible , where λ is a regular cardinal, prov ided tha t (1) K has λ -filtered colimits, (2) K has a set A of λ -presen table o b jects suc h tha t ev ery ob ject of K is a λ -filtered colimit of o b j ects from A . An ob ject K in K is called λ - pr esentable if its hom-functor hom( K , − ) : K → Set preserv es λ -filtered colimits. A category is ac c essible if it is λ -accessible for some regular cardinal λ . A co complete ( λ - )accessible category is called lo c al ly ( λ -)pr esentable . All needed facts ab out lo cally presen table and accessible categories can b e found in [1] or [21]. CLASS-LOCALL Y PRESENT ABLE CA TEGORIES 3 Definition 2.1. A categor y K is called cla ss- λ -ac c essible , where λ is a r egula r cardinal, provide d that (1) K has λ -filtered colimits, (2) K has a class A o f λ - presen table o b jects such that ev ery ob ject of K is a λ -filtered colimit of o b j ects from A . A categor y is clas s-ac c essible if it is class- λ -accessible for some regu- lar cardinal λ . A complete and co complete class- λ -accessible category is called class-lo c al ly λ -pr esentable . A category is class-lo c al ly pr esentable if it is class-lo cally λ -presen table for some regular cardinal λ . Finally , a category K is called cl a ss-pr e ac c essib l e if it satisfies (2 ) for some r egula r cardinal λ . Example 2.2. (1) Eac h λ -accessible category is class- λ -accessible. Sin- ce eac h lo cally presen table catego ry is complete, eac h lo cally λ -presen- table categor y is class-lo cally λ - presen table. (2) Given a category A , P ( A ) will denote the category of smal l pr e she aves on A . Th ese are functors A op → Set whic h are small colimits of hom-functors. F or a small category A , w e hav e P ( A ) = Set A op . The category P ( A ) is a lw a ys class-finitely-accessible (= class- ω - accessible) b ecause eac h small presheaf is a small filtered colimit of finite colimits of hom-functors a nd the latter are finitely presen table. This relies on a g eneral fact that arbitrary colimits ma y b e expres sed as filtered colimits of finite colimits. P ( A ) is alwa ys co complete but not necessarily complete. F or instance, it do es not hav e a terminal ob ject in the case when A is a large discrete category (it means that it has a prop er class of ob jects and the only morphisms are the iden tities). This explains why w e added completeness in to the definition of a class- lo cally presen table category . (3) The category T op of top ological space s is not class-lo cally pre- sen table. The reason is that the only presen table ob jects are discrete spaces. (4) Give n a category A , let Ind( A ) b e the full sub cat ego ry of P ( A ) consisting of small filtered colimits of hom-functors. This construction w as in tro duced b y Grothendiec k (see [3]) and Ind( A ) is alwa ys class - finitely-accessible . In fact, eac h class-finitely-accessible category K is equiv alen t to Ind( A ) fo r A b eing the full sub category of K consisting of finitely presen table ob jects. The pro of is the same as in the case of finitely a ccessible categories. This can b e generalized to each regular cardina l λ b y intro ducing Ind λ ( A ) as the full sub category of P ( A ) consisting of small λ -filtered colimits of hom-functors. Then Ind λ ( A ) is alw ay s class- λ -accessible and, conv erse ly , eac h class- λ -accessible category K is equiv alent to 4 B. CHORNY AND J. ROSICK ´ Y Ind λ (pres λ K ) where pres λ K is the full sub category o f K consisting of λ -presen table ob jects. The pro of is the same a s in [1] 2.26. Remark 2.3. (1) F or eac h category A , the Y oneda em b edding yields the functor Y : A → P ( A ) making P ( A ) a free completion of A under small colimits. Analog o usly , Y : A → Ind λ ( A ) is a free completion of A under small λ -filtered colimits (the second Y is the co domain restriction o f the fir st one). (2) The cat ego ry Ind λ ( A ) is co complete if and only if A is λ - c oc o m- plete in the sense tha t it has λ - smal l c olimits , i.e., colimits of diagrams D : D → A suc h that the category D ha s less than λ morphisms. The pro of is the same as in the case of accessible catego ries, i.e., when A is small (see, e.g., [1], 1.46) . One can also pro ceed as follows. (3) Giv en a class- λ -accessible category K , w e can express the class pres λ K as a union of an increasing chain of small sub categories indexed b y all ordinals A 0 ⊆ A 1 ⊆ . . . A i ⊆ . . . This results in writing K as a union of a chain of λ -accessible categories Ind λ A 0 ⊆ Ind λ A 1 ⊆ . . . Ind λ A i ⊆ . . . and f unctors pres erving λ -filtered colimits and λ - presen table ob jects. If K is class-lo cally λ -presen table, w e can assume that the first chain con- sists of sub categor ies closed under λ -small colimits. Then the second c hain consists of colimit preserving functors. This prov es (2). (4) Recall that, giv en a small full sub category A o f a category K and an ob ject K in K , the c anonic al di a gr a m of K (with resp ect to A ) is the fo rgetful functor D : A ↓ K → K . Here, A ↓ K consists of all morphisms a : A → K with A in A and D sends a to A . W e say that K is a c a n onic al c olimit of A -ob jects if the family a : A → K form a colimit co cone from t he canonical diagram. If K is λ -a ccessib le than eac h ob ject of K is a canonical colimit of its canonical dia gram with resp ect to pres λ K and this canonical diag ram is λ -filtered (see [1], Prop osition 2.8). No w, let K b e a class- λ -accessible category . A consequ ence of (3) is that, f or each o b ject K of K , there is a small full sub category A of pres λ K suc h that K is a canonical λ -filtered colimit of A -ob jects. (5) The category A ↓ K used ab ov e is a sp ecial case of a general c o mma-c ate gory F 1 ↓ F 2 where F 1 : K 1 → L and K 2 → L are functors: w e tak e F 1 as t he inclusion of A to K and F 2 the functor from the CLASS-LOCALL Y PRESENT ABLE CA TEGORIES 5 one-morphism category to K with t he v alue K . The category F 1 ↓ F 2 has morphisms f : F 1 K 1 → F 2 K 2 as ob jects and morphisms f → f ′ are pair s of morphisms k 1 , k 2 for whic h the square F 1 K 1 f / / F 1 k 1   F 2 K 2 F 2 k 2   F 1 K ′ 1 f ′ / / F 2 K ′ 2 comm utes. Another sp ecial case is the catego ry of morphisms K → = Id ↓ Id. (6) Ev ery lo cally presen table category is cow ellpow ered, whic h do es not generalize to class-lo cally presen table ones. F o r example, the or- dered class K of ordinals with the added largest elemen t is class-lo cally ω - presen table with isolated ordinals as ω -presen table o b jects. But K is no t cow ellp ow ered. Hence a class-lo cally presen table category do es not need to b e lo cally r a nk ed in the sense of [2]. Th us The orem I I I.7 there do es not imply our 4.3. W e will no w giv e a criterion for when Ind λ ( A ) is class-lo cally pre- sen table, if A is λ -co complete, i.e., when Ind λ ( A ) is complete. W e will need the follo wing concepts. A set X of ob jects of a category K is called we akly initial if eac h ob ject K of K admits a morphism X → K with X ∈ X . A catego ry K is called appr o x i m ately c omplete if, for eac h diagra m D : D → K , the categor y of cones K → D o v er D has a w eakly initial set. Theorem 2.4. L e t A b e a λ -c o c omplete c ate g o ry (wher e λ is a r e gular c a r din al). Then Ind λ ( A ) is c omplete if and only if A is app r oxi m ately c o mplete. Pr o of. F ollo wing [18] and [22], P ( A ) is complete if and o nly if A is appro ximately complete. Since A is λ -co complete, Ind λ ( A ) is co com- plete (see 2.3 (2) ). Since P ( A ) is a free completion of A under colimits, there is a colimit preserving functor F : P ( A ) → Ind λ ( A ) suc h that the comp osition A Y − − − − → P ( A ) F − − − − → Ind λ ( A ) is naturally isomorphic to the Y oneda em b edding. Consequen tly , F is left adjoint to the inclusion of Ind λ ( A ) into P ( A ) (see the pro of of [1], 1.45). Th us Ind λ ( A ) is a reflectiv e full sub category of P ( A ). Hence Ind λ ( A ) is complete whenev er A is appro ximately complete. 6 B. CHORNY AND J. ROSICK ´ Y Con v ersely , let Ind λ ( A ) b e complete and consider a diagram D : D → A . W e expres s its limit K in Ind λ ( A ) as a filtered colimit ( k e : A e → K ) e ∈E of ob jects from A . No w, each cone A → D with A ∈ A , uniquely factorizes t hrough the limit cone via the mor phism t : A → K . Since t factorizes through some k e , the cones A e → D o btained by precom- p osing the limit cone with k e , e ∈ E form a w eakly initial set of cones o v er D . Th us A is appro ximately complete.  Remark 2.5. F ollowing 2.4, a class- λ -accessible category K is class- lo cally λ -presen table if and only if pres λ K is λ -co complete and ap- pro ximately complete. F ollowing [18] and [22], P ( A ) is class-lo cally finitely-presen table iff A is appro ximately complete. Theorem 2.6. L et K b e a c ate gory and λ a r e gular c ar dinal. Then K is class-lo c al ly λ -pr esentable if and only if it is e quival e nt to a ful l, r efle ctive s ub c ate gory of P ( A ) clo se d under λ -filter e d c olimits f o r some appr oximately c omplete c ate gory A . Pr o of. Giv en a class-lo cally λ -presen table category K , we put A = pres λ K and define the c anonic al functor E : K → P ( A ) b y taking E ( K ) : A op → Set to b e the restriction o f ho m ( − , K ) on A . Since E (colim D ) ∼ = colim E D for each λ - filtered dia g ram D : D → K , eac h E ( K ) is a λ - filtered colimit of hom-functors and thu s b elongs to P ( A ). Moreo v er, the functor E preserv es λ -filtered colimits. Since P ( A ) is a free completion of A under colimits, there is a colimit preserving functor F : P ( A ) → K suc h that the comp osition K E − − − − → P ( A ) F − − − − → K is nat ura lly isomorphic to Id K . Moreov er, F is left adjoint to the inclu- sion of K into P ( A ). Thus K is a reflectiv e full sub cat ego ry of P ( A ). Finally , fo llowing the pro of of 2.4, A is a ppro ximately complete. Con v ersely , let A b e an appro ximately complete category and K a full reflectiv e sub category of P ( A ) closed under λ -filtered colim- its. Since the reflection F : P ( A ) → K is left adjoint to t he inclu- sion of K in P ( A ) whic h preserv es λ -filtered colimits, F preserv es λ - presen table ob jects. Since eac h ob ject o f P ( A ) is a λ -filtered colimit of λ -presen table ob jects, K has the same prop erty . Th us K is class-lo cally λ -presen table.  CLASS-LOCALL Y PRESENT ABLE CA TEGORIES 7 Definition 2.7. A functor F : K → L is called c l a ss- λ -ac c essible (where λ is a regular cardinal) if K and L are class- λ -accessible cate- gories and F preserv es λ - filtered colimits. A class- λ -accessible functor preserving λ -presen table ob j ects is called str ongly clas s - λ -ac c essible . F is called (str ongly) class-ac c essible if it is (strongly) class- λ - acce- ssible for some regular cardinal λ . The uniformization theorem [21], 2.4.9 (see [1], 2.19) implies that eac h accessible functor is strongly a ccessible. Among o thers, this uses the fact that, giv en a set A of ob jects of an accessible categor y K , there is a r egula r cardinal λ suc h that eac h A ∈ A is λ -presen table. This do es not generalize to class-access ible case and we hav e to distinguish b et w een class -accessible and strongly class-accessible functors here. F or example, the large discrete categor y D is class- ω -accessible and an y functor from D in to a class- ω -accessible category is class - ω -accessible but not alw a ys strongly class- ω -a ccessib le. Remark 2.8. In the same wa y as for accessible categories, o ne can replace λ - filtered colimits in 2.1 b y λ - directed ones. Moreov er, one can show that, given a regular cardinals λ ⊳ µ , eac h class- λ -accessible category K is class- µ -accessible. The a rgumen t (see [1], 2.1 1) g o es as follo ws. One writes K ∈ K a s a λ -directed colimit ( a i : A i → K ) i ∈ I of λ -presen table ob jects. Let ˆ I b e the p oset of all λ -directed subsets of I of cardinality less then µ (ordered by inclusion). Due to λ ⊳ µ , ˆ I is µ -directed. Colimits of ( A i ) i ∈ M , M ∈ ˆ I are µ -presen table and K is their µ -directed colimit. If K is µ -presen table then K is a retract of some λ - directed colimit of ( A i ) i ∈ M . Consequen tly , a strongly class- λ -accessible functor F : K → L is strongly class- µ -accessible. Recall that the successor λ + of eac h cardinal λ is alw ay s regular and λ ⊳ λ + . 3. Limits of class-accessible ca tegorie s The fundamen tal discov ery of [2 1] is that accessible categories are closed under constructions of limit t yp e. In particular, they a re closed under pseudopullbac ks. The distinction b et w een pullbac ks and pseu- dopullbac ks is that the latter use isomorphisms instead of iden tities (see the pro of o f 3.1 b elow ). W e are going to sho w that this generalizes 8 B. CHORNY AND J. ROSICK ´ Y to class-access ible categories. F or a pseudopullbac k P ¯ F / / ¯ G   L G   K F / / M w e will use the notatio n P = PsPb( F , G ). Prop osition 3.1. L et λ b e a r e gular c ar dinal an d F : K → M and G : L → M str ong l y cla s s- λ -ac c essible functors. Th e n their pseu- dopul lb ack PsPb ( F , G ) is a class- λ + -ac c essible c ate gory and ¯ F , ¯ G ar e str ong l y class- λ + -ac c essible functors. Pr o of. Let F and G b e strongly class- λ - accessible . Ob jects of t heir pseudopullbac k are 5-tuples ( K , L, M , f , g ) where K ∈ K , L ∈ L , M ∈ M and f : F K → M , g : GL → M a re isomorphisms (morphisms are ob vious). Since b oth F and G preserv e λ -filtered colimits, PsPb( F , G ) has λ - filtered colimits and ¯ F and ¯ G preserv e them. It remains to sho w that eac h ob ject ( K , L, M , f , g ) from PsPb( F , G ) is a λ + -filtered colimit of o b jects f r om PsPb( F , G ) whic h a r e λ + -presen table in K × L . F ollo wing 2.8, K , L and M are class- λ + -accessible and F and G are strongly class- λ + -accessible. F ollo wing 2.3 (4), t here is a small full sub category A 1 of pres λ + K , a small full sub category A 2 of pres λ + L and a small full sub category A 3 of pres λ + M , suc h that K is a cano nical λ + -filtered colimit of A 1 -ob jects, L is a canonical λ + -filtered colimit of A 2 -ob jects and M is a canonical λ + -filtered colimit of A 3 -ob jects. W e can a lso assume that A 1 , A 2 and A 3 are closed under λ + -small λ -filtered colimits. W e will denote the canonical diagrams as C : C → K , D : D → L and E : E → M and their canonical colimits as ( k c : C c → K ) c ∈C , ( l d : D d → L ) d ∈D and ( m e : E e → M ) e ∈E . Let c 0 ∈ C , d 0 ∈ D and e 0 ∈ E . Since F C c 0 and GD d 0 are λ + -presen table, there is m 01 : e 0 → e 1 in E , f 0 : F C c 0 → E e 1 and g 0 : GD d 0 → E e 1 suc h that f F ( k c 0 ) = m e 1 f 0 and g G ( l d 0 ) = m e 1 g 0 . Analogously , there is k 01 : c 0 → c 1 in C , l 01 : d 0 → d 1 in D and morphisms f ′ 1 : E e 1 → F C c 1 , g ′ 1 : E e 1 → GD d 1 suc h that f − 1 m e 1 = F ( k c 1 ) f ′ 1 , f ′ 1 f 0 = F C ( k 01 ), g − 1 m e 1 = G ( l d 1 ) g ′ 1 and g ′ 1 g 0 = GD ( l 01 ). There is m 12 : e 1 → e 2 in E , f 1 : F C c 1 → E e 2 and g 1 : GD d 1 → E e 2 suc h that f F ( k c 1 ) = m e 2 f 1 , f 1 f ′ 1 = E ( m 12 ), g G ( l d 1 ) = m e 2 g 1 and g 1 g ′ 1 = E ( m 12 ). Contin uing this pro cedure, w e get c hains ( k ij : c i → c j ) i O O ` h / / ` ( u,h,v ) D W e o btain a fa cto r izat io n A = A 0 f 01 − − − − − → A 1 f 1 − − − − → B . No w, one applies this construction to f 1 and contin ues up to a regular cardinal λ suc h that eac h morphism from C ha s a λ -presen table domain (one tak es colimits in limit steps). This implies that f λ is in C  and th us A f 0 λ − − − − − → A λ f λ − − − − − → B CLASS-LOCALL Y PRESENT ABLE CA TEGORIES 15 is the desired factorization of f . Let us stress that the mo r phism f 0 λ is a transfinite comp osition of pushouts of elemen ts of C . Suc h morphisms are called C - c el lular . The resulting factorization of f dep ends on the subset C ∗ f = [ i<λ C f i . Giv en a morphism ( a, b ) : f → f ′ where f : A → B and f ′ : A ′ → B ′ in K → , w e get the induced mor phism a λ : A λ → A ′ λ with a λ f 0 λ = f ′ 0 λ a and f ′ λ a λ = bf λ . The reason is tha t , giv en a triple ( u, h, v ) fo r f , the comp osition ( au, bv ) : h → f ′ factorizes through a tr iple ( u ′ , h ′ , v ′ ) for f ′ . Th us our factorization K → → K → acts b oth on ob jects and on morphisms. But w e cannot expect that it is functorial, i.e., that it preserv es comp osition. The problem is in finding compatible c hoices of factorizations of triples ab ov e. There is a canonical c hoice of a triple a b o v e in the case that C ∗ f ⊂ C ∗ f ′ . Th us w e can mak e o ur facto r izat io n functorial on each small full sub category A of K → . It suffices to use C ∗ A = [ f ∈A C ∗ f for factor izing. Of course, for differen t full sub categories A , the result- ing fa ctorizations are differen t. (4) Let K b e class-lo cally µ -presen table category and C b e a cone- coreflectiv e class of morphisms whose domains and co domains are λ - presen table. F ollo wing 2 .8, we can assume that λ < µ . Consider a small full sub category A of pres µ K . Since (Ind µ A ) → = Ind µ ( A → ), C A = [ f ∈A C f can be used as C h for each morphism h in Ind µ A . Since pushouts comm ute with µ -filtered colimits in K , C ∗ A can b e used as C ∗ h . Th us w e can mak e our factorization functoria l on Ind µ A . Moreo v er, the corresp onding functor is µ -accessible. Th us it is strongly ν -a ccessible for some ν but this cardinal dep ends o n A in g eneral. Lik e in (3), the factorization itself dep ends on A . In the case when K is lo cally presen table and C is a set, w e ha v e K = Ind µ A fo r some µ and w e get a strongly accessible functorial factorization on K . This w as claimed by J. H. Smith and our pro of completes those from [16], 7.1 and [23] 3.1 (here, one should use o ur triples ( u, h, v ) instead of pairs ( u, h )). W e hav e pro v ed the f o llo wing theorem. 16 B. CHORNY AND J. ROSICK ´ Y Theorem 4.3. L e t K b e a c la ss-lo c al ly pr esentable c ate g o ry, C a c one- c o r efle ctive class of morphis ms of K and ass ume that ther e is a r e gular c a r din al λ such that e ach morphism fr om C has the λ -pr e sentable do- main. Then (cof ( C ) , C  ) is a we ak factorization system in K . A full sub category L of K is called we akly r efle ctive in K if, for eac h K in K , the comma-category K ↓ L has a we akly initial ob ject. It means the existence of a morphism r : K → K ∗ with K ∗ ∈ L such that eac h morphism K → L with L ∈ L factorizes through r . Corollary 4.4. L et K b e a cla ss-lo c al ly pr esentable c ate gory, C a c one- c o r efle ctive class of λ -pr esentable morphism s of K . Then Inj( C ) is we akly r efle ctive and close d unde r λ -filter e d c olimits in K . Pr o of. A w eak reflection o f K is give n b y a (cof ( C ) , C  ) factorization K r − − − − → K ∗ − − − → 1 . Assume tha t ( k d : K d → K ) d ∈D is a λ -filtered colimit of ob j ects K d ∈ Inj( C ). Give n h : C → D in C and u : C → K , there is d ∈ D and u ′ : C → K d with u = k d u ′ . There is v : D → K d with v h = u ′ . Since k d v h = u , K ∈ Inj( C ).  Corollary 4.5. L et K b e a cla ss-lo c al ly pr esentable c ate gory, C a c one- c o r efle ctive cla s s of mo rphisms of K . L et λ b e a r e gular c ar dinal such that e ach morph ism fr om C is λ -p r ese ntable. Then C  is clo s e d under λ -filter e d c olimits in K → . Pr o of. It suffices to observ e tha t g has a rig h t lifting prop erty w.r.t. f : A → B if and only if g is injectiv e in K → to the morphism ( f , id B ) : f → id B . Since f is λ -presen table in K → , the r esult follows from 4.4.  The following example show s that, in 4 .4, C  and Inj( C ) do not need to b e class-accessible. Example 4.6. Let O b e the category whose ob jects are ordinal n um- b ers (considered as w ell-ordered sets) and isotone maps. The category P ( O ) from 2.2 (2) can b e understo o d as a transfinite extension of sim- plicial sets b ecause hom( − , α + 1) can be tak en as the α -simplex ∆ α . Since O is approximately complete, P ( O ) is class-lo cally presen table. Let ∆ 1 s b e the symmetric 1 - simplex. This means t he ob ject having t w o p oin ts 0 and 1, t w o non-degenerated 1-simplices [0 , 1] and [1 , 0] and all α -simplices made out from thes e. In more detail, ∆ 1 s is a co equalizer of morphisms f , g : ∆ 1 → ∆ 2 where f sends ∆ 1 to the face [0 , 2] of ∆ 2 and g is the constan t morphism on 0. Let j : ∆ 1 → ∆ 1 s b e the inclusion CLASS-LOCALL Y PRESENT ABLE CA TEGORIES 17 on [0 , 1]. Then, follow ing 4.3, (cof ( j ) , j  ) is a weak f a ctorization system cofibran tly generated by a single morphism. Injectivit y with resp ect to j means that each 1 -simplex is symmetrized. W e will sho w that Inj ( j ) is not class-accessible. The reason is that, giv en a regula r cardinal α , eac h w eak reflection of ∆ α to Inj( j ) adds a t least α -man y 1-simplices b ecause w e hav e to symmetrize eac h 1-simplex in ∆ α . Let ( ∆ α ) ∗ denote the w eak reflection whic h adds to eac h 1- simplex just one symmetric 1-simplex. W e will show that, for eac h regular cardina l α , (∆ α ) ∗ is not α -presen table in Inj( j ). Let α b e a cardinal and (∆ α ) ∗ β , β ≤ α extends eac h 1-simplex in ∆ α b y β -many symmetric 1- simplices indexed b y ordinals i < β . Then (∆ α ) ∗ α is a n α -directed colimit of (∆ α ) ∗ β , β < α in Inj( j ). W e index all 1-simplices in ∆ α b y ordinals i < α and consider the morphism f : ∆ ∗ α → (∆ α ) ∗ α sending the symmetric 1-simplex added to the i - t h 1-simplex e i in ∆ α to the i -th added symmetric 1-simplex extending e i . Clearly , f do es not factorize through any (∆ α ) ∗ β , β < α . No w, assume that Inj( j ) is class- λ - a ccessible . Then ∆ ∗ λ is a λ - directed colimit k d : K d → ∆ ∗ λ of λ -presen table ob jects K d in Inj( j ). Since Inj( j ) is closed under filtered colimits in P ( O ), the w eak reflec - tion r : ∆ λ → ∆ ∗ λ factorizes through some k d , i.e., r = k d f . Since r is a w eak reflection, t here exists g : ∆ ∗ λ → K d with g r = f . Th us k d g r = r . Consider a non-degenerated 1-simplex [ i, j ] in ∆ λ . Then, ∆ ∗ λ con tains the new 1- simplex [ j, i ] suc h t ha t [ i, j, i ] is a 2- simplex with [ i, i ] degen- erated. Since [ j, i ] is the only 1-simplex with this pro p ert y , k d g m ust send it to itself. Th us k d g = id ∆ ∗ λ . Therefore ∆ ∗ λ is λ - presen table as a retract of K d , whic h is a contradiction. Consequen tly , Inj( j ) is not class-access ible. Recall that a morphism is called λ -presen table if it has the λ -presen- table domain and the λ -presen table co domain (see 3.8 (2)). Definition 4.7. Let K b e a class -lo cally λ -presen table category . A w eak factorization system ( L , R ) in K is called c ofibr antly class - λ - gener ate d if L = cof ( C ) for a cone-coreflectiv e class C of mor phisms suc h that (1) morphisms from C are λ -presen table and (2) an y morphism b et w een λ -presen table ob jects has a w eak fac- torization with the middle ob ject λ -presen table. A cone-coreflectiv e class C of morphisms will b e called λ - b ounde d if (cof ( C ) , cof ( C )  ) satisfies conditions (1) and (2 ) a b o v e. W e say that ( L , R ) is c ofibr a n tly c l a ss-gener ate d if there is a regular cardinal λ suc h that ( L , R ) is cofibrantly class- λ -generated. The same for b ounde d . 18 B. CHORNY AND J. ROSICK ´ Y Remark 4.8. (1) An y set C is a cone-coreflectiv e class. In this case, the factorizat io n is alwa ys functorial (b ecause, in 4.2, C f = C fo r eac h f ). But, 4.6 shows that C do es not need to b e b ounded. This follows from 4.10 but there is a direct verific ation. Assume t ha t ∆ λ → ∆ 0 has a w eak factorization with the λ -presen table middle ob j ect K . There are morphisms k : K → ∆ ∗ λ and g : ∆ ∗ λ → K and, lik e in 4.6, k g = id ∆ ∗ λ . Th us ∆ ∗ λ is λ -presen table in P ( O ) and, therefore, in Inj( j ), whic h cannot happ en. (2) Let C b e a cone-coreflectiv e class of mor phisms in a class-lo cally λ -presen table category . Giv en f : A → B a nd C f , w e denote by T f a set of all triples ( u, h, v ) from 4.2 (3) with h ∈ C f whic h are needed f or cone-coreflectivit y . Let T ∗ f = [ i<λ T f i . Assume the existence of a regula r cardinal µ > λ suc h that t ha t the cardinalit y of T ∗ f is smaller than µ fo r each mor phism f : A → B with A and B µ -presen table. Then for suc h an f , all ob j ects A i , i ≤ λ from 4.2(3) are µ -presen table and A λ is the middle ob j ect in a weak factorization of f . Th us C is µ - b ounded. D ue to c hoices of sets T f , we cannot exp ect to get a functorial factorization in this wa y . Ev en, w e cannot make it functorial on small full sub categories. (3) Given a we ak factor izat io n f = f 2 f 1 from 4.7 (2), w e can c ho ose C f = { f 1 } and T f = { (id A , f 1 , f 2 ) } . Then (2) ab ov e implies that 4.2 (2) yields a w eak factorizatio n fro m 4.7 (2). (4) W e do not know whether a λ - b ounded C is µ -b ounded for λ ⊳ µ . This is tr ue pro vided that the factorization is functorial – then it is giv en by a strongly class- λ -accessible functor whic h is strongly class- µ - accessible (see 2.8). (5) W e say tha t C is ( λ, λ + )- b ounde d if it satisfies 4 .7 (1) fo r λ and (2) f or λ + . Each ( λ, λ + )-b ounded class is λ + -b ounded. W e will show that a unio n C ∪ C ′ of t w o ( λ , λ + )-b ounded classes is λ + -b ounded. The union is cone-coreflectiv e ( see 4.2 (1) ) . Let f : A → B b e a morphism b et w een λ + -presen table ob jects. W e pro ceed b y a transfinite construction for i ≤ λ . W e start with its (cof ( C ) , C  ) factorization A 0 = A f 01 − − − − − → A 1 g 0 − − − − → B Then we take a (cof ( C ′ ) , ( C ′ )  ) fa ctorization of g 0 A 1 f 12 − − − − − → A 2 g 1 − − − − → B W e put f 02 = f 12 f 01 and con tinue the pro cedure by a (cof ( C ) , C  ) factorization of g 1 . In a limit step i , f j i : A j → A i is giv en b y a CLASS-LOCALL Y PRESENT ABLE CA TEGORIES 19 transfinite comp osition and g i : A i → B is the induced mo r phism. W e finish a t λ and get a factorization f : A f 0 λ − − − − − → A λ g λ − − − − − → B The ob ject A λ is λ + -presen table b ecause λ + -presen table o b jects are closed under λ + -small colimits and b oth C and C ′ satisfy (2) f or λ + . W e ha v e f 0 λ ∈ cof ( C ∪ C ′ ). Finally , g λ is b oth a λ -filtered colimit of morphisms from C  and a λ -filtered colimit of morphisms f rom ( C ′ )  . F ollo wing 4.5, g λ ∈ C  ∩ ( C ′ )  = ( C ∪ C ′ )  . Theorem 4.9. L et K b e a class-l o c al ly λ -pr esentable c ate gory and ( L , R ) a c ofibr a ntly class- λ -gener ate d we ak factorization system. Then R is a class- λ -ac c essib le c ate gory str ongly λ -ac c essibly emb e dde d in K → . Pr o of. F ollo wing 3.8 ( 2), K → is class-lo cally λ -presen table and the pro- jections P 1 , P 2 : K → → K ar e strongly class- λ -accessible. F ollo wing 4.5, R is closed in K → under λ -filtered colimits. Consider a morphism f : K → L in R . W e ha v e to sho w that f is a λ -filtered colimit of λ -presen table ob j ects f r o m R . F ollo wing 2.3 (2), K → is a union of a c hain Ind λ ( A → 0 ) ⊆ Ind λ ( A → 1 ) ⊆ . . . Ind λ ( A → i ) ⊆ . . . of lo cally λ -presen table categories and strongly class- λ -accessible func- tors. There is i 0 suc h that f b elongs to Ind λ ( A → i 0 ). F ollo wing 4.4 and 4.7 ( 2 ), giv en a λ -presen table ob ject h of K → , its w eak factorization h 2 h 1 has h 2 λ -presen table in K → and thus in R . Since A → i 0 is small, there is i 1 suc h that A → i 1 con tains all h 2 for h fro m A → i 0 . Analogously , there is i 2 suc h that A → i 2 con tains all h 2 for h fro m A → i 1 . W e will con- tin ue this pro cedure, in limit steps we tak e i j as the suprem um of a ll i k with k < j . Let B b e the in tersection of A → i λ and R . Consider a morphism ( a, b ) : h → f with h : A → B in A → i λ . Then h b elongs to A → i j for some j < λ . Thus B contains h 2 . Since f is in R and h 1 in L , there is c : C → K suc h that f c = bh 2 and ch 1 = a ; here C is the mid- dle o b ject in t he weak factor izat io n h = h 2 h 1 . Hence ( a, b ) factorizes through h 2 . It remains to sho w that B ↓ f is λ -filtered. Then it will b e cofinal in A → i λ ↓ f and f will b e the canonical colimit of B - ob jects. Let X b e a λ -small subcategory of B ↓ f . There is j < λ suc h that X is a sub category of A i j ↓ f . Th us X has an upp er b ound ( a X , b X ) : h → f , h ∈ X in A → i j ↓ f . Then h 2 is an upp er b o und of X in B ↓ f . This prov es that the latter category is λ - filtered.  20 B. CHORNY AND J. ROSICK ´ Y Corollary 4.10. L et K b e a class -lo c al ly λ -p r esentable c ate gory with a λ -pr esentable terminal obje ct and ( L , R ) b e a c ofibr antly cl a ss- λ - gener ate d we ak factorization system. Then Inj( L ) is a class-ac c e ssible c a te gory str ongly ac c essibly emb e dd e d in K . Pr o of. Since a w eak reflection of an ob j ect K in Inj( L ) is given b y a w eak factorization of its morphism K → 1 to a terminal ob ject, the result f ollo ws from the pro of of 4.9.  Theorem 4.11. L et K b e a class-lo c al l y pr esentable c ate gory, C a c on e - c o r efle ctive class of morphis ms of K and ass ume that ther e is a r e gular c a r din al λ such that e ach morphism fr om C is λ -pr esentable. Then (colim( C ) , C ⊥ ) is a factorization system in K . Pr o of. Giv en a morphism f : A → B , w e for m the pushout of f and f and denote by f ∗ a unique morphism making the following diagram comm utativ e A f / / f   B p 2   id B   ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ B p 1 / / id B ' ' P P P P P P P P P P P P P P P P P P P P P P P P P P A ∗ f ∗ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ B Let ¯ C = C ∪ C ∗ where C ∗ = { f ∗ | f ∈ C } . Since pushouts of λ -presen table ob jects are λ -presen t a ble, each morphism from ¯ C has a λ -presen table domain. It suffices to sho w tha t the class ¯ C is cone-coreflectiv e. Then, follo wing 4.3, ( cof ( ¯ C ) , ¯ C  ) is a w eak factor izat io n system. F ollow ing the pro o f of 4.1 in [7], w e conclude that ¯ C  = C ⊥ and (cof ( ¯ C ) , C ⊥ ) is a factorization system. Finally , the pro of of 2.2 in [1 7] sho ws that cof ( ¯ C ) = colim( C ). W e hav e to pro v e that the class C ∗ is cone-coreflectiv e. L et g b e a morphism in K and tak e the pullback D ∗ q 1 / / q 2   C g   C g / / D CLASS-LOCALL Y PRESENT ABLE CA TEGORIES 21 W e ha v e a unique g ∗ : C → D ∗ suc h that q i g ∗ = id C for i = 1 , 2. W e will sho w that comm utative squares A ∗ u / / f ∗   C g   B v / / D with f ∈ C uniquely corresp ond to commutativ e sq uares A t / / f   C g ∗   B h / / D ∗ The corresp ondence assigns to u and v the pair t, h where t = up 1 f and h is give n b y q i h = up i for i = 1 , 2 (b ecause g up 1 = v = g up 2 ). Since q i hf = u p i f = t = q i g ∗ t for i = 1 , 2, we ha v e hf = g ∗ t . Conv ersely , giv en t and h , w e get u and v b y means o f up i = q i h for i = 1 , 2 and v = g q 1 h . Since g up i = g q i h = v = v f ∗ p i for i = 1 , 2, w e hav e g u = v f ∗ . The one-to- one corresp ondence b et w een u and h is eviden t. Since v = v f ∗ p 1 = g u p 1 = g q 1 h and t = q 1 g ∗ t = q 1 hf = up 1 f , there is a one-to-one corresp ondence b et w een v and t as well. Since the class C is cone-coreflectiv e, the comma-category C ↓ g ∗ has a we akly terminal set C g ∗ . W e will show that the corresp onding set ( C g ∗ ) ∗ is w eakly terminal in C ∗ ↓ g . Giv en ( u , v ) : f ∗ → g , w e take the corresp onding ( t, h ) : f → g ∗ . There is a factorization ( t, h ) : f ( t 1 ,h 1 ) − − − − − − − → e ( t 2 ,h 2 ) − − − − − − − → g ∗ with e : X → Y in C g ∗ . Let ( u 2 , v 2 ) : e ∗ → g correspo nds to ( t 2 , h 2 ), v 1 = h 1 and u 1 : A ∗ → X ∗ is induced b y u 1 p i = p i h 1 for i = 1 , 2 where p i are from the pushout defining e ∗ . Since u 2 u 1 p i = u 2 p i h 1 = q i h 2 h 1 = q i h = up i for i = 1 , 2, w e hav e u 2 u 1 = u . F ur t her, v 2 v 1 = ( g q 1 h 2 ) h 1 = g q 1 h = v . Finally , since e ∗ u 1 p i = e ∗ p i h 1 = h 1 = v 1 = v 1 f ∗ p i 22 B. CHORNY AND J. ROSICK ´ Y for i = 1 , 2, w e hav e d ∗ u 1 = v 1 f ∗ . Hence we get a factorizatio n ( u, v ) : f ∗ ( u 1 ,v 1 ) − − − − − − − → e ∗ ( u 2 ,v 2 ) − − − − − − − → g with e ∗ ∈ ( C g ∗ ) ∗ .  Corollary 4.12. L et K b e a class- l o c al ly λ -pr esentable c ate gory and C a c one-c or efle ctive class of λ -pr e s e ntable morp h isms of K . Then Ort( C ) is r efle ctive and c l o se d under λ -filter e d c olimits in K . Mor e ove r, Ort( C ) is cl a s s-lo c al ly λ -pr esentable. Pr o of. A reflection of R ( K ) is giv en by a (colim( C ) , C ⊥ ) factorization K r − − − − → R ( K ) − − − → 1 . Since Ort( C ) = Inj( ¯ C ) where ¯ C is fr o m the proo f of 4.11 , Ort( C ) is closed under λ -filtered colimits in K (fo llowing 4.4). F ollow ing 2.6, Ort( C ) is class-lo cally λ -presen table.  Remark 4.13. (1) Giv en C from 4 .1 2, w e sho w in the same w a y as in 4.9 that C ⊥ is class-lo cally λ - presen table. Concerning colim( C ), we know that it is coreflectiv e in K → and the coreflector R : K → → colim ( C ) preserv es λ - filtered colimits. Hence colim( C ) is a full image of a class- accessible f unctor. But w e do not kno w whether R : K → → K → is strongly class-accessible and w e thus do not kno w whether colim( C ) is class-lo cally presen table. What is missing is the condition 4.7 (2). (2) Let K b e a class-lo cally λ -presen table category written as a union of a c hain of lo cally λ -pr esen table full strongly λ -accessibly embedded sub categories K i (see 3.10). Let C b e a class- λ -access ible full subcat- egory of K suc h that ( C ∩ K i ) ⊥ (calculated in K i ) is reflectiv e in K i . Then C ⊥ is reflectiv e in K . 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Rosick ´ y Dep ar tment o f Ma thema tics and St a tistics Masar yk University, F acul ty of Sciences Brno, Czech Republic rosicky@ma th.muni.cz