Colimits of accessible categories

COLIMITS OF A CCESSIBLE CA TEGORIES R. P AR ´ E AND J. R OSICK ´ Y ∗ Abstract. W e show that any dire cted colimit of a cessible c a te- gories and accessible full em beddings is accessible and, assuming the existence of arbitra rily la rge s trongly compact car dinals, an y directed colimit of acessible categ o ries and ac cessible embeddings is accessible. 1. Introduction Accessib le c ategories are close d under constructions of “a limit type”. More precisely , the 2-category of accessible categories and accessible functors has all limits appropriate for 2-catego r ies calculated in the 2- category of categories and functors ( see [5]). The situation is m uc h less satisfactory for colimits. The only general res ult is t ha t lax colimits of strong diagrams of accessible categories and accessible functors exist and are calculated as the idemp o ten t completion of the lax colimit of categories (see [5 ], Theorem 5.4.7). In this pap er w e show that any directed colimit of accessible categories and accessible full em b eddings is accessible and, assuming the existence of a prop er class of strongly compact cardinals, access ible categories are closed under directed col- imits of em beddings. W e do not know whether set theory is really necessary for the second r esult. W e also do not kno w anything ab out general directed colimits. W e will start with an example of a colimit of accessible categories whic h is not accessible (but has split idemp ot ents). All undefined concepts concerning access ible catego r ies can b e found in [1] or [5]. Recall that a functor F : K → L is called λ -accessible if K and L are λ -a ccessible categories and F preserv es λ -directed colimits. F will b e called str ongly λ - ac c essible if, in addition, it preserv es λ - presen table ob jects. An y λ -accessible functor is strongly µ -a ccessible for some regular cardinal µ . F is (strongly) accessible if it is (strongly) Date : Septem be r 21, 201 1. 1991 Mathematics Su bje ct Classific ation. 18C3 5, 03E55. Key wor ds and phr ases. acce s sible category , dir ected colimit, compact cardina l. ∗ Suppo rted by MSM 00 2 1622 409. 1 2 R. P AR ´ E AND J. ROSICK ´ Y λ -accessible for some regular cardinal λ . CA T will denote the (non- legitimate) category of categories and functors while A CC is the (non- legitimate) category o f a ccessible categories and accessible functors. Example 1.1. Let K b e a com binatorial mo del category and W its class of w eak equiv alences. Then W is an accessible category and its em bedding G : W → K → in to the category of mo r phisms of K is ac- cessible (see [4] A.2.6.6 o r [6 ] 4.1). Let dom , co d : W → K b e t he functors assigning to eac h f ∈ W its domain or co domain. These func- tors a re a ccessible and let ϕ : dom → cod the natural tra nsforma t ion suc h that ϕ f = f . Then the coin v erter of ϕ is the homotop y category Ho K = K [ W − 1 ] of K . T he homotopy category has v ery often split idemp oten ts (for instance if K is stable) and is almost neve r accessi- ble; e.g., if K is the mo de l cat ego ry of sp ectra then Ho K has split idemp oten ts and is not acces sible. 2. Dire cted colimits of accessible full embeddings Theorem 2.1. L et F ij : K i → K j , i ≤ j ∈ I b e a di r e cte d diagr am of ac c essible c ate gories and a c c essible ful l emb e ddings. Then its c olimi t in CA T is ac c essible as ar e the c oli mit inje ctions, and is in fact the c olim it in A CC . Pr o of. W e can assume the F ij are full inclusions. Then w e w an t to show that K = S i ∈ I K i is acces sible. Let κ b e such that eac h K i is κ -acces sible, eac h inclusion K i ⊆ K j is strongly κ -accessible , and κ > | I | . Let d mn : K m → K n , m ≤ n ∈ M , b e a κ -directed diagram in K . W e claim that there is an i 0 ∈ I and a cofinal su bset M 0 ⊆ M suc h that K m ∈ K i 0 for all m ∈ M 0 . Otherwise there w ould b e, for ev ery i , an m i ∈ M suc h that K p / ∈ K i for all p ≥ m i . Then as M is κ -directed and κ > | I | , there is one p ≥ m i for all m i , and the corresp onding K p is not in an y K i , a con tradiction. No w M 0 is κ -dir ected as it is cofinal in M so colim m ∈ M 0 K m exists in K i 0 . No w for any co cone h k m : K m → L i m ∈ M in K , L will b e in some K i and if w e take i 1 ≥ i, i 0 then w e get a co c one h k m : K m → L i m ∈ M 0 in K i 1 . As K i 0 ⊆ K i 1 preserv e s κ - directed colimits, this co cone factors uniquely through colim m ∈ M 0 K m , so colim m ∈ M 0 K m is the colimit in K as we ll. If our diagr am lies entirely in one K i to start with w e can take i 0 = i and M 0 = M , so the inclusion K i ⊆ K preserv es κ -directed colimits. COLIMITS OF A CCESSIBLE CA TEGORIES 3 If K is κ -presen table in K i , and h K m i m ∈ M is a κ -directed diagram in K , then w e can c hoose the i 0 ab ov e so that it is also ≥ i . Then K (colim K m , K ) = K i 0 (colim K m , K ) ∼ = colim K i 0 ( K m , K ) = colim K ( K m , K ) b ecause K is a lso κ -presen table in K i 0 . So K is κ - presen table in K i.e. the inclus ions K i ⊆ K are strongly κ -a ccessible . Ev ery ob ject of K is a κ -directed colimit of κ -presen tables in some K i so also in K . Thu s K is κ -acces sible. Finally , K is the colimit of K i in ACC , for if h G i : K i → Li i ∈ L is a compatible family of κ i -accessible functors, w e can choo se the κ in the abov e a rgumen t to b e la r ger than all κ i , and then the ex tension G : K → L will preserv e κ -directed colimits.  Example 2.2. L et n be the ordered set { 1 < 2 < · · · < n } and consider the chain of accessib le em b eddings Set → Set 2 → Set 3 → · · · → Set n → · · · where the tra nsition for n to n + 1 extends a path of length n to one of length n + 1 b y adding an iden tit y at the end. The colimit can b e iden tified with the category of infinite paths A 1 → A 2 → A 3 → · · · whic h ar e ev en tually constant, i.e. there is an N suc h that A n → A n +1 is an iden tit y for all n ≥ N . It is ω 1 -accessible but not ω -acces sible. Remark 2.3. Theorem 2.1 can b e extended to directed colimits of em beddings F ij suc h that fo r each commu tativ e triangle F ij A F ij ( h ) / / F ij ( f ) " " E E E E E E E E E E E E F ij C F ij B g < < y y y y y y y y y y y y there is g : B → C such that F ij ( g ) = g . In fa ct, w e can c hoose m 0 ∈ M 0 and r ep eat the argument ab ov e to get i 1 > i 0 and a cofinal subs et M 1 ⊆ M 0 suc h that d m 0 m ∈ K i 1 for eac h m ∈ M 1 . Since d mn ∈ K i 1 for each m ≤ n from M 1 (b ecause d mn d m 0 m = d m 0 n ), colim m ∈ M 1 K m exists in K i 1 . Using this colimit in the proo f ab o v e instead of colim m ∈ M 0 K m , w e get the ex tension of 2 .1 . 3. Directe d colimits of accessible embeddings A cardinal µ is called str on gly c om p act if f o r ev ery set I , ev ery µ - complete filter on I is con tained in a µ -complete ultrafilter on I . Often, 4 R. P AR ´ E AND J. ROSICK ´ Y compact cardinals are called strong ly compact. A cardina l µ is called α - str ongly c o mp a ct if for ev ery set I , ev ery µ -complete filter on I is con tained in an α -complete ultra filter on I ( µ is called L αω -compact in [2]). Clearly , µ is compact if and only if it is strongly µ -compact. Theorem 3.1. L e t F ij : K i → K j , i ≤ j ∈ I b e a dir e cte d dia gr am of str ongly λ -ac c essible emb e ddings a nd F i : K i → K its c olimit in CA T . L et λ ⊳ µ b e a str ongly α -c omp act c ar dinal wher e α = max { λ, | I | + } . Then K is µ -ac c essible and F i ar e str ongly µ -ac c essible. Pr o of. First, w e will sho w that K has µ -directed colimits. Let d mn : K m → K n , m ≤ n ∈ M b e a µ - directed diagram in K . Let F b e the filter on M generated b y sets M m = { k ∈ M | m ≤ k } , m ∈ M . Since M is µ -directed, F is µ -complete and th us it is con tained in a n α -complete ultrafilter U on M . Put M i m = { k ∈ M m | d mk ∈ K i } for m ∈ M and i ∈ I . Since M m = S i ∈ I M i m , | I | < α and U is α -complete, there is ˜ m ∈ I suc h that M ˜ m m ∈ U . Using the α -completeness of U again, w e get U ∈ U suc h that ˜ m = ˜ n for eac h m, n ∈ U . W e denote this common v alue of ˜ m b y ˜ U . Restrict our starting diagram b y taking d mn : K m → K n suc h that m, n ∈ U and d mn ∈ K ˜ U . W e get a sub diagram of the starting diagram and we will sho w that this sub diagram is λ -directed. Consider a subset X ⊆ U with | X | < λ . Then V = U ∩ \ x ∈ X M ˜ U x b elongs to U . Th us V 6 = ∅ and for m ∈ V , w e hav e d xm ∈ K ˜ U for eac h x ∈ X . Th us our sub diagram is λ -directed. Let K b e its colimit in K ˜ U . Since F ˜ U preserv e s λ -directed colimits, K is a colimit of our sub diagram in K . F or eac h m ∈ M , there is n ∈ U ∩ M m b ecause the in tersection b elongs to U . Th us our sub diagram is cofinal in the whole diagram, whic h means that K is a colimit of the starting diagram in K . W e ha v e prov ed that K has µ -directed colimits. Moreov er, since an y µ -directed colimit in K is calculated in some K i , the em beddings F i preserv e µ -directed colimits. Since λ ⊳ µ , the em b eddings F ij are strongly µ -a ccessible . This means that if A is µ -presen table in K i then it is µ -presen table in K j for all i ≤ j ∈ I . Since any µ -directed colimit in K is calculated in some K i , A is µ -presen table in K as w ell. Th us the em b eddings F i preserv e µ - presen table ob jects. Since eac h K ∈ K b elongs to some K i and it is a µ -directed colimit of µ -presen table ob jects in K i , K is a µ -directed colimit of µ -presen table ob jects in K . Th us K is µ -accessible.  COLIMITS OF A CCESSIBLE CA TEGORIES 5 Corollary 3.2. Assuming the existenc e o f arbitr arily lar ge c omp act c ar dinals, A CC is close d in CA T under dir e cte d c olimits of embb e d- ings. Pr o of. Let F ij : K i → K j , i ≤ j ∈ I b e a directed diagram of em b ed- dings in ACC . The re is a regular cardinal λ suc h that all functors F ij are strongly λ -accessible (see [1] 2.19). There is a compact cardi- nal µ > λ, | I | + . Since µ is inaccess ible (see [3]), [5] 2.3.4 implies that λ ⊳ µ . F ollow ing 3.1, a colimit K of F ij in CA T is µ -accessible and F i : K i → K are µ -access ible as w ell. Th us F i : K i → K is a colimit in A CC ( see the proo f of 2.1).  Example 3.3. Consider the follo wing coun table ch ain of lo cally finitely presen table categories a nd finitely acces sible functors Set F 01 − − − − − → Set 2 F 12 − − − − − → . . . Set n F nn +1 − − − − − − − → . . . Here, Set is the category of sets, F nn +1 ( X 1 , . . . , X n ) = ( X 1 , . . . , X n , X n ) and F nn +1 ( f 1 , . . . , f n ) = ( f 1 , . . . , f n , f n ) is the action of F nn +1 on ob- jects and morphisms. The colimit Set <ω in CA T consists of sequenc es ( X n ) n ∈ ω whic h are ev en tually constan t, i.e., there is n ∈ ω suc h that X n = X m for all n ≤ m . Similarly , morphisms are ev en tually constan t sequence s ( f n ) n ∈ ω of mappings. F ollow ing 2.1, Set <ω is accessible as- suming the existence of a strongly ω 1 -compact cardinal. W e do not know whether the access ibilit y of Set <ω dep ends on set theory . Reference s [1] J. Ad´ amek and J. Rosick´ y, L o c al ly Pr esentable and A c c essib le Cate go ries , Cambridge Universit y Press 1994. [2] P .C. Eklof and A. H. Mekler, Almost F r e e Mo d ules , Nor th-Holland 1990 . [3] T. Jech S et The ory , Academic Press 1978. [4] J. Lurie, Higher T op os The ory , Princeton Univ. Press 2009 . [5] M. Makk ai and R. Par´ e, A c c essible Cate gories: The F oundations of Cate gor- ic al Mo del The ory , AMS 19 89. [6] J. Rosick´ y, On c ombinatorial mo del c ate gories , Appl. Cat. Str. 17 (20 09), 303-3 16. 6 R. P AR ´ E AND J. ROSICK ´ Y Dep ar tment of Ma thema tics and St a tistics Dalhousie University Halif ax, NS, Canada, B3H 3J5 p are@ma thst a t. dal.ca Dep ar tment of Ma thema tics and St a tistics Masar yk U niversity, F acul ty of S ciences Kotl ´ a ˇ rsk ´ a 2, 611 37 Brno, Czech Rep ublic rosicky@ma th. muni.cz