quasi-coproducts and accessible categories with wide pullbacks

Applied Categorical Structures 4: 387-402, 1996. 387 (~) 1996 Kluwer Academic Publishers. Printed in the Netherlands.

Quasi-Coproducts and Accessible Categories with Wide Pullbacks

HONGDE HU and WALTER THOLEN* Department of Mathematics and Statistics, York University, North York, Ontario, Canada,

M3J IP3

(Received: 5 December 1994; accepted: 12 September 1995)

Abstract. We establish a 2-categorical duality involving the 2-category .A,~ of all to-accessible categories with wide pullbacks, also known as locally tc-polypresentable categories, and of functors preserving ~¢-filtered colimits and wide puUbacks. Commutation of wide pullbacks with so-called quasi-coproducts in Set is the basic ingredient to this duality, which leads to a full characterization of categories of type Wdpb Filt,~(A, Set) = .A,~

Mathematics Subject Classifications (1991). 18A35, 18B40, 18C99.

Key words: Accessible category, wide pullback, quasi-coproduct, quasi-based category, free group action, polylimit, flat functor.

Introduction

Accessible categories as considered in [22, 27, 1] (with a predecessor present already in SGA4) merge the interests of people working in Algebra, Category Theory, Model Theory, Logic and Domain Theory and have therefore become a focus of research. Accessible categories with (co)limits are well understood since they are exactly the locally presentable categories of Gabriel and Ulmer [14], which have been characterized model-theoretically and syntactically by Volger [33] and Coste [9], respectively, in the finitary case; see also [21, 28]. However, accessible categories with fewer limits have been investigated more closely only in the recent past, in particular accessible categories with

- connected limits = locally multipresentable categories (cf. [11, 19, 15, 8, 18]) - wide pullbacks = locally polypresentable categories (cf. [23, 31, 3, 16]; see

also [12] for background).

The difference between these two types of categories may appear to be slight at first sight (see, however, [29]). Commutation of connected limits with coproducts in Set is the true reason that this first type of category allows for a smooth duality theory (see [25, 26, 17]). It is therefore natural to ask: which type of colimits, in

* The first author acknowledges financial assistance from a special research grant of the Faculty of Arts at York University. The second author is partially supported by an NSERC operating grant.

J Presented at the European Colloquium of Category Theory, Tours, France, 25-31 July 1994.

388 HONGDE HU AND WALTER THOLEN

addition to coproducts, commutes with wide pullbacks (i.e., fibred products) in Set? The answer is given in Section 1 of this paper: colimits of some diagrams C: G --~ Set , with a small groupoid G; here G may be assumed to be skeletal, so that it is just a disjoint union of groups Gi, and "some" means that each group action given by C must be free. We call such colimits quasi-coproducts; they appear less explicity in conjunction with "Galois sketches" in [3] and with "pullback theories" in [16].

Universal properties that give factorizations which are unique only "up to a unique automorphism which leaves the free generators fixed" have been studied predominantly in the realm of generalized right-adjoint functors (cf. [20, 6, 32, 30, 2]), with Paul Taylor's domain-theoretic notion of stable functor having become widely accepted. The paper [32] took Mac Lane's approach to adjointness and defines quasi-initial and locally initial objects first. We recall these definitions (in the dual situation) in Section 2 in order to define the notion of polylimit as in [23] (which allows for easy comparison with Diers' [10] notion of multilimit), with the immediate goal of showing that the category n-Flat(C) of ~;-flat functors on a small category C with polylimits for diagrams of size less than ~; has wide pullbacks (see 2.4). When translated into the language of ~;-accessible categories, this result tells us that a K-accessible category has wide pullbacks if its full subcategory A~ of K-presentable objects has polycolimits for diagrams of size less than ~. It is then natural to ask whether this condition is also necessary for the existence of wide pullbacks.

The positive answer to this question is obtained most conveniently from the duality established in Section 4 which gives a full characterization of the categories of type

Wdpb Filt,~(A, Set) ,

i.e., the category of all Set-valued functors on a K-accessible category A with wide pullbacks which preserve these limits and K-filtered colimits. In [18] we considered the easier problem when wide pullbacks are traded for arbitrary connected limits. We showed that the categories obtained this way are equivalent to formal coproduct completions of small categories with multilimits of diagrams of size < ~; these are the K-complete based categories (cf. [7]), i.e., categories with ~;-limits and coproducts in which every object is the coproduct of coprime objects (so that their representables preserve coproducts), and the coprime objects form (essentially) a small set. We show that this result remains true in the wide- pullback case, provided coproducts get traded for quasi-coproducts everywhere. The biadjoint of the passage A ~ Wdpb Filt~ (A, Set) is given by assigning to every quasi-based category B with ~;-limits the category

Lim~ I_[~ (B, Se t )

of Set-valued functors preserving ~;-limits and quasi-coproducts. In Section 3 we show that such a category is equivalent to the category ~-Flat(Bq) (with Bq the

QUASI-COPRODUCTS AND ACCESSIBLE CATEGORIES WITH WIDE PULLBACKS 389

full subcategory of B of quasi-coprime objects), which is a-accessible and has wide pullbacks.

The connected-limit case which we considered in [18] is technically easier to handle since every category has an easily described formal coproduct completion. The results of this paper suggested to us to investigate completions of categories under quasi-coproducts and their connections with coproduct completions, but these topics go beyond the scope of this paper and will therefore be treated in a separate paper.

I. Quasi-Coproducts in Set

1.1. An action of a group G on a set X is mapping G × X --+ X, denoted by (g,x) ~ g . x and satisfying 1 • x = x and g . (h . x) = (9" h) • x. This is equivalently described by its Cayley representation, i.e., by a functor C: G --+Set (with C(1) = idx and C(g)(x) = 9" x). The colimit of C is the joint coequalizer q: X --+ Q of the maps (C(g): X --~ X)g~c. When considering Q a trivial G- set (so that g • z = z for all z E Q), then q becomes a C-map, in fact: it is characterized as a universal trivial G-map with domain X , since any other map f : X --+ Y with f ( g . x) = f (x ) for all 9 E G and x E X factors uniquely through q. We may also think of q as the map that collapses two points of X if and only if they belong to the same orbit Gx = {g • x I 9 E G} (for some x E X); in other words, Q is just a set of labels for the distinct orbits of the action, and q maps each point to the label of its orbit. In particular, if the action is transitive (so that X = Gx for some and then every x E X), q is constant. If the action is free (so that every stabilizer Gz = {g E G I g" x = x} is trivial), then the maps G --+ Gx, g ~ g . x, are bijective, and we may think of X as a disjoint union of copies of G, and these copies represent the distinct fibres of q. An action is called regular if it is both transitive and free.

1.2. More generally, let G be a groupoid, that is a skeletally small category in which every arrow is an isomorphism. We wish to describe a particular construction of a colimit of a functor G: G --+ B for a cocomplete category B. We can, without loss of generality, suppose that G is skeletal, in which case, if I denotes the set of objects, G is a disjoint union of groups Gi, i E I. Let qi: X~(= C(i)) --+ Qi denote the coequalizer of the action of G~ on Xi. Then~ for Q = IIQ~ with injections ki: Qi --~ Q, (Q, kiq~) is a colimit of C.

In case B = Set, for a family (mi: Xi --+ Q)ieI to form the colimit of the diagram D simply means that each mi is a trivial Gi-map, and that every family (f~: X~ --+ R)iex of trivial Gi-maps factors jointly through (mi)iei by a unique map Q --+ R. The construction in terms of coproducts and coequalizers then shows that this universal property holds if and only if

(1) Q, is the disjoint union of the ranges mi(Xi) , i E I, and


(2) for all i E I and x, y E Xi , mi (x) = mi(y) holds i f and only i f y = g . x for some 9 E Gi.

1.3. A diagram C: G --4 B (with a groupoid G) is called quasi-discrete if (in the notation of 1.2) for every i E / and for every non-initial object B E B, the group action

Gi ¢-~ G --~ B B(_~-) Se t

is free (so that for any g E Gi and s: B -+ Xi = C(i), one can have C(g)s = s only if g = 1 or B = 0); in this case we call the colimit of C a quasi-coproduct of the objects Xi, i E I.

For B = Set , it suffices to consider a singleton set B when checking quasi- discreteness. Hence C is quasi-discrete if and only if the action of Gi on Xi is free, for every i E I. This property can be built into the characterization given in 1.2, by replacing the word "some" in 1.2(2) by "a unique".

A category B is said to have quasi-coproducts if every small quasi-discrete diagram has a colimit in B. We say that a functor F : B ~ C preserves quasi- coproducts if for every quasi-discrete diagram C: G --~ B with a colimit in B, also F C is quasi-discrete and colim F C TM F (colim C).

1.4. By wide pullbacks (or fibred products) we mean limits of small diagrams of the form (fi: Ai --+ B)i~x. In Set coproducts commute with connected limits. For quasi-coproducts we have:

PROPOSITION. In Set quasi-coproducts commute with wide (but small) pullbacks.

Proof Let J = (dk: k -+ e}keK be the diagram scheme for a wide pullback (with K small and, in order to avoid triviality, non-void), and let G be a small groupoid, which may be assumed skeletal so that G is the disjoint union of groups Gi, i E I = obG. For a diagram D: G × J --+ S e t such that D ( - , j ) is quasi-discrete for all j E d = obJ, we must show

colimi limj D ( i , j ) ~- limj col imiD(i , j ) . (*)

Here each Xij := D ( i , j ) is a Gi-set, and (with j fixed), col imiD(i , j ) is given by the quasi-coproduct (mij: X q ~ Qj)ieI. For every i E I and k E K uik := D(i, dk): Xik ~ Xie is a Gi-map, due to functoriality of D. Hence, for every k E K, one has a map vk: Qk --+ Qe such that

X i k mik ) Qk

(1)


commutes. For every i ~ I, let (Pik: Xi ~ X i k ) k ~ K be the wide pullback of (ui~)keK, and let (Pk: Q --+ Q/¢)keK be the wide pullback of (v~)~eK, hence Q = limj colimiD(i , j) . In order to show (.), we must verify that the maps (mi: Xi --+ Q)ieI which make the diagrams

X i mi > Q

(2)

commute, form a quasi-coproduct (i.e., the colimit of a quasi-discrete diagram). For that, we just check properties (1) and (2) of 1.2.

(1) Consider z = (zk)keK E Q. (The wide pullbacks are assumed to be formed as subsets of direct products.) Each zk belongs to the range of a unique rnikk; in fact, the ilc's all coincide: for zk~ = mhlc~(Xl) and zlc2 = mi2k2(x2), since Vkl(Zkl) = Vk2(Zk2) diagram (1) gives mile(Uilkl(Xl) ) = mi2e(Uiz~z(x2)), and this implies il = i2 since (raie)iez is a quasi-coproduct. Hence we have a unique i C 1 and a family (xk)keK with Zk = rnik(Xk) , but we are not sure yet that (xk)keK E Xi. But with a fixed/co E K, one has m~e(Ui~(xN)) = mie(uik(Xk)), hence u~,(xk(,) = gk" Uik(Xk) = uik(gk" xk) with eL E Gi for all k E K. Now we have y = (gk " xk)keK E Xi, and mi(y) = z.

(2) Consider x = (xk)keg, Y = (Yk)keK E Xi (for some i E I) with mi(x) = mi(y), hence mik(xk) = mik(Yk) and therefore Yk = gk .xk with gk E G~ for all k C K. But, in fact, since the group action Xie is free, all gk's coincide: for kl, k2 E K, since uik, (xk~) = u~k2(xk:) and ~.tik I ( y k l ) : ~ik2(Yk2), one has

gk~ " Uik2(Xk:) = Uik,(gk, " Xkl) = Uik2(gk2 " Xk2) = gk2" Uik2(Xkz),

hence gk~ = gkz. []

1.5. We remark that, in the proof of Proposition 1.4 quasi-discreteness of the diagrams D ( - , j ) was exploited "only" in case j = e. However, in the pres- ence of Gi-maps Xik --~ X~e, freeness of the group action on )fie automatically,, gives freeness of the group action on each Xitc. The question remains, however, whether in the proposition the restriction to quasi-discrete diagrams could be omitted; also, whether wide pullbacks could be traded for arbitrary connected limits (as in the case of coproducts). The answer to both questions is negative, as the following two examples show; in both examples, the indexing system I (in the notation of 1.2) is a singleton set, that is: the groupoid G is just a group G.


1. Every group G acts on itself by left translation, i.e. the group action is the group multiplication. The pullback of two copies of the only map G --+ 1 onto a singleton set is G × G, considered as a G-set by left translation. (Note that the action of G on itself and on G × G is free, but not on 1, unless G is trivial.) The (singleton-indexed) quasi-coproduct of G is given by (G --+ 1), while the quasi-coproduct of G × G is described by m: G × G --+ G, (x, y) ~-+ x - ly . But the pullback of two copies of 1 --+ 1 is 1, not G, unless G was trivial. Consequently, in Set, colimits of diagrams of groups (i.e., of bijective self-maps) may fail to commute with (ordinary) pullbacks, even with kernel pairs.

2. Next we show that, in Set, (singleton-indexed) quasi-coproducts may fail to commute with equalizers. We fix a non-trivial element h in a group G, let G act on G × G by left translation and consider the C-map f : G × G --+ G × G, (x, y) (xh, yh). The equalizer of f and the identity map is 0. However, with m the quasi-coproduct of G × G as above, the map f : G --+ G induced by f is given by (x ~-+ h- lxh) , and its equalizer with the identity map is G whenever G is abelian, not (~. (Note that, in this example, G acts freely on both equalizers; if one simplifies this example by trading G x G for G, even this property is lost.)

2. Polylimits and Their Preservation by Set-Functors

2.1. Recall that an object T of a category C is quasiterminal (cf. [32]) if it is weakly terminal and has the property that for all x, y: B --+ T in C there is exactly one morphism with tx = y; necessarily, t is an automorphism. Hence T is quasiterminal iff the group G = Aut C (T) acts regularly on the non-empty sets C(B, T), for all B E C. T is called locally terminal in C if it is quasiterminal in its connected component of C; equivalently, if every C-morphism x: B --+ T is terminal in the co-sliced category B / C of morphisms with domain B (cf. [32]). Finally, a (small) family (T/)iel of objects in C is called polyterminal if I labels the connected components of C one-to-one and if each T~ is locally terminal in C; explicitly, this means for all B C C that

(1) there is a unique i E I such that C(B, T~) # 0, and (2) for all x, y ~ C(B, Ti) there is a unique (auto-)morphism t: Ti ~ T~ with

tx = y (see [23]).

Note that (Ti)~ei is multiterminal in the sense of Diers [10, 11] iff (T~)ici is polyterminal and each group Gi = A u t c (T/) is trivial.

2.2. A family (Li, Ai: ALi --+ D)ieI of cones in a category C is a polylimit of a diagram D: D --+ C if (L~, A~)icz is a polyterminal object in the comma category (A $ D) of cones over D. A functor F: C --+ S e t merges the polylimit (L~, Ai)~cI if

QUASI-COPRODUCTS AND ACCESSIBLE CATEGORIF_.S WITH WIDE PULLBACKS 393

(1) for every i E 1, the group action

C~(A,~D)-~cF S e t

is free, with Gi =Aut(A$D)(L~,Ai) and U the forgetful functor (so that (Ft)(x) = x with t E Gi and x E FL~ is possible only for t = l, i.e., the map Ft is without fixed points for t ¢ 1), and

(2) the quasi-coproduct (mi: FLi --+ Q)~ex gives a limit of FD; more precisely, since FA~d: FLi --+ F D d is a trivial G~-map for all i E I and every d E D, one obtains a cone )~: A Q ~ F D with Ad. mi = FAid, and this cone is required to be limiting.

In case (L~, Ai)~ex is a multilimit, this notion of preservation coincides with the ones used in [ l l , 18].

LEMMA. Every representable functor merges polylimits. Proof We just need to check conditions (1) and (2) above, with F = C ( B , - ) .

But (1) is an immediate consequence of 2.1 (2), and for (2) one considers a cone c~: A A --+ C ( B , - ) D . Then, for every a E A, the cone O~a: A B --+ D (with c~d = (~d)(a)) factors as O~a = ,ki(AXa), with a unique i E I and some Xa: B --+ Li in C. But xa is "unique up to a unique automorphism" t E Gi, and since mi(Xa) = mi(tx~) for all t E Ga, the only map ~o: A -4 Q with A(Aqo) -- o~ is described by ~p(a) -- mi(x~). []

2.3. Recall that a functor F: C ~ Se t is n-flat (for an infinite regular cardinal n) if F is a n-filtered colimit of representable functors (in the functor category (C, Set)); equivalently, if the category el(F) °p is n-filtered, with e l (F) = (1 $ F ) the element category of F (see [21]). The category C is called n-polycomplete if every diagram D: D --+C with # D < n has a polylimit; if F: C --+ S e t merges such polylimits, then it is called n-polycontinuous.

PROPOSITION. For a small n-polycomplete category C, a functor F: C --4 Set is n-flat if and only if F is n-polycontinuous.

Proof "if". We must show that e l (F) °p is n-filtered, that is: for every diagram E: D --+ e l (F) with # D < n there is a cone. One forms the polylimit (Li, Ai)ie/ of the diagram D = UE in C, with U: el(F) --~ (3 the forgetful functor. Since F merges this polylimit, the canonical natural transformation t: A1 --+ FU gives us an element z: 1 --+ Q with A(Az) --- t E and Q, A as in 2.2. There is only one i E I such that z can be written as z = mi(x) with x E FLi. Since A(Am~) = FAi, the cone Ai can be lifted to a cone A(Li, x) --+ E, as desired.

"only if". Conversely, let F be a n-filtered colimit of representable functors C(Cj , - ) , j E J ; without loss of generality, J may be assumed to be x-directed. The preservation by F of a polylimit (Li, ~i)iEI of D in C now follows from its preservation by each C ( C / , - ) and the fact that n-filtered colimits commute


with limits in (C,Set). In fact, one can easily check condition (1) of 2.2 using the explicit construction of directed colimits in Set, and the following computation shows the validity of condition (2):

colimi F Li "~ l imjC(Cj ) = colimico , Li

-~ col imjcol imiC(Cj , Li)

="~ colimj (lim C(Cj , D - ) )

='~ lim(colimjC(Cj , D - ))

~- lim FD.

2.4. For a small category C, let n-Flat(C) be the full subcategory of the functor category (C,Set) containing all n-fiat functors.

COROLLARY. I f C is n-polycomplete, then n-Flat(C) has wide pullbacks. Proof According to Proposition 2.3, one just needs to show that the wide

pullback F of n-polycontinuous functors Fk -4 G (k E K) is again n- polycontinuous, i.e., preserves the polylimit (Li, Ai)ici of any diagram D in C of size < n. In fact, F satisfies the freeness condition (1) of 2.2 since G does and since there is a natural transformation F -4 G. Condition (2) follows from the commutation of quasi-coproducts with wide pullbacks in Set (see Proposi- tion 1.3):

limk colimiFkLi =~ limk(lim FkD) = lim F D . colim~ F Li = []

3. Quasi-Based Categories

3.1. Extending the terminology used in [7] and [18] we call an object B of a category B (quasi-)coprime if the representable functor B ( B , - ) : B -+ S e t preserves all existing (quasi-)coproducts of B. (Since coproducts are special quasi- coproducts, a quasi-coprime object is coprime, not vice versa, in general.)

If in a quasi-coproduct (mi: Bi --+ Q)~ex in B (with I = obG, G = (J~cx Gi) we have a quasi-coprime object Bj, then the group Gj is isomorphic to the group AUtB/Q(Bj, mj) of automorphisms t: Bj -4 Bj in B with rnjt = mj. Indeed, since (C(B j , mi): C(Bj , Bi) -4 C(Bj , Q))iel is a quasi-coproduct in Set, any endomorphism t: Bj -4 Bj in B with mj t = mj can be written uniquely as t = C(g), g E Gj, with C: G -4 B the given quasi-discrete diagram. Hence C induces an isomorphism

Gj -+ (B /Q) ( (B j , mj), (Bj ,mj)) .

3.2. We call a category B (quasi-)based if B has (quasi-)coproducts and if there is a small subcategory C of (quasi-)coprime objects in B such that every object


of B is a (quasi-)coproduct of objects in C. Bq denotes the full subcategory of quasi-coprime objects in B.

PROPOSITION. For B quasi-based, Bq is essentially small and dense in B. Proof Let C be as in the definition of quasi-based category. We show that

Bq is equivalent to C. Indeed, every B E Bq appears in a quasi-coproduct (mi: Ci --+ B)i~1 with objects C~ E (3, which is preserved by B ( B , - ) . Hence 1B E B ( B , B) can be written as mis = 1B with s E B ( B , Ci) and a unique i E 1. But also B ( C i , - ) preserves the quasi-coproduct, so that mi(smi) = mi implies that smi must be an automorphism of Ci. Hence mi is an isomorphism. Consequently, essential smallness of Bq follows from smallness of C.

In order to show that Bq is dense in B, we pick for every A E B a diagram C: (~ --+ B with a (skeletal) groupoid G and values in Bq such that A is the colimit of C. It then suffices to show that there is a cofinal functor

F: G -+ B q / A

whose composite with the forgetful Bq/A -+ B is C. But since the colimit of C is a quasi-coproduct (mi: Ci -+ A)i~i, we may define F by Fi = mi (i E I = obB). Any object (B, f : B -+ A) in Bq/A can be written as f = m i f I with a unique i E I , by quasi-coprimity of B. Hence there is an arrow (B, f ) -+ Fi in Bq/A. Furthermore, any two arrows (B, f ) -+ F j and (B, f ) ~ Fk lead to factorizations mju = f = inky, which is possible only for j = k and if there is a g E Gi = G(i , i) with C(g)u = v. Hence the comma-category ((B, f ) .1. F ) is (non-empty and strongly) connected. []

3.3. Proposition 3.2 leads to the following representation of quasi-based categories.

PROPOSITION. Every quasi-based category B is equivalent to the full subcategory of quasi-coproducts o f representable functors in (Bq °p, Set).

Proof Since Bq is dense in B the functor

E: B ~ (Bq°P, Se t ) , A ~ B ( J - , A ) ,

(with J : Bq ~ B) is full and faithful. E preserves any quasi-coproduct of (Bi)ieI in B, since colimits in (Bq °p, Se t ) are computed pointwise and since B(C, - ) preserves quasi-coproducts for every C E Bq. But since every object in B is a quasi-coproduct of objects in Bq which E maps to representable functors in (B~ p, Se t ) , B is in fact equivalent to the full subcategory of quasi-coproducts of representable functors, as claimed. []

COROLLARY. In a quasi-based category, coproducts are stable under pullback, and quasi-coproducts commute with wide pullbacks (to the extent that these limits exist).


Proof The full and faithful functor E preserves quasi-coproducts (in particular: coproducts) as well as all existing limits. The properties claimed for the quasi-based category B therefore follow from the respective properties of the functor category (B~ p, Set ) . []

3.4. Finally, we give a necessary (and, as we shall show in 4.4 below, sufficient) condition for a quasi-based category to have n-limits (i.e., limits of diagrams of size < ~, with a regular cardinal ~).

PROPOSITION. For a quasi-based category B with e;-limits Bq is t,;-polycomplete.

Proof For a diagram D: D ~ Bq with #D < ~; we may form the limit (L, A) of J D in B (with J : Bq ~ B) and then present L as a quasi-coproduct (mi: Li --+ L)iei of objects Li E Bq. We claim that (Li,)~(Ami))i~I is a polylimit of D. Indeed, every cone fl: A B --4 D in Bq factors uniquely as J/~ = ,~(Af), and f : B ~ L factors as f = miu for a u n i q u e i E l a n d some u: B --+ Li, by quasi-coprimity of B. If we have miu ~- miv for another v: B ~ Li, then v = C(g). u for a unique g C Gi = G(i , i); here C: G ~ B is the quasi-discrete diagram of which L is the colimit. Since Li is quasi-coprime, any morphism t: L~ ~ Li with mit = m~ is of the form t = C(g), so that the universal property of a polylimit follows. []

3.5. From 2.4 and 3.4 one has that ~-Flat(Bq) has wide pullbacks. This category can now be described more conceptually, as follows.

PROPOSITION. For a quasi-based category B with n-limits, the category

Lim~ IF(B, Se t )

of functors preserving ~-limits and quasi-coproducts is equivalent to the category ~-Flat(Bq) and has wide pullbacks.

Proof We consider the restriction functor

R: Lim~ ]_ I - (B , Se t ) --~ (Bq, Se t ) , M ~-~ M J,

with J: Bq ~ B. Density of J makes R full and faithful. Hence the proof will be complete once we have shown that the essential image of R is ~-Flat(Bq). Certainly, every R-flat functor F: Bq -~ S e t is in the essential image of R, since F TM col imiBq(Bi , - ) is a ~;-filtered colimit of representables; Bi E Bq implies B(Bi, - ) E Lim,~ [I"~(B, Se t ) , and R preserves the existing(!) pointwise colimit: R ( c o l i m i B ( B i , - ) ) ~- F.

Conversely, we must show that for every M E Lim~ LI ~ (B, Se t ) , the restriction M J is n-flat. In Bq, n-polylimits are formed from ~;-limits and quasi-


coproducts of B (see 3.4), both of which are preserved by M. Hence M J is ~-polycontinuous and therefore r;-flat (according to 2.3). [3

4. Duality for Accessible Categories with Wide Pullbacks

4.1. Let ~; be an infinite regular cardinal. Recall from [14] that an object A of a category A is n-presentable if the representable functor A(A, - ) : A --+ S e t preserves n-filtered colimits. A is n-accessible if A has ~-filtered colimits and if there is a small subcategory C of A consisting of t~-presentable objects such that every object of A is a n-filtered colimit of a diagram of objects in C. A category is accessible if it is n-accessible for some ~. A functor between accessible categories is accessible if it preserves n-filtered colimits for some n (see [27]). For any small category C, the category ~;-Flat(C) of n-fiat functors from C to Set is n-accessible (see [27]). If C is ~;-polycomplete, then ~-Flat(C) has wide pullbacks (see 2.4). Our first goal is to show that every ~-accessible category A with wide pullbacks arises this way. Since every t~-accessible A satisfies

A ~ ~;-Flat ( A T )

with A~ the full subcategory of a-presentable objects, we just need to show that A ~ is ~;-polycomplete.

4.2. It is convenient to prove the following important proposition first:

PROPOSITION. For a ~-accessible category A with wide pullbacks, the category

Wdpb Filth(A, Se t )

of functors preserving wide puUbacks and ~-filtered colimits is quasi-based and has r~-limits, and its full subcategory of quasi-coprime objects is equivalent to a ~ .

Proof. Quasi-coproducts and ~-limits exist in B := Wdpb Filt,~(A, Se t ) since they commute in Set with wide pullbacks and t~-filtered colimits. As a ~- accessible functor, each F E B satisfies the solution set condition (see [13] and [27]), hence preservation of wide pullbacks makes F have a left polyadjoint, i.e., for all X E S e t there is a polyinitial (= dual to polyterminal) object in (X -1, F) , (cf. [23]). In particular, for X = 1, the element category el(F) = (1 ,1. F ) has a polyinitial object (Ai, zi)i~t. By the Yoneda Lemma, each z~ E FAi corre- sponds to a natural transformation ~i: A(Ai , - ) ~ F . We first want to show that these natural transformations present F as a quasi-coproduct of representables in (A, Set).

Let Gi = AUtel(F)(Ai, x~) be the group of automorphisms t: Ai --+ Ai in A with (Ft)(z~) = x~ or, equivalently, ~ • A ( t , - ) = ~ , for each i C / , and


define C: G = Ui~I Gi --+ (A, Set) by C(t) = A ( t , - ) . This diagram is quasi-discrete. In fact, for every non-initial H E (A, Set) and any natural transformation a: H ~ A(Ai, - ) with C(t).~r = a, there is B E A and b E H B ~ ~) such that aB(b), t = orB(b) with aB(b): Ai -+ B in A, hence aB(b): (Ai, xi) --+ (B,F(crB(b))(xi)) in el(F); this implies t = 1 since (Ai,xi)i~r is polyinitial. For every A E A one must now check that the family

(~iA: A(Ai, A) --+ FA)icI

satisfies the quasi-coproduct properties (1), (2) of 1.2. But since (~iA)(f) = (Ff)(x i ) for every f E A(Ai, A), these follow immediately from the polyini- tiality of (Ai, xi)iEt in el(F). Hence F is the quasi-coproduct of (A(Ai, - ) ) ies in (A, Set).

Next one has to show that each Ai is n-presentable, so that we actually have a quasi-coproduct in B. But for a n-filtered colimit X --- colimjXj in A, every h E A(Ai, X) gives an element ~i(h) E F X ~- colimjFXj, which then gives a morphism in A(Ai, Xj) through which h factors since F X j is a quasi- coproduct of the family (A(Ai, Xj))i~i. This sketches the proof of A(Ai, X) colimjA(Ai, Xj), the details of which are straight-forward and therefore omitted here.

Finally, each representable functor A(Ai, - ) is quasi-coprime in (A, Set), since if H is a quasi-coproduct of Hk (k E K), then HAi -~ Nat (A(Ai , - ) , H) is a quasi-coproduct of HkA -~ Nat (A(Ai, - ) , Hk) (k C K). Consequently, since Ai C An, A ( A i , - ) is quasi-coprime in B. This completes the proof that B is quasi-based. The assertion that Bq is equivalent to A °v now follows from (the proof of) Proposition 3.3.

4.3. We now obtain the following characterization theorem for n-accessible categories with wide pullbacks (see also Theorem 0.20 of [23] and Theorem 4.11 of [3]):

THEOREM. For an infinite regular cardinal n, a category A is n-accessible with wide pullbacks if and only if A is equivalent to the category n-Flat(C), for a small n-polycomplete category C.

Proof According to 4.1, we just need to show that A °p is n-polycomplete for a n-accessible category A. But by Proposition 4.2, B = Wdpb Filth(A, Set) is n-complete and quasi-based with Bq = A °v, and Bq is n-polycomplete by Proposition 3.4. []

4.4. We can now proceed and define the duality we have in mind. Let .A,~ be the 2-category of n-accessible categories with wide pullbacks, of functors preserving n-filtered colimits and wide pullbacks, and of all natural transformations between them. B,~ is the 2-category of quasi-based categories with n-limits, of functors preserving quasi-coproducts and n-limits, and of all natural transforma-


tions between them. Propositions 3.5 and 4.2 and Theorem 4.3 guarantee that the assignment

A ~ Wdpb Filt,~(A, Set)

is the object-part of a 2-functor if: ,A~ --+ op /3~, and that the assignment

B ~ Lim~ I I ~ ( B , Set)

is the object-part of a 2-functor 9: B °p --+ ,A~. (Here B ~ dualizes B~ w.r.t. 1-cells.) In fact/I) is a left 2-adjoint of 9, with the unit of the adjunction given by the evaluation functor

7/A: A -+ kO~A, A ~-~ (F ~-~ FA) ;

similarly, the counit is the evaluation functor

CB: B --+ ~ ffy B , B ~-~ ( M ~-~ M B ) .

(The details are easily checked, analogously to §4.2 of [27].)

THEOREM. The 2-functors • and • define a biequivalence between ,A~ and B°P; in other words,

(1) for every AE ,A~, the functor T1A is an equivalence, and

(2) for every BE 13~, the functor EB is an equivalence.

Proof. (1) With B := ffA, one has an equivalence R: tI, B -+ n-Flat(B), M ~-~ M J, from 3.5. The composite

R~TA: A -+ ~-Flat(Bq) _~ ~-Flat(A °p)

is the canonical equivalence A ~ A( i ( ) , A), with i: A,~ ~ A (see Prop. 2.18 of [27]). Consequently, ~/A is an equivalence of categories.

(2) From the presentation of a-Flat(Bq) as the free completion of B~ v under n-filtered colimits one has an equivalence of categories

Z*" Filt~(~-Flat(Bq), Set) -+ (Bq p, Set),

given by F ~-+ F Z , with Z: B~ p -+ a-Flat(Bq) the restricted Yoneda embedding (cf. [27, Prop. 1.2.4]). The quasi-inverse S: ~-Flat(Bq) -+ kVB of/? is given by left Kan-extension along J: Bq '-+ B and induces an equivalence

S*: Filt~(kVB, Set) --+ Filt~(~;-Flat(Bq), Set),

given by G ~ GS. Its restriction S # to Og2B is still (full and) faithful, and the composite S#SB is isomorphic to the full and faithful functor E: B --+ (B~ p, Set) of 3.3. Hence ~B is full and faithful.


Finally, eB is also essentially surjective on objects. Indeed, with A -- kVB, any functor F E C A is a quasi-coproduct of representable objects Nat(B(Bi, - ) , - ) -~ eBBi, with Bi E Bq. But the evaluation functor e B preserves quasi-coproducts, hence F ------ eB (colimiBi). []

COROLLARY. (1) Two n-accessible categories A, A l with wide pullbacks are equivalent if and only if the categories ~A, ~A I are equivalent.

(2) Two n-complete quasi-based categories B, B I are equivalent if and only if the categories #B, qB' are equivalent.

4.5. Part (1) of the theorem can be freed of the fixation of the cardinal n, as follows. For an accessible category A with wide pullbacks, let

Wdpb Acc(A, Se t )

be the category of accessible functors A -4 Se t preserving wide pullbacks. Index-raising shows that this category has all small limits and quasi-coproducts.

THEOREM. For an accessible category A with wide pullbacks, the category

Lim H~(Wdpb Acc(A, Set ) , Se t )

of functors preserving small limits and quasi-coproducts is equivalent to A. Proof. A is n-accessible for some n. The inclusion functor

i: ~ A --4 Wdpb Acc(A, Se t )

induces a functor

i*" LimH~(Wdpb Acc(A, Set ) , Se t ) -4 g/~A, M ~-~ Mi.

The evaluation functor 77A: A -4 9 & A factors through i*, so that there is a functor

CA: A --+ L i m H ~ ( W d p b Acc(A, S e t ) , S e t ) .

Since ~A is an equivalence of categories, it suffices to show that i* is full and faithful in order to conclude that also e A is an equivalence. Hence we consider functors M, N in the domain of i* and a natural transformation/3: Mi -4 Ni and must show that fl is of the form/3 = ai for a unique a: M -4 N.

In order to define aF: M(F) -4 N(F) for F E Wdpb Acc(A, Set) , one first considers the case F = A ( B , - ) . Since B is a n-filtered colimit B --- colimkAk of objects Ak E A,~, one has F -~ limk A(Ak, - ) in this case, and M, N preserve this limit. Consequently, since each A(Ak, - ) belongs to CA, a F is determined by/3A(Ak,-) and the limit property. In the general case, we use the fact that F is a quasi-coproduct of representable functors A ( B i , - ) , by Proposition 4.2. Since


M, N preserve quasi-coproducts and since each OtA(Bi,_ ) is already uniquely defined, c~F is given by the universal property of the colimit. []

COROLLARY. An accessible functor F: A --+ B of accessible categories with wide pullbacks is an equivalence of categories if and only if F preserves wide pullbacks and the induced functor F*: Wdpb Acc(B, Set) ~ Wdpb Acc(A, Set) is an equivalence of categories.

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quasi-coproducts and accessible categories with wide pullbacks

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