on diagrams and flatness of functors

Journal of Pure and Applied Algebra 154 (2000) 247–256www.elsevier.com/locate/jpaa

On diagrams and atness of functors

Ye�m KatsovDepartment of Mathematics and Computer Science, Hanover College, Hanover, IN 47243-0890, USA

Communicated by I. Moerdijk

Dedicated to Professor Bill Lawvere on the occasion of his sixtieth birthday

Abstract

Characterizations of set-valued weakly at (in B. Stenstr�om’s sense) functors over small cat-egories C, describing the structure of those functors from di�erent points of view (categoricaland universal algebraic), have been obtained. This result not only generalizes, but also combinesand supplements, all related results so far known for S-acts. The approach taken is based onexploiting diagrams of functors similar to one used by Mac Lane and Moerdijk for describing at functors. c© 2000 Elsevier Science B.V. All rights reserved.

MSC: 18A30; 18A40; 18B99; 18G99; 20M50

1. Introduction

This paper concerns the representations of a small category C by set-valued functors(or presheaves). Particularly we examine di�erent concepts of atness, their connectionswith the idea of diagrams of functors and “structural” theorems for functors.From now on, let SetsC and SetsC

op

stand (respectively) for the categories of co-variant and contravariant functors, de�ned on a small category C to the category Setsof sets, and natural transformations between them. As is well known (see, for example,[7,8], or [12]), a tensor product bifunctor −⊗− : SetsC×SetsCop → Sets may be de-�ned in such a way that it becomes a left adjoint of the Hom functor; more precisely,for any functor F ∈ SetsC there is a natural adjunction F ⊗ − : SetsCop � Sets :HomSets(F(−);−). Thus, the functor F⊗− is always right exact, i.e. preserves colimits

E-mail address: [email protected] (Y. Katsov).

0022-4049/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S0022 -4049(99)00190 -5

248 Y. Katsov / Journal of Pure and Applied Algebra 154 (2000) 247–256

[11, Section V.5]. However, just as for R-modules, the tensor product F ⊗ − is notalways left exact. In the case of modules, a module A is called at when A ⊗ −is left exact, i.e. preserves short exact sequences, which is equivalent to preservingmonomorphisms (embeddings), or �nite limits. In a non-additive setting like the presentone, all those cases are quite di�erent and lead to di�erent, interesting and usefulnotions of atness contingent upon the types of �nite limits that the correspondingtensor product supposedly preserves. Almost all those notions originally appeared inthe context of S-Sets over the monoid S (or, as the standard term is now, S-Acts) —a particular case of functor categories SetsC when C is a single-object category, i.e.a monoid. (One may consult, for instance, [15,13,2], and the references in them forhistorical or chronological matters.)The most interest and activity have occurred around the following notions of atness

for a functor F ∈ SetsC:• F is at [12, Section VII.5, De�nition 1; 4] [6] i� F ⊗− preserves �nite limits,i.e. is left exact;• F is weakly at [15] (now, in the context of S-Acts, the term ‘strongly at’ [9]is more popular) i� F ⊗− preserves equalizers and pullbacks;

• F is pullback- at [13] i� F ⊗− preserves pullbacks;• F is equalizer- at [13] i� F ⊗− preserves equalizers;• F is mono- at [9,7] i� F ⊗− preserves monomorphisms (embeddings).We have listed these notions in such a way that each of them (except pullback- and

equalizer- atness) sounds stronger than the subsequent ones. Indeed, it has been knownthat, in order as they are presented, each of those notions (except pullback- and weak atness) in general leads to wider classes of S-acts (or functors). As to the classesof weakly at and pullback- at functors, it turned out [2] (see also Corollary 3.7)that they coincide (this question was left open in [13]).Flat functors, being geometric morphisms [12, Section VII.5, Theorem 2], not only

proved to be very important in topos theory, but also play a signi�cant role in thepresent paper and constitute the chief “building blocks” of one of the main (structural)results of this paper (Theorem 3.6), describing from di�erent points of view the struc-ture of weakly at functors. There exist ([12, Section VII.6, Theorem 3]; also, for atleast a “colimit” description, see [4]) very good and practicable categorical (using acolimit procedure) as well as universal algebraic descriptions of the structure of atfunctors. The idea of the diagram of (also known as the Grothendieck constructionon, or the category of elements of) a functor F was found to be useful in obtainingthose descriptions. (The idea of diagrams of functors is solely based on the Yonedalemma, which (in our view) is a functorial variation of A. Robinson’s idea of diagramin model theory [5, Lemma 1:4:2]; amazingly, both ideas obviously appeared inde-pendently and at about the same time. Perhaps this observation may explain, in someway, why Robinson used ‘diagram’, at that time so unusual for a model-theorist’s term(cf. [5, p. 17]).)In any case, exploiting further the “diagram” approach of [12] as well as com-

bining methods of categorical and universal algebra, we obtain a structural result

Y. Katsov / Journal of Pure and Applied Algebra 154 (2000) 247–256 249

(Theorem 3.6) on weakly at functors. It is worth mentioning that this result notonly generalizes (contains as immediate consequences) and combines Theorems 2:3and 2:4 of [2] and Theorem 5:3 of [15], but it also supplements them (e.g. the condi-tions (2) and (5) are of importance) as well as provides for them an alternative proof,based on di�erent ideas and techniques. (Thus we do not need to appeal to Theorem1:2 of [10] as it was done in the proof of the implication (e)⇒(f) in Theorem 5:3of [15].) Among other results of Section 3, we would single out Proposition 3.3 andits Corollary 3.4 that provide new useful characterizations of at functors and are ofinterest on their own.Finally, all notions and facts of categorical algebra, used here without any comments,

can be found in [11,14], and in the recently published [1]; for notions and facts fromuniversal algebra, we refer to [3].

2. Preliminaries

A category A is �ltered [12] if(i) ∃A ∈A (“A is not empty”);(ii) ∀A; B ∈A∃C ∈A∃� :C → A ∃� :C → B;(iii) ∀A; B ∈A∀�; � :A→ B ∃C ∈A∃ :C → A � = � .A category A satisfying conditions (i) and (iii) above is pseudo�ltered when it also

satis�es the following condition:(iipsf ) ∀A1; A2; B; C ∈ A∀�1 :A1 → B ∀�2 :A2 → B ∀�1 :A1 → C ∀�2 :A2 → C

∃M ∈A ∃�1 :M → A1 ∃�2 :M → A2 �1�1 = �2�2; �1�1 = �2�2.Note that in the terminology of [14, p. 72, De�nition 9:3:4], our �ltered and pseudo-

�ltered categories are co-�ltered and co-pseudo�ltered, respectively.A non-empty category A is connected (cf. [14, De�nition 9:1:1]) when for any two

objects A; B ∈A there exists a �nite sequence of arrows A= A0 ← A1 → A2 ← · · · ←A2n−1 → A2n = B joining A to B.On the objects of any category A, there exists the equivalence relation, connected-

ness, that is the smallest equivalence relation on them, with respect to which any twoobjects connected by a morphism are equivalent. The full subcategories of A de�nedby the equivalence classes of the connectedness relation are called connected compo-nents of A. It is obvious (see also [14, 9:1:2]) that every category A is a disjointunion (or coproduct) of its connected components.

Lemma 2.1. (1) A category A is pseudo�ltered i� it satis�es the conditions (i); (iii)above; and the next(ii∗psf ) ∀�1 :A1 → B ∀�2 :A2 → B ∃�1 :M → A1 ∃�2 :M → A2 �1�1 = �2�2.(2) A category A is �ltered i� it is pseudo�ltered and has a single connected com-

ponent (i.e. connected).(3) A category A is pseudo�ltered i� it is a coproduct of its �ltered connected

components.


Proof. (1) ⇒. It follows immediately from (iipsf ) by putting �1 = �1 and �2 = �2.⇐. In the notation of (iipsf ), there exist 1 :N → A1 and 2 :N → A2 such that

�1 1 =�2 2. If, by (iii), :M → N is such that �1 1 =�2 2 then, apparently, �1 = 1 and �2 = 2 .(2) ⇒. Applying consequently (ii) and twice (iii) to corresponding assumption of(iipsf ), we have the conclusion of the latter and the implication.⇐. It follows directly from (1) by using induction on the length of a sequence ofarrows jointing objects A and B:(3). It is obvious.

Recall (see [1, p. 22] or [12, p. 386]), the category∫C F of elements of a functor

F∈SetsC has as objects the pairs (A; a) where A∈ |C|, and a∈F(A), and as morphismsfrom one such pair (A; a) to another (B; b) those morphisms f :A → B for whichF(f)(a) = b or f · a = b (the action of F(f) on an element a is often denoted by adot (cf. [12, p. 25])). There is the evident forgetful functor �F :

∫C F → C. Also (see

[1, Theorem 2:15:6]), F is the colimit of representable functors, i.e. the colimit of the

composite functor∫C F→

�F C Y∗→SetsC where Y ∗ is the contravariant Yoneda embedding.

Then, denoting by∫C Fj connected components of

∫C F (which, for simplicity, we shall

also call connected components of F), and by Fj colimits of the composite functors∫C Fj −→

�FjC Y∗−→SetsC, where �Fj is the restriction of �F on the subcategory

∫C Fj, and

using [14, Proposition 9:1:4], we have

Lemma 2.2. F is the coproduct of Fj, i.e. F =⊔j Fj.

The following useful observation follows directly from de�nitions.

Lemma 2.3. Any functor between categories preserves connectedness. In particular; anatural transformation � :F → T induces the obvious functor �� :

∫C F →

∫C T de�ned

on objects of∫C F by (A; a) 7→ (A; �A(a)); and �� preserves connected subcategories of∫

C F .

From [8, Remark 1], the de�nition of −⊗−, Lemmas 2.2 and 2.3 we have

Lemma 2.4. The functor of taking of colimits of functors F ∈ SetsC; i.e. the functorColim : SetsC → Sets; is naturally isomorphic to the functor of taking of the sets ofconnected components of functors F ∈ SetsC.

3. Weakly at functors

Recall some terminology (cf. [15]). The functor F ∈ SetsC is free if it is isomorphicto a small coproduct

⊔i H

Ai of representable functors HAi = HomC(Ai;−); i ∈ I . F isprojective if and only if it is a retract of a free functor. F is �nitely generated if


there exists an epimorphism⊔i H

Ai → F; i ∈ I , and |I |¡∞; F is cyclic if there existsan epimorphism HA → F . Further, a sequence of the form of a “fork” K

��F

→Rin SetsC is said to be exact if is a coequalizer of � and �. A congruence � on afunctor F ∈ SetsC is a set of such equivalence relations �A on F(A) for each objectA ∈ |C| that whenever x�Ay and f :A→ B then f ·x�Bf ·y. Finally, R ∈ SetsC is saidto be �nitely presented if there exists an exact sequence K � F → R with F �nitelygenerated free and K �nitely generated.The following observation is useful.

Lemma 3.1. A functor R ∈ SetsC is �nitely presented i� there exist a �nitely gener-ated free functor F and a �nitely generated congruence � on F such that R ∼= F=�in SetsC (the congruence � is �nitely generated if it is the smallest congruence onF that contains some �nite number of pairs (x1; y1) ∈ F(A1) × F(A1); : : : ; (xn; yn) ∈F(An)× F(An) and F=� is the corresponding factor functor).

Proof. ⇒: Let K ��F

→R be a corresponding exact sequence with F �nitely

generated free, and � :HA1 t · · · t HAm→K an epimorphism. Then, by the Yonedalemma, the elements a1 = �(1A1 ) ∈ K(A1); : : : ; am = �(1Am) ∈ K(Am) generate (in ob-vious sense) the functor K . Let � be the congruence on F generated by the pairs(�A1 (a1); �A1 (a1)); : : : ; (�Am(am); �Am(am)), and � :F → F=� the canonical epimorphism.Clearly �� = ��, and therefore there exists �′ :R → F=� such that �′ = �. Since is a coequalizer of � and �; Ai�Ai(ai) = Ai�Ai(ai); i = 1; : : : ; m; hence by the generalalgebraic arguments — homomorphism theorems (see, for instance, [3]) and the natureof �, there exists ′ :F=� → R such that ′� = . As and � are epimorphisms, itfollows that �′ and ′ are natural isomorphisms, i.e. R ∼= F=�.⇐ : Let pairs (x1; y1); : : : ; (xn; yn), where (xi; yi) ∈ F(Ai) × F(Ai); i = 1; : : : ; n,

generate the congruence � on F ∼= HB1 t · · · tHBm and � :F → F=� ∼= R the canonicalepimorphism. Then it is easy to see that the sequence HA1 t· · ·tHAn �

��F

→R, where�(1Ai) = xi and �(1Ai) = yi; i = 1; : : : ; n, is exact.

Using Lemma 3.1 we obtain a generalization to the category of functors SetsC (andeven an alternative proof) of Proposition 4:1 of [15].

Proposition 3.2. Every functor F ∈ SetsC is a co�ltered colimit (directed limit) of�nitely presented functors.

Proof. It is obvious (also one can easily see that by applying the Yoneda lemma,descriptions of colimits in Sets and SetsC (see, for example, [14, 8:4:4 and 8:5:1])and [14, Proposition 10:2:1]; see also [1, Theorem 2:15:6] or [12, Proposition 1,p. 41]) that any functor F ∈ SetsC is isomorphic in SetsC to some factor functor


(⊔i∈I H

Ai)=� of the free functor⊔i∈I H

Ai modulo the congruence � generated by pairs(xj; yj); j ∈ J . If J is a directed limit of its �nite subsets Jk , and �Jk is the congruenceon

⊔i∈I H

Ai generated by pairs of Jk , then, as usual, we have (⊔i∈I H

Ai)=�=ColimJk→J

(⊔i∈I H

Ai)=�Jk . Now, if I is a directed limit of its �nite subsets In, then clearly⊔i∈I H

Ai = ColimIn→I (⊔i∈In H

Ai), and consequently (⊔i∈I H

Ai)=�Jk = ColimIn→I

(⊔i∈In H

Ai)=�Jk . Thus, from Lemma 3.1 and the equation (⊔i∈I H

Ai)=� = ColimJk→J

(ColimIn→I (⊔i∈In H

Ai)=�Jk ) we obtain the assertion.

Proposition 3.3. A functor F; having only one connected component; is at i� everynatural transformation R→ F; where R is �nitely presented; can be factored througha representable functor.

Proof. ⇐=: By [12, Section VII. 6, Theorem 3], it is enough to show that∫C F is

�ltered. As∫C F has a single connected component, F is nonempty and the condition

(i) for∫C F to be �ltered is satis�ed.

Now, if (A; a) and (B; b) are objects of∫C F , then there is the morphism � :HA t

HB → F de�ned by the equations �(1A) = a and �(1B) = b. So, there exist C ∈ |C|and c ∈ F(C); �c :HC → F; �′ :HA t HB → HC such that �c(1C) = c and � = �c�′.From the latter we have the existence of � :C → A and � :C → B such that � · c = aand � · c= b, and therefore the morphisms � : (C; c)→ (A; a) and � : (C; c)→ (B; b) inthe category of elements

∫C F . Thus,

∫C F satis�es (ii), too.

Finally, suppose in∫C F we have two morphisms �; � : (A; a) → (B; b); hence,

�; � :A → B, and � · a = b = � · a. Also, there is the morphism � :HA → F de�nedby �(1A) = a, and clearly �B(�) = �B(�). Let � be the congruence on HA generatedby the pair (�; �), and � :HA → HA=� the canonical epimorphism. By Lemma 3.1HA=� is �nitely presented; therefore, there exists �′ :HA=� → F such that �′� = �.Now, there exist C ∈ |C|; :HA=�→ HC , and �′′ :HC → F such that �′′ = �′′. So, if A(1A�)=u :C → A, we have a=�(1A)=�′′ �(1A)=�′′ (1A�)=�′′A(u)=u · c for someelement c ∈ F(C). Moreover, B(��)= B(�(1A�))=�( A(1A�))=�u= B(��)=· · ·=�u.Consequently in

∫C F we have found u : (C; c) → (A; a) such that �u = �u; and thus

proved that∫C F satis�es (iii).

⇒: First observe (cf. [1, Theorem 2:15:6]; [12, p. 41]) that for f : (A; a) → (B; b)to be a morphism in

∫C F is equivalent to the relation �aC(f;−) = �b (where the

natural transformations �a :HA → F and �b :HB → F are de�ned by �a(1A) = a and�B(1B) = b, respectively; and C(f;−) :HB → HA corresponds to f by the Yonedaembedding) in SetsC .By Lemma 3.1 and without loss of generality, let us consider in SetsC a morphism

� : (HA1 t · · · tHAm)=�→ F , where � is the congruence on HA1 t · · · tHAm generatedby a �nite number of pairs of morphisms (�ik ; �

jk); �

ik :Ai → Xk; �

jk :Aj → Xk; and

Ai; Aj ∈ {A1; A2; : : : ; Am}; k = 1; : : : ; n. Let a1 = �(1A1�) ∈ F(A1); : : : ; am = �(1Am�) ∈F(Am); xk = �ik · �(1Ai�) = �ik · ai = �(�ik�) = �(�jk�) = �jk · �(1Aj�) = �jk · aj ∈ F(Xk) foreach k = 1; : : : ; n. Therefore, in the category of elements

∫C F we have the following

objects (A1; a1); : : : ; (Am; am), and morphisms �ik : (Ai; ai) → (Xx; xk) ← (Aj; aj) :�jk for


each of the pairs (�ik ; �jk); k = 1; : : : ; n. From the observation above, the latter induces

in SetsC the morphisms C(�ik ;−) :HXk → HAi and C(�jk ;−) :HXk → HAj for eachof the pairs (�ik ; �

jk); k = 1; : : : ; n, as well as �a1 :H

A1 → F; : : : ; �am :HAm → F . Since

by [12, Section VII. 6, Theorem 3] the category∫C F is �ltered, one may easily �nd

(by simple induction arguments on the number of objects (Al; al); l = 1; : : : ; m, andpairs (�ik ; �

jk); k =1; : : : ; n) in

∫C F an object (A; a) and such morphisms �1 : (A; a)→

(A1; a1); : : : ; �m : (A; a)→ (Am; am), that �ik�i = �jk�j for each pair (�

ik ; �

jk); k = 1; : : : ; n.

From this one easily obtains the existence of such �′ :HA1 t · · · tHAm → HA in SetsC

that �′(�ik)=�′(�jk); k=1; : : : ; n; �a�

′=��, where � :HA1t· · ·tHAm → (HA1t· · ·tHAm)=�is the canonical epimorphism, and �a :HA → F is de�ned by �a(1A)= a: In turn, fromthe nature of the congruence � follows the existence of �′ : (HA1 t · · · tHAm)=�→ HA

such that �′= �′�. Consequently and because � is an epimorphism, one gets �a�′= �,and concludes the proof.

Corollary 3.4. Let a functor F have only one connected component and every naturaltransformation R→ F from a �nitely presented functor R can be factored through a�nitely generated free functor. Then F is at.

Proof. By Proposition 3.2 we may assume that F is a directed limit of �nitely presentedfunctors Ri with canonical morphisms �i :Ri → F; i ∈ I , i.e. F = Colimi∈IRi. First weshall show that any �i can be factored through a representable functor. Indeed, byLemma 2.4 and [1, Proposition 2:12:1∗] (see also [11, Section IX.2], or [14, 8:6:2]),1= ColimF = Colim(Colimi∈I Ri) = Colimi∈I (ColimRi) = Colimi∈I ni, where 1 is thesingleton and Colimi∈I ni is a directed limit of the �nite sets ni of connected componentsof the functors Ri; i ∈ I . From this and a description (see, for instance, [14, 9:4:2]) ofdirected limits in Sets, one may easily �nd a morphism �ij : ni → nj in the directedsystem under which the whole set ni is mapped into one element of nj, and thereforeconclude that �ij :Ri → Rj maps Ri into a subfunctor, let say R′j, of Rj that corresponds(in the notations of Lemma 2.2) to the connected components

∫C R

′j of Rj. Let �j :Rj →

F = Rj�′j→HA1 t · · · t HAm �j→F . Then, by Lemma 2.3, the restriction of �′j to R′j goes

through just one, let us say HA1 , of the representable functors, and hence �i can befactored through HA1 .

Now let R be �nitely presentable and R �→F=R �→HA1t· · ·tHAn →F . Because iscompletely de�ned by the elements (1A1 ); : : : ; (1An); F is a directed limit of Ri; i ∈ I ,and HA1t· · ·tHAn is free, there exists a factorization HA1t· · ·tHAn →F=HA1t· · ·tHAn

′→Ri �i→F for some �i :Ri → F . Since �i can be factored through a representablefunctor, we have that � can also be factored through a representable functor. Thus, Fis at by Proposition 3.3.

By using Lemma 3.1 and arguments similar to the ones of [15, Proposition 4:3], onecan easily generalize that result to the category SetsC.


Proposition 3.5. For an epimorphism � : P → F in SetsC the following propertiesare equivalent:(a) Finitely presented functors are �-projective; i.e. for every morphism � from a

�nitely presented functor K to F there exists �′ : K → P such that ��′ = �.(b) For every family of a1 ∈ F(A1); : : : ; am ∈ F(Am) and relations �i · ak(i)

=�i ·al(i); �i : Ak(i) → Xi ← Al(i) :�i; k(i); l(i) ∈ {1; : : : ; m}; i=1; : : : ; n; there existx1 ∈ P(A1); : : : ; xm ∈ P(Am) such that �(xj)=aj; j=1; : : : ; m; and �i ·xk(i)=�i ·xl(i)for all i.

Following [15], we call an epimorphism � : P → F in SetsC pure if it satis�es theequivalent conditions of Proposition 3.5.The desired explicit description of weak atness can now be stated as

follows:

Theorem 3.6. For a nonempty functor F ∈ SetsC the following statements areequivalent:(1) F is weakly at.(2) F is a small coproduct of at functors.(3) F is pullback- at.(4) The category

∫C F of elements is pseudo�ltered.

(5) F is a small copseudo�ltered colimit of representable functors.(6) Every epimorphism � : P → F in SetsC is pure.(7) There exists a pure epimorphism � :

⊔i∈I H

Ai → F for some Ai; i ∈ I .(8) Every natural transformation R → F; where R is �nitely presented; can be

factored through a �nitely generated free functor.(9) F is a small co�ltered (directed) colimit of �nitely generated free functors.

Proof. (1) ⇒ (2): In notations of Lemma 2.2, F =⊔j Fj . Since in Sets coproducts

commute with pullbacks and equalizers, and by Theorem 1 of [8] (or [12, p. 357,Theorem 1]) a functor −⊗G has a right adjoint, for any functor G ∈ SetsCop , we haveF ⊗G ' ⊔

j Fj ⊗G '⊔j(Fj ⊗G), and therefore all functors Fj are weakly at. Then,

denoting by �1 the constant functor corresponding to the singleton 1, and taking intoconsideration [8, Remark 1], the de�nition of − ⊗ −, and connectness of ∫C Fj, wehave Fj⊗�1=ColimFj=1. As �1 and 1 are terminal objects of the categories SetsC

op

and Sets, respectively, we have obtained that the functors Fj ⊗ − preserve pullbacksand terminal objects. Hence, by [1, Proposition 2:8:2] they preserve �nite limits, andall Fj are at.(2) ⇒ (3): Let F =

⊔j Fj, where all Fj are at. So, all functors Fj ⊗ − preserve

pullbacks, and as it was shown above F⊗G ' ⊔j Fj⊗G '

⊔j(Fj⊗G) for any functor

G ∈ SetsCop . From these facts and since in Sets coproducts commute with pullbacks,we obtain the implication.(3) ⇒ (4): Again, let F =

⊔j Fj be as in Lemma 2.2 and pullback- at. Then,

repeating the same arguments as in (1) ⇒ (2), we get that all functors Fj are at.


Hence, by [12, p. 386, Theorem 1] each category∫C Fj is �ltered. Now, noting that∫

C F =⊔j

∫C Fj, the assertion follows from part (3) of Lemma 2.1.

(4) ⇒ (5): By Lemma 2.1,∫C F =

⊔j 0

∫C Fj, where each

∫C Fj is �ltered. As

colimits commute with coproducts, and each Fj is co�ltered colimit of the composite

functors∫C Fj→

�FjC Y∗→SetsC, the statement follows from the dual of (3) of Lemma

2.1.(5)⇒ (1): Because colimits commute with coproducts, the functor F is a coproduct

of functors Fj that, by Lemma 2.1∗, are co�ltered colimits of representable functors.Since in SetsC colimits can be calculated “pointwise”, and from a description of co�l-tered colimits in Sets (see, for instance, [1, Proposition 2:13:3]), it is easy to see that∫C Fj is �ltered. Therefore, by [12, p. 386, Theorem 1], all functors Fj are at, andhence, weakly at. Thus, since in Sets coproducts commute with pullbacks and equal-izers, and using again the isomorphisms F ⊗ G ' ⊔

j Fj ⊗ G '⊔j(Fj ⊗ G) for any

functor G ∈ SetsCop , we have obtained that F =⊔j Fj is weakly at.

(6)⇒ (7)⇒ (8) are obvious.(8)⇒ (2): In the notation of Lemma 2.2, it is enough to show that all subfunctors

Fj of F are at. But all Fj have a single connected component and obviously satisfy(8), so by Corollary 3.4 they are at.(2)⇒ (6): In the notations of Lemma 2.2, consider the inverse images Pj=�−1(Fj).

Clearly P=⊔j Pj; and �=

⊔j �j; where �j : Pj → Fj is the restriction of � to Pj. From

condition (b) of Proposition 3.5 it is easy to see that a coproduct of pure epimorphismsis pure. Therefore it is enough to show that all �j are pure. Indeed, by Proposition3.3 any morphism R �→Fj from a �nitely presented functor R can be factored through

a representable functor, let say HA; i.e. R �→Fj = R �→HA →Fj for some � and . Inturn, since HA is a projective object of SetsC

op

; can be factored through �j; andconsequently � is factored through �j.(5)⇒ (9) is obvious.(9) ⇒ (2) follows immediately since in Sets both coproducts and directed limits

commute with pullbacks.

As a corollary of this theorem we obtain one more time the answer to P. Normak’squestion [13].

Corollary 3.7. For functors of SetsC the notions of weak atness and pullback- atnesscoincide.

From Proposition 3.3 and condition (8) we have (cf. [15, Corollary 5:4])

Corollary 3.8. Finitely presented at functors are retracts of representable functors;�nitely presented weakly at functors are projective.

Also from the condition (ii) on a category to be �ltered [12, Section VII. 6,Theorem 3] and Theorem 3.6 we get (cf. [15, Proposition 5:5]).


Corollary 3.9. Every �nitely generated weakly at functor is a small coproduct ofcyclic at functors.

References

[1] F. Borceux, Handbook of Categorical Algebra 1: Basic Category Theory, Cambridge University Press,Cambridge, 1994.

[2] S. Bulman-Fleming, Pullback- at acts are strongly at, Canad. Math. Bull. 34 (1991) 456–461.[3] G. Gr�atzer, Universal Algebra, 2nd ed., Springer, New York, 1979.[4] A. Grothendieck, J.L. Verdier, Prefaisceaux, Lecture Notes in Math., Vol. 269, Springer, Berlin, 1972.[5] W. Hodges, Model Theory, Cambridge University Press, Cambridge, 1993.[6] J.R. Isbell, Perfect categories, Proc. Edinburgh Math. Soc. 20 (1976) 95–97.[7] E.B. Katsov, Flat functors, Mat. Zametki 19 (1976) 577–586 (in Russian).[8] E.B. Katsov, The tensor product of functors, Sibirsk. Mat. �Z. 19 (1978) 318–327 (in Russian).[9] M. Kil’p, On homological classi�cation of monoids, Siber. Math. J. 13 (1972) 396–401.[10] D. Lazard, Autour de la platitude, Bull. Soc. Math. France 97 (1969) 81–128.[11] S. Mac Lane, Categories for the Working Mathematician, Springer, New York, 1971.[12] S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory,

Springer, New York, 1992.[13] P. Normak, On equalizer- at and pullback- at acts, Semigroup Forum 36 (1987) 293–313.[14] H. Schubert, Categories, Springer, New York, 1972.[15] B. Stenstr�om, Flatness and localization over monoids, Math. Nachr. 48 (1971) 315–334.

on diagrams and flatness of functors

Documents