Theory and Applications of Categories, Vol. 2, No. 10, 1996, pp. 113–132.

A NOTE ON FREE REGULAR AND EXACT COMPLETIONS AND THEIR INFINITARY GENERALIZATIONS

HONGDE HU AND WALTER THOLEN

Transmitted by Michael Barr

ABSTRACT. Free regular and exact completions of categories with various ranks of weak limits are presented as subcategories of presheaf categories. Their universal properties can then be derived with standard techniques as used in duality theory.

Introduction

A category A with finite limits is regular (cf. [2]) if every morphism factors into a regular epimorphism followed by a monomorphism, with the regular epimorphisms being stable under pullback; it is exact if, in addition, equivalence relations are effective, that is, if every equivalence relation in A is a kernel pair. It was noted by Joyal that, in the definition of regular category, one may replace “regular epimorphism” by the weaker notion of “strong epimorphism” in the sense of Kelly [11].

In [4] Carboni and Magno presented a one-step construction of the free exact completion C_{ex} of a category C with finite limits (=lex), in terms of so-called pseudo-equivalence relations. Recently, Carboni and Vitale [5] have constructed the free regular completion C_{reg} of C with weak finite limits (= weakly lex), and the free exact completion A_{ex/reg} for a regular category A, so that C_{ex} can be obtained as (C_{reg})_{ex/reg} for C with weak finite limits. The objects of C_{reg} are given by finite sources (f_{i} : X → X_{i})_{i∈I} of arrows in C, and the morphisms are defined to be equivalence classes of suitably compatible Cmorphisms between the domains of the given sources. Also for A_{ex/reg} the description of objects is simple, as A-objects with a fixed equivalence relation, while morphisms are less easily described: they are given by relations between the underlying A-objects satisfying certain compatibility conditions with the structure-preserving equivalence relations.

Quite a different approach to C_{ex} for C weakly lex and C small was given by Hu [9] who generalized a result by Makkai [14]: the exact completion C_{ex} is given by the category

C^{∗+} = ∏Filt(C^{∗}, Set)

of product- and filtered-colimit-preserving set-valued functors on

Partial financial assistance by the Natural Sciences and Engineering Research Council of Canada is acknowledged.

Received by the editors 31 January 1996 and, in revised form, 4 December 1996.

Published on 18 December 1996

1991 Mathematics Subject Classification : 18A35, 18E10, 18G05.

Key words and phrases: regular category, exact category, regular completion, exact completion, flat functor, accessible category.

\begin{matrix}c \\ ◯ \end{matrix} Hongde Hu and Walter Tholen 1996. Permission to copy for private use granted.

C^{∗} = Flat(C, Set),

the category of flat functors on C; note that C^{∗} is a finitely accessible category since C is small, and that it has products since C is weakly lex. We show here that for any category C with finite limits, C_{reg} and C_{ex} are full subcategories of C^{∗+} with some additional properties (see Remark 2.5).

In a talk in the Sydney Category Seminar in February 1995, Max Kelly proposed to make better use of the Yoneda embedding

y : C → (C^{op}, Set)

when constructing C_{reg} and C_{ex}. Because of the correspondences

\underline{X ⟶ Y_{i}}

\underline{y(X) ⟶ y(Y_{i})}

\underline{y(X) ⟶ ∏_{i∈I} y(Y_{i})}

y(X) ⟶^{e} A ⟶^{m} ∏_{i∈I} y(Y_{i})

(with e being a regular epimorphism and m being a monomorphism) the Carboni-Vitale construction suggests to take as objects of C_{reg} those F ∈ (C^{op}, Set) which appear at the same time as quotients of representables and as subobjects of finite products of representables.

This paper outlines Kelly’s construction of C_{reg} and shows that C_{ex} may be described conveniently within (C^{op}, Set) as well, as Kelly had anticipated in his talk: take those F ∈ (C^{op}, Set) which admit a regular epimorphism e : y(X) → F whose kernel pair K is again covered by a representable functor, so that there is a regular epimorphism y(Y) → K. The proof that C_{ex} constructed this way is indeed finitely complete remains a bit laborious, but we find it convenient that most proofs reduce to checking closedness properties of C_{reg} and C_{ex} within the familiar presheaf environment, which allows to present the intrinsic connection between both categories more directly.

We also stress the point that it takes no additional effort to prove all results for a category C with weak κ-limits, rather than just weak finite limits; here κ is a regular infinite cardinal or the symbol ∞. Recall that a (weak) κ-limit in C is a (weak) limit of a diagram D : J → C with #J < κ, which in case κ = ∞ simply means that J must be small. Then the notion of regularity and exactness must be “κ-fied” as follows: C is κ-regular (κ-exact) if C is regular (exact), has κ-limits, and if κ-products of regular epimorphisms are regular epimorphisms. Note that for κ = ℵ_{0}, the latter property comes for free:

f × g = (f × id_{D})(id_{A} × g) : A × C → B × D

is the composition of a pullback of g with a pullback of f. Notice furthermore that κ-Barr-exactness in the sense of [13] and [9] implies κ-exactness; ∞-regular is called “completely regular” in [5].

A simple description of C_{ex} with C co-accessible is also given at the end of the paper. We show that for a co-accessible category C with weak limits, the objects of C_{ex} are exactly all those F ∈ (C^{op}, Set) which admit a regular epimorphism into F with representable domain.

Acknowledgment: We thank Enrico Vitale for detecting a gap in an earlier version of the proof of Theorem 3.4. We are also grateful for valuable comments by Max Kelly and the anonymous referee.

1. Flat Functors

Let κ be an infinite regular cardinal or the symbol ∞. A category C is said to be weakly κcomplete if it has weak κ-limits, i.e., for any κ-diagram D : I → C (so that the morphism set of I has cardinality less than κ which, in case κ = ∞, just means that I must be small), a weak limit of D exists in C.

1.1. Definition. ([9]) Let C be a weakly κ-complete category, and B a category with κ-limits. A functor F : C → B is called κ-flat, if for any κ-diagram G : I → C, and for each weak limit cone (f_{i} : D → G(i))_{i∈I} on G, the morphism k : F (D) → limF ∘ G with F (f_{i}) = p_{i} ∘ k for all i ∈ I, is a regular epi; here the morphisms p_{i} are limit projections.

1.2. Remark. (i) For C small, and B the category Set of sets, as pointed out in [9], F is κ-flat iff it is a κ-filtered colimit of representable functors. (ii) For B having (regular epi, mono)-factorization, we notice that F is κ-flat iff there is a cone (f_{i} : D → G(i))_{i∈I} on G with F (f_{i}) = p_{i} ∘ k as in 1.1, so that k is a regular epi. (iii) ℵ_{0}-flat functors were called left covering functors in [5].

1.3. Proposition. For any locally small category C, let y : C → (C^{op}, Set) be the Yoneda embedding. If C is weakly κ-complete, then y is κ-flat.

Proof. Follows from 1.1 and the fact that a morphism a : M → N in (C^{op}, Set) is regular epi iff a_{C} : M(C) → N(C) is surjective for each C ∈ C.

1.4. Proposition. Let F : C → B be any functor. If C and B have κ-limits, then F is κ-flat iff it preserves κ-limits.

Proof. One only needs to show that a κ-flat functor F preserves κ-limits, that is, equalizers and κ-products. Since similar proofs can be found in [7] and [5], we can omit the proof here.

1.5. Proposition. For any weakly κ-complete category C, the category C^{∗} (= κ − Flat(C, Set)) of set-valued κ-flat functors has products. Moreover, for κ an infinite regular cardinal, C^{∗} has κ-filtered colimits.

Proof. Let (F_{i})_{i∈I} be a small family of functors of C^{∗}. Since F_{i} is κ-flat, for any κdiagram G : J → C, the morphism k_{i} : F_{i}(D) → lim F_{i} ∘ G is surjective for each i ∈ I. Therefore, the morphism ∏ k_{i} : ∏ F_{i}(D) → ∏ lim F_{i} ∘ G is surjective. ∏ lim F_{i} ∘ G is isomorphic to lim ∏ F_{i} ∘ G in Set, so we have that C^{∗} is closed under products in (C, Set).

Suppose that κ is an infinite regular cardinal. Let M : I → C^{∗} be a κ-filtered diagram. Since M(i) is κ-flat, the morphism k_{i} : M(i)(D) → lim M(i) ∘ G is surjective for each i ∈ I; here G : J → C is a κ-diagram. For any u ∈ colim_{J} lim_{I} M(i) ∘ G, there are i ∈ I and u_{i} ∈ lim M(i) ∘ G such that u = f_{i}(u_{i}) with f_{i} the colimit injection. Denote colim_{I} lim_{J} M(i) ∘ G by Q, and consider the following commutative diagram:

[figure: colim M(i)(D); M(i)(D); Q; lim M(i) ∘ G; k; f_{i}; f_{i}(D); k_{i}]

where k is a morphism induced by the family ⟨k_{i}⟩_{i∈I}. Each k_{i} is surjective, so there is x ∈ M(i)(D) such that k_{i}(x) = u_{i}; we let y = f_{i}(D)(x) and we have k(y) = u. Since κ-filtered colimits commute with κ-limits in Set, Q is isomorphic to lim_{J} colim_{I} M(i) ∘ G. Therefore, C^{∗} is closed under κ-filtered colimits in (C, Set).

2. C_{reg} and C_{ex}

2.1. Definition. Let y : C → (C^{op}, Set) be the Yoneda embedding. A functor F : C^{op} → Set is weakly representable if there is a regular epimorphism y(C) → F with C in C; we also say that F is regularly covered by C or y(C) in this case. We define extensions C_{κ−reg} and C_{κ−ex} of C as follows, for any weakly κ-complete category C:

(i) C_{κ−reg} is the full subcategory of (C^{op}, Set) whose objects are these weakly representable functors which are subfunctors of κ-products of representable functors.

(ii) C_{κ−ex} is the full subcategory of (C^{op}, Set) of functors F such that there is a regular epimorphism e : y(C) → F whose kernel pair (m, n : G → y(C)) has the property that G again is weakly representable.

Since we keep κ fixed, we shall write C_{reg}, C_{ex} for C_{κ−reg}, C_{κ−ex}, respectively.

2.2. Remark. Given B ∈ C_{reg}, we have a mono m : B → ∏_{i∈I} y(C_{i}) with #I < κ. Let the p_{i}’s be the projections of the product ∏ y(C_{i}), and let (u, v : A → B) be the kernel pair of a regular epi e : y(D) → B with D ∈ C. Then A is a limit of the family of (p_{i} ∘ m ∘ e, p_{i} ∘ m ∘ e)_{i∈I}. By 1.3, A can be regularly covered by y(S), where S is a weak limit of the family (p_{i} ∘ m ∘ e, p_{i} ∘ m ∘ e). Thus, C_{reg} is a full subcategory of C_{ex}.

2.3. Theorem. If C is weakly κ-complete, then C_{reg} is κ-regular, and C_{ex} is κ-exact; moreover, the inclusions from C_{reg} and C_{ex} into (C^{op}, Set) are κ-regular.

Proof. Step 1. C_{reg} has κ-products.

Let (F_{i})_{i∈I} be a family of objects of C_{reg} with #I < κ. If, in (C^{op}, Set), F_{i} is a subobject of the product ∏_{j∈I_{i}} y(C_{j}), then ∏ F_{i} is a subobject of the product of all ∏_{j∈I_{i}} y(C_{j}), with i ∈ I. If the morphisms p_{i} : y(D_{i}) → F_{i} are regular epis, also ∏ p_{i} : ∏ y(D_{i}) → ∏ F_{i} is a regular epi. Let Q be a weak product of all D_{i} in C; by 1.1, the unique arrow y(Q) → ∏ y(D_{i}) is a regular epi. Consequently, we have a regular epi from y(Q) onto ∏ F_{i}. This shows that ∏ F_{i} is in C_{reg}.

Step 2. C_{reg} has equalizers.

Let (u, v : M → N) be a pair of arrows in C_{reg}, and m : P → M be an equalizer of u and v in (C^{op}, Set). P is a subobject of M, and M is a subobject of a κ-product of representable functors, so P is a subobject of the product. To show that P can be regularly covered by a representable functor, let s : y(C) → M and t : y(D) → N be regular epis. Let m′ : P′ → y(C) be an equalizer of u ∘ s and v ∘ s. We then have a unique arrow w : P′ → P making the following diagram a pullback:

[figure: P′; P; y(C); M; m′; s; w; m]

where w is a regular. Also, P′ can be regularly covered by a representable functor. Indeed, let k : N → ∏_{i∈I} y(B_{i}) be mono, and q_{i} : ∏ y(B_{i}) → y(B_{i}) be the product projections. Then P′ is a joint equalizer of the family (q_{i} ∘ u ∘ s, q_{i} ∘ v ∘ s : y(C) → y(B_{i}))_{i∈I}. Since C and B_{i} are in C and since y is full and faithful, we can write y(u_{i}) = q_{i} ∘ u ∘ s and y(v_{i}) = q_{i} ∘v ∘s. By 1.2, P′ is regularly covered by y(W); here W is a weak joint equalizer of the family (u_{i}, v_{i} : C → B_{i})_{i∈I}.

This completes the proof that C_{reg} has κ-limits.

Step 3. C_{ex} has κ-products.

Let (F_{i})_{i∈I} be in C_{ex}, with #I < κ. There are regular epis p_{i} : y(D_{i}) → F_{i} such that the kernel pair M_{i} of p_{i}

M_{i} f_{i} ; \overrightarrow{⟶} ; g_{i} y(D_{i}) \underrightarrow{p_{i}} F_{i}

can be regularly covered by some y(B_{i}). Given the regular epis q_{i} : y(B_{i}) → M_{i} (for all i), then also ∏ q_{i} : ∏ y(B_{i}) → ∏ M_{i} is a regular epi. So ∏ M_{i} is regularly covered by y(W); here W is a weak product of the family of B_{i}. M_{i} is a subobject of the product y(D_{i})×y(D_{i}), so ∏ M_{i} is a subobject of the product of all y(D_{i})×y(D_{i}). This shows that ∏ M_{i} is in C_{reg}. Let s : y(Z) → ∏ y(D_{i}) be a regular epi; here Z is a weak product of D_{i}.

Thus, we have the regular epi ∏ p_{i} ∘ s : y(Z) → ∏ F_{i}. Forming pullbacks, we obtain the following diagram:

[figure: y(Z); G; H; ∏ y(D_{i}); ∏ M_{i}; G′; ∏ F_{i}; ∏ y(D_{i}); y(Z); b′; g; b; ∏ f_{i}; s; ∏ p_{i}; a′; f; a; ∏ g_{i}; s; ∏ p_{i}]

Note that y(Z), ∏ M_{i} and ∏ y(D_{i}) are in C_{reg}. Hence, also H is in C_{reg} and can be regularly covered by a representable functor. But H is the domain of the kernel pair of the regular epi ∏ p_{i} ∘ s. This shows that ∏ F_{i} is in C_{ex}.

Step 4. C_{ex} has pullbacks.

Case 1. First we deal with the case of a kernel pair

M f ; \overrightarrow{⟶} ; g y(D) \underrightarrow{p} G

of a regular epi p : y(C) → G with G ∈ C_{ex} such that M is regularly covered by a representable functor. Also, M is a subobject of the product y(C) × y(C), so M ∈ C_{reg}. From Remark 2.2, we have that M is in C_{ex}.

Case 2. For arrows s : y(A) → G and t : y(B) → G with G ∈ C_{ex}, we can write s = p ∘ s′ and t = p ∘ t′ for the regular epimorphism p of Case 1. We form the following diagram of pullbacks:

[figure: H; M; y(B); t′; N; G′; y(C); p; y(A); s′; y(C); p; G]

Since G′, y(A) and y(B) are in C_{reg}, so is H. Thus, H is in C_{ex}, i.e., the pullback of s and t is in C_{ex}.

Case 3. For any arrows s : F → G and t : F′ → G of C_{ex}, let p : y(A) → F and q : y(B) → F′ be regular epis. We form the following diagram of pullbacks:

[figure: H; M; y(B); q; N; G′; F′; t; y(A); p; F; s; G]

As in Case 2, H is in C_{reg}. Also, G′ is regularly covered by H as p and q are regular epis. Say that e : H → G′ is the regular epi given by the diagram, and m : G′ → F × F′ is the mono determined by the pullback of s and t. Note that F × F′ is in C_{ex}. We form the kernel pair of m ∘ e:

H′ u ; \overrightarrow{⟶} ; v H \underrightarrow{m ∘ e} F × F′

That H′ can be regularly covered by a representable functor follows from the same property for H. Also, H′ is a subobject of H × H, so H′ ∈ C_{reg}. This shows that the kernel pair of e is in C_{reg}. Using the same argument as in Step 3, we can show that G′ is in C_{ex}. This completes the proof that C_{ex} has pullbacks.

Step 5. Every kernel pair of any morphism in C_{reg} has a coequalizer.

In fact, let (u, v : M → N) be the kernel pair of f : N → G in C_{reg}. Take the coequalizer of u and v:

M u ; \overrightarrow{⟶} ; v N \underrightarrow{g} G′

Then G′ is a subobject of G. Thus, G′ is in C_{reg}.

Step 6. Every equivalence relation in C_{ex} has a coequalizer.

Let (u, v : M → N) be an equivalence relation of objects in C_{ex}, and g : N → G be the coequalizer of u and v in (C^{op}, Set); we show that G is in C_{ex}, as follows. Given the regular epi s : y(A) → N, we form the following diagram of pullbacks:

[figure: H; M′′; y(A); s; M′; M; u; N; v; g; y(A); s; N; g; G]

Since y(A), N and M are in C_{ex}, so is H. Thus, the kernel pair H of the regular epi g ∘ s is regularly covered by a representable functor. This shows that G is in C_{ex}.

Finally, that regular epis in C_{reg} and C_{ex} are stable under pullback and under κproducts follows immediately from the corresponding properties of (C^{op}, Set).

2.4. Corollary. Let k : C → C_{reg} and l : C → C_{ex} be the restricted Yoneda embeddings. If C is weakly κ-complete, then

(i) k and l are κ-flat;

(ii) for every C ∈ C, k(C) and l(C) are regularly projective in C_{reg} and C_{ex}, respectively;

(iii) for each A ∈ C_{ex}, there are C ∈ C and a regular epi l(C) → A in C_{ex}. Moreover, A is in C_{reg} iff A is a subobject of a κ-product of objects in C.

Proof. By 1.3 and 2.3.

2.5. Remark. (i) In the next section we only use the properties of 2.4 to show the universal properties of C_{reg} and C_{ex}. Consequently, these are necessary and sufficient conditions describing the free regular and exact completions of C. (ii) For any weakly κ-complete category, C^{∗} of 1.5 has products. Let

∏(C^{∗}, Set)

be the category of set-valued functors preserving products. Then ∏(C^{∗}, Set) is ∞-exact, since products commute with regular epimorphisms and limits in Set. Consider the evaluation functor

e_{C} : C → ∏(C^{∗}, Set)

It is clear that e_{C} is full and faithful. From 2.3 and (i) above, C_{ex} is equivalent to the full subcategory of ∏(C^{∗}, Set) whose objects F are covered by a regular epimorphism e_{C}(C) → F whose kernel pairs have the same property again. Likewise, we can describe C_{reg} as a full subcategory of ∏(C^{∗}, Set).

(iii) For κ an infinite regular cardinal, let

C^{∗+} = ∏ Filt_{κ}(C^{∗}, Set)

be the category of set-valued functors preserving products and κ-filtered colimits. C^{∗+} is κ-exact, as κ-filtered colimits commute with regular epimorphisms and κ-limits in Set. Therefore, C_{reg} and C_{ex} can be described as full subcategories of C^{∗+} as in (ii).

3. Universal properties of C_{reg} and C_{ex}

3.1. Proposition. Let C be a weakly κ-complete category. With k and l as in 2.4 one has:

(i) If B is κ-regular, then every κ-flat functor F : C → B has a left Kan extension F! along k, and F! preserves regular epimorphisms.

(ii) If B is κ-exact, then every κ-flat functor F : C → B has a left Kan extension F! along l, and F! preserves regular epimorphisms.

Proof. We only give the proof of part (ii), as (i) can be done in the same manner. For convenience, we assume that l is the inclusion functor.

The proof here follows the same argumentation as in 3.5 of [9]. For the existence of F!, by the dual of Theorem X.3.1. in [12], it suffices to show that the composite F ∘ P : l/C′ → C → B has a colimit in B for each C′ ∈ C_{ex}, where P is the projection ⟨C, C → C′⟩ ↦ C. Since C′ ∈ C_{ex}, we have a regular epimorphism e : A → C′ with A in C. Let

D u′ ; \overrightarrow{⟶} ; v′ A

be the kernel pair of e; so e is the coequalizer of (u′, v′), and there is a regular epimorphism d : S → D with S ∈ C. Then e is a coequalizer of the morphisms (u′ ∘ d, v′ ∘ d). Denote u′ ∘ d by u, and v′ ∘ d by v. Define a category E whose only non-trivial arrows are

e u ; \overrightarrow{⟶} ; v e ∘ u

Let i : E → l/C′ be the inclusion functor. One then has:

3.2. Lemma. i is final.

Proof. Firstly, f/i is non-empty, for any f : C → C′ with C ∈ C. Indeed, by the projectivity of C, there is a morphism w : C → A such that f = e ∘ w.

To show that f/i is connected, let m, n be any two morphisms in f/i. Then we only need to consider the following three cases.

Case 1: m, n : f → e, i.e., e ∘ m = e ∘ n = f. Since ⟨u′, v′⟩ is the kernel pair of e, there is a unique morphism k′ : C → D such that m = u′ ∘ k′ and n = v′ ∘ k′. By the projectivity of C , we obtain a morphism k : C → S with k′ = d ∘ k. Thus, m = u ∘ k and n = v ∘ k, i.e., u : k → m and v : k → n; here k : f → e ∘ u is in f/i.

Case 2: m, n : f → e ∘ u, i.e., e ∘ u′ ∘ d ∘ m = e ∘ v′ ∘ d ∘ n = f. Since ⟨u′, v′⟩ is the kernel pair of e, there is a unique morphism k′ : C → D such that u′ ∘ d ∘ m = u′ ∘ k′ and v′ ∘ d ∘ n = v′ ∘ k′. By the projectivity of C, we have a morphism k : C → S with k = d∘k′. We conclude that u∘m = u∘k and v ∘n = v ∘k. Thus, we have four morphisms u : m → u ∘ k, u : k → u ∘ k, v : n → v ∘ k, and v : k → v ∘ k joining m and n.

Case 3: m : f → e and n : f → e ∘ u. By the projectivity of C, there is a morphism m′ : f → e ∘ u such that m = u ∘ m′, because u is regular epi. Thus, we have morphisms m′, n : f → e ∘ u. That f/i is connected now follows from Case 2. This completes the proof of that f/i is connected.

We continue with the proof of 3.1. Since i is final, according to Theorem IX.3.1 in [12], to prove that F! exists, we only need to show that the pair of morphisms (F (u), F (v)) has a coequalizer in B.

Let (p, q) be the product projections of F (A)×F (A), and let ϵ : F (S) → F (A)×F (A) be the unique morphism so that F (u) = p ∘ ϵ and F (v) = q ∘ ϵ. Since B is κ–exact, ϵ has a factorization ϵ = y ∘ x with y : Q → F (A) × F (A) mono and x : F (S) → Q regular epi, for some Q ∈ B.

3.3. Lemma. y is an equivalence relation on F (A).

Proof. (i) (Reflexivity) The diagonal Δ : F (A) → F (A) × F (A) factors through y.

Let k′ : A → D be the morphism so that id_{A} = u′ ∘ k′ = v′ ∘ k′. Since S is projective, one obtains a morphism k : A → S with k′ = d ∘ k, hence id_{A} = u ∘ k = v ∘ k. It follows that

id_{F (A)} = F (u) ∘ F (k) = p ∘ y ∘ x ∘ F (k) = q ∘ y ∘ x ∘ F (k).

Consequently, Δ = y ∘ (x ∘ F (k)).

(ii)(Symmetry) There exists a morphism t : Q → Q such that p ∘ y = q ∘ y ∘ t and q ∘ y = p ∘ y ∘ t. Let (π_{1}, π_{2}) be the product projections of A × A, and let m : D → A × A be the induced morphism of (u′, v′). Since the kernel pair of a morphism always yields an equivalence relation, there exists n : D → D such that π_{1} ∘ m = π_{2} ∘ m ∘ n and π_{2} ∘ m = π_{1} ∘ m ∘ n. Since d : S → D is regular epi, by the projectivity of S, there is a morphism n′ : S → S such that n ∘ d = d ∘ n′. Thus, we have π_{1} ∘ m ∘ d ∘ n′ = π_{2} ∘ m ∘ d and π_{2} ∘ m ∘ d ∘ n′ = π_{1} ∘ m ∘ d, i.e., u ∘ n′ = v and v ∘ n′ = u. Applying F to the above equalities, we obtain F (u) ∘ F (n′) = F (v) and F (v) ∘ F (n′) = F (u). Hence

p ∘ y ∘ x ∘ F (n′) = q ∘ y ∘ x, and q ∘ y ∘ x ∘ F (n′) = p ∘ y ∘ x.

Let (f, g) be the kernel pair of x. Then,

p ∘ y ∘ x ∘ F (n′) ∘ f = q ∘ y ∘ x ∘ f = q ∘ y ∘ x ∘ g = p ∘ y ∘ x ∘ F (n′) ∘ g.

Similarly,

q ∘ y ∘ x ∘ F (n′) ∘ f = q ∘ y ∘ x ∘ F (n′) ∘ g.

Consequently, y∘x∘F (n′)∘f = y∘x∘F (n′)∘g. But y is monic, so x∘F (n′)∘f = x∘F (n′)∘g. Since (f, g) is the coequalizer of x, there is a unique morphism t : Q → Q such that t ∘ x = x ∘ F (n′). It is easily seen that t is the required morphism.

(iii) (Transitivity) For the pullback diagram of p ∘ y and q ∘ y

[figure: Q; P; F (A); Q; p ∘ y; q ∘ y; b; a]

the morphism δ = ⟨(p ∘ y) ∘ a, (q ∘ y) ∘ b⟩ : P → F (A) × F (A) factors through y.

Let (z, w : U → F (S)) be the pullback of F (u) and F (v). There is a unique morphism α : U → P such that x ∘ z = b ∘ α and x ∘ w = a ∘ α. Then, α is a regular epi. In fact, let b′ : U_{1} → F (S) and x_{1} : U_{1} → P be the pullback of x and b, and let x_{2} : U_{2} → P and a′ : U_{2} → F (S) be the pullback of a and x. Since x is regular epi, so are x_{1} and x_{2}. If x′_{2} and x′_{1} is the pullback of x_{1} and x_{2}, then b′ ∘ x′_{2} and a′ ∘ x′_{1} is the pullback of F (u) and F (v). Therefore, α = x_{1} ∘ x′_{2} = x_{2} ∘ x′_{1}. That α is regular epi follows from the fact that x_{1} and x′_{2} are regular epis.

Since F is κ-flat, there are two morphisms s, t : V → S in C with u ∘ s = v ∘ t such that F (s) = z ∘ β and f(t) = w ∘ β for some regular epi β : F (V) → U in B. Thus, α ∘ β : F (V) → P is a regular epi in B.

Note that (u′, v′) is an equivalence relation on A. Let

[figure: D; T; A; D; u′; v′; c_{1}; c_{2}]

be the pullback diagram of u′ and v′. Thus, the morphism c = ⟨u′ ∘c_{2}, v′ ∘c_{1}⟩ : T → A×A factors as c = m ∘ r′ for some r′ : T → D; here m is as in the proof for symmetry. Since u ∘ s = v ∘ t, i.e., u′ ∘ d ∘ s = v′ ∘ d ∘ t, there exists a unique morphism d′ : V → T such that d ∘ s = c_{1} ∘ d′ and d ∘ t = c_{2} ∘ d′. By the projectivity of V, we have r : V → S with d ∘ r = r′ ∘ d′. Since

u′ ∘ d ∘ t = u′ ∘ c_{2} ∘ d′ = π_{1} ∘ c ∘ d′ = π_{1} ∘ m ∘ r′ ∘ d′ = u′ ∘ d ∘ r,

one obtains u ∘ t = u ∘ r. Similarly, v ∘ s = v ∘ r. Applying F to the above equalities, one gets F (u) ∘ F (t) = F (u) ∘ F (r) and F (v) ∘ F (s) = F (v) ∘ F (r). Since F (u) = p ∘ y ∘ x and F (v) = q ∘ y ∘ x, it follows that

p ∘ y ∘ x ∘ F (r) = p ∘ y ∘ x ∘ F (t) = p ∘ y ∘ x ∘ w ∘ β = p ∘ y ∘ a ∘ α ∘ β

and q ∘ y ∘ x ∘ F (r) = q ∘ y ∘ b ∘ α ∘ β. Let (f′, g′) be the kernel pair of α ∘ β. Then

p ∘ y ∘ x ∘ F (r) ∘ f′ = p ∘ y ∘ x ∘ F (r) ∘ g′,
q ∘ y ∘ x ∘ F (r) ∘ f′ = q ∘ y ∘ x ∘ F (r) ∘ g′,

and consequently y∘x∘F (r)∘f′ = y∘x∘F (r)∘g′, which implies and x∘F (r)∘f′ = x∘F (r)∘g′ as y is monic. Since α ∘ β is the coequalizer of f′ and g′, there is a unique morphism η : P → Q such that η ∘ α ∘ β = x ∘ F (r). Thus, p ∘ y ∘ a = p ∘ y ∘ η and q ∘ y ∘ b = q ∘ y ∘ η. Consequently, δ = y ∘ η.

Finally we can complete the proof of 3.1. Since B is κ-exact, every equivalence relation is effective, so (p ∘ y, q ∘ y) has a coequalizer. Since F (u) = p ∘ y ∘ x and F (v) = q ∘ y ∘ x, it follows that (F (u), F (v)) has a coequalizer as x is regular epi. This completes the proof of the existence of F!.

From the above proof, we see that F! takes any regular epi with domain in C into a regular epi. Indeed, given a regular epi e : P → Q in C_{ex}, we take a regular epi d : C → P with C ∈ C. Since F (e) ∘ F (d) is a regular epi, so is F (e).

For κ-regular categories A and B, recall that a functor F : A → B is κ-regular if F preserves κ-limits and regular epimorphisms. We denote the category of κ-regular functors from A into B by κ-Reg(A, B). For C and B as in 2.1, κ-Flat(C, B) is the category of κ-flat functors from C into B.

3.4. Theorem. Let C be a locally small category with weak κ-limits.

(a) C_{reg} has the following universal property which characterizes it as the free κ-regular completion of C:

(i) For any κ-regular B, the functor

Σ : κ−Reg(C_{reg}, B) → κ−Flat(C, B), M ↦ M ∘ k

induced by k of 2.4 is an equivalence of categories.

(ii) The quasi-inverse of the equivalence Σ of (i) takes a κ-flat functor F : C → B to its left Kan extension F! along k.

(b) C_{ex} has the following universal property which characterizes it as the free κ-exact completion of C:

(i) For any κ-exact category B, the functor

Σ : κ−Reg(C_{ex}, B) → κ−Flat(C, B), M ↦ M ∘ l

induced by l is an equivalence of categories.

(ii) The quasi-inverse of the equivalence Σ of (i) takes a κ-flat functor F : C → B to its left Kan extension F! along l.

Proof. We give a proof of part (b); the proof of (a) proceeds similarly.

The fullness and faithfulness of Σ follow from the properties of C_{ex} described in 2.5. For details, see the proof of Proposition 5.8 in [8]. We now prove that Σ is essentially surjective on objects. Since Σ is full and faithful, by Corollary X.3.3 in [10], it suffices to show that for any κ-flat functor F : C → B, F has a left Kan extension F! of F along e_{C}, and F! is κ-regular. The existence of F! was shown in Proposition 3.1. Since F! preserves regular epimorphisms, it remains to be shown that F! preserves κ-limits, and by 1.4, we only need to show that F! is κ-flat. We proceed in several steps.

Step 1. F! is flat w.r.t. κ-products. Indeed, let (B_{i})_{i∈I} be a family of objects in C_{ex} with #I < κ, and p_{i} : C_{i} → B_{i} be regular epis with C_{i} in C. By 3.1, F!(p_{i}) is regular epi for every i ∈ I. Since regular epis are stable under κ-products in B, ∏ F!(p_{i}) : ∏ F!(C_{i}) → ∏ F!(B_{i}) is a regular epi. Let W be a weak product of C_{i} in C. The induced arrow t : F (W) → ∏ F!(C_{i}) is a regular epi as F is κ-flat. There is a canonical arrow s : F (∏ B_{i}) → ∏ F!(B_{i}), and the weak projections e_{i} : W → C_{i} of W in C, when composed with p_{i} induce an m : W → ∏ B_{i}. Since s ∘ F!(m) = ∏ F!(q_{i}) ∘ t, with ∏ F!(q_{i}) ∘ t regular epi, also s must be a regular epimorphism.

Step 2. F! is flat w.r.t. pullbacks. We proceed as in 2.3.

Case 1. Let e : A → B be a regular epi in C_{ex} with A ∈ C. Consider the kernel pair of F!(e):

Q f ; \overrightarrow{⟶} ; g F!(A) \underrightarrow{F!(e)} F!(B)

Let (u, v : D → A) be the kernel pair of e in C_{ex}, and a : S → D be a regular epi with S ∈ C. From the proof of 3.1, we can see that the unique arrow x : F!(S) → Q is a regular epi in B. Let b : F!(D) → Q be the unique arrow so that F!(u) = f ∘ b and F!(v) = g ∘ b. Then x = b ∘ F!(a), and this implies that b is a regular epi.

Case 2. Let e be the morphism of Case 1, and f : C → B be any arrow with C ∈ C. Since C is regular projective, f = e ∘ g for some g : C → A. We form the following diagram of pullbacks:

[figure: Q′; c′; F!(C); g′; F!(g); Q; c; F!(A); c; F!(e); F!(C); F!(e); F!(B)]

As before, we let D be the kernel pair of e in C_{ex}, and b : F!(D) → Q and a : S → D be regular epis in Case 1. We form a weak pullback S′ of g and u∘a in C_{ex}; here F!(u) = f ∘b. Thus, we have the following diagram of pullbacks:

[figure: Q′′; s; Q′; c′; F!(C); F!(d); g′; F!(g); F!(S); a ∘ b; Q; c; F!(A)]

where s is a regular epi as a ∘ b is a regular epi. Let S′ be a weak pullback of g and u ∘ a in C. Since F is κ-flat, the unique arrow p : F!(S′) → Q′′ is a regular epi. Thus, we have a regular epi s ∘ p : F!(S′) → Q′. Let D′ be the pullback of e and f in C_{ex}, and b′ : F!(D′) → Q′ be the unique arrow determined by the universal property of the pullback Q′. Then s ∘ p = b′ ∘ F!(s′); here s′ : S′ → D′ is the unique arrow determined by the universal property of the pullback D′. This shows that b′ is a regular epi.

Case 3. Now we consider arrows f : K → B and g : A → B of C_{ex} with K, A ∈ C. Let e : C → B be a regular epi with C ∈ C. Then f = e ∘ f′ and g = e ∘ g′. We form the following pullback diagrams

[figure: Q′; m′; F!(K); F!(g′); Q; m; F!(C); e′; F!(e); F!(A); F!(f); F!(B)]

Let (u : S → C, v : S → A) be the pullback of e and f. From the argument of Case 2, the unique arrow b : F!(S) → Q is a regular epi. Let a : D → S be a regular epi with D ∈ C. Then b ∘ F!(a) : F!(D) → Q is a regular epi. We form the consecutive pullback diagrams

[figure: Q′′; b′; Q′; m′; F!(K); F!(g′); F!(D); b ∘ F!(a); Q; m; F!(C)]

Let D′ be a weak pullback of u∘a and g′. Since F is κ-flat, the unique arrow b′ : F!(D′) → Q′′ is a regular epi. Thus, c ∘ b′ : F!(D′) → Q′ is a regular epi. Consequently, the unique arrow F!(S′) → Q′ is a regular epi; here S′ is the pullback of f and g.

Case 4. Next we consider arrows f : A → B and g : K → B with A ∈ C. Let e : C → K be a regular epi with C ∈ C. We form the pullback diagrams

[figure: Q′; e′; Q; g′; F!(A); F!(f); F!(C); F!(e); F!(K); F!(g); F!(B)]

Let S′ be the pullback of f and g ∘ e. From the Case 3, the unique arrow b : F!(S′) → Q′ is a regular epi. Thus, we have a regular epi e′ ∘ b : F!(S′) → Q as e′ is a regular epi. Consequently, the unique arrow a : F!(S) → Q is a regular epi; here S be the pullback of f and g.

Case 5. Finally, for arbitrary arrows f : A → B and g : K → B of C_{ex}, we can reduce the proof to the case just mentioned.

Step 3. F! is flat w.r.t. equalizers.

Given f, g : A → B in C_{ex}, let m : Q → F!(A) be the equalizer of F!(f), F!(g) in B. With a regular epi u : C → A in C_{ex} with C ∈ C one forms the pullback

[figure: P; n; F!(C); k; F!(u); Q; m; F!(A)]

so that n is the equalizer of F!(f ∘ u), F!(g ∘ u). The pullback

[figure: R; s; F!(C); t; F!(g ∘ u); F!(C); F!(f ∘ u); F!(B)]

gives an arrow q : P → R which is an equalizer of s, t. There is a weak pullback of f ∘ u, g ∘ u, with projections x, y : E → C in C, and the induced arrow p : F (E) → R is regular epi. For the pullback diagram

[figure: R′; s′; P; t′; q; F (E); p; R]

t′ is easily recognized as the equalizer of F (x), F (y). But since F is flat, there is a weak equalizer Z of x and y such that the induced arrow i : F (Z) → R′ is regular epi. As pullbacks of regular epis, also s′ and k are regular epi. Consequently, k ∘ s′ ∘ i : F (Z) → Q is a regular epi.

Step 4. F! preserves equalizers. Looking at the kernel pair of a monomorphism f in C_{ex}, one sees immediately that F!(f) is mono as well, since F! is flat w.r.t. pullbacks. But flatness w.r.t. equalizers and preservation of monos makes F! preserve equalizers.

Step 5. F! is κ-flat. As usual, one presents the limit of a κ-diagram in C_{ex} as an equalizer of two arrows between κ-products. A routine diagram chase shows that flatness w.r.t. κ-products and preservation of equalizers yield the desired result.

4. Simplifying the description of C_{ex}

We have seen that the objects of C_{ex} consist of functors regularly covered by representable functors which satisfy the property: the kernel pair of any regular epi (see the proof of 2.3) from a representable functor into an object of C_{ex} is regularly covered by a representable functor. The question is under which condition on C this latter condition can be omitted from the definition of the objects of C_{ex}.

First we consider a small category C and the evaluation functor

e_{C} : C → (κ−Flat(C), Set)

It was shown in [9] that κ-Flat(C) has products and κ-filtered colimits (as a κ-accessible category), and C_{ex} is equivalent to the category of functors from κ-Flat(C) into Set which preserve products and κ-filtered colimits. The new embedding e_{C} gives that if a functor of C → Set can be regularly covered by a representable functor, then it satisfies the additional property mentioned above automatically.

If C is not small, this is no longer true, for the reason that κ-Flat(C) may no longer be the κ-filtered colimit completion of C^{op}. Consequently, one may have a functor F : κ-Flat(C) → Set which preserves κ-filtered colimits and products and satisfies the solution set condition, such that F is regularly covered by a representable functor on κ-Flat(C), but this representable fails to be of the form e(C) although it preserves κfiltered colimits. In what follows we give sufficient conditions on C for overcoming these difficulties.

As a first preparatory step we consider a category A with products. Since regular epis commute with products in Set, a weakly representable functor F : A → Set preserves products. The following characterization of weakly representable functors is analgous to Theorem V.6.3 of [12].

4.1. Proposition. Let A be a locally small category with products, and let F : A → Set be a functor. The following conditions are equivalent:

(i) F is weakly representable.

(ii) F preserves products and satisfies the solution set condition on 1 ∈ Set.

Proof. If F is weakly representable, we already saw that F preserves products. To verify that F satisfies the solution set condition on 1, it is easy to show that 1/F has a weakly initial object: take a regular epimorphism x : A(A, −) → F in (A, Set); then x_{B} : A(A, B) → F (B) is surjective for any B ∈ A. This says that (A, x ∈ F (A)) is a weakly initial object of 1/F.

Conversely, let F preserve products and satisfy the solution set condition on 1 ∈ Set, so that there is a weakly initial family (A_{i}, x_{i} ∈ F (A_{i}))_{i∈I} in 1/F. Since F preserves products, we obtain a weakly initial object (∏ A_{i}, x ∈ F (∏ A_{i})) with x = ⟨x_{i}⟩_{i∈I} in 1/F. By the Yoneda Lemma, such an x corresponds to a natural transformation x : A(A, −) → F with A = ∏ A_{i}, and this implies that x_{B} : A(A, B) → F (B) is surjective for any B ∈ A. We conclude that x is a regular epimorphism in (A, Set).

Recall that an object A of a category A is said to be κ-presentable if the representable functor A(A, −) preserves κ-filtered colimits existing in A. A is κ-accessible if: (i) A has κ-filtered colimits; (ii) there is a small subcategory C of A consisting of κ-presentable objects such that every object of A is a κ-filtered colimit of a diagram of objects in C. The full subcategory of A whose objects are the κ-presentable ones is denoted by A_{κ}. A category is accessible if it is κ-accessible for some κ. A functor between accessible categories is accessible if it preserves κ-filtered colimits for some κ ([1] and [14]).

Recall that every accessible functor F : A → B satisfies the solution set condition for any B ∈ B (cf. Proposition 6.1.2 of [14])), with the validity of the converse proposition being logically equivalent to the set-theroetic Vopěnka’s Principle, see [15]. It is natural to ask whether preservation of products makes the accessibility of a functor equivalent to the solution set condition.

4.2. Proposition. Let A be an accessible category with products (called a weakly locally presentable category in the sense of [1]), and let F : A → Set be a functor preserving products. Then the following conditions are equivalent:

(1) F is weakly representable.

(2) F is accesssible.

(3) F satisfies the solution set condition for any X ∈ Set.

Proof. If F is weakly representable, take a regular epimorphism A(A, −) → F with A ∈ A_{κ} for some κ, then it is easy to see that F is κ-accessible. That (2) implies (3) follows from Proposition 6.1.2 of [14]. Proposition 4.1 shows that (3) implies (1).

4.3. Corollary. For an accessible category A with products, a product-preserving functor F : A → B into an accessible category B is accessible iff F satisfies the solution set condition for any B ∈ B.

Proof. Only the “if” part still needs to be proved. For an arbitrary accessible category B, each representable functor B(B, −) satisfies the solution set condition (since it is accessible). Hence B(B, −)∘F satisfies the solution set condition and is therefore accessible (see Proposition 4.2), for any B ∈ B. But since the representables collectively detect accessibility (see [14]), the proof is complete now.

For a κ-accessible category A with products, as we mentioned before, the category ∏ Filt_{κ}(A, Set) is κ-exact. Furthermore, for any accessible category A with products, let

∏ Acc(A, Set)

be the category of set-valued accessible functors preserving products, i.e., of weakly representable functors. Then ∏ Acc(A, Set) is ∞-exact. A typical example of this kind of category is the opposite of the category of R-modules, for any commutative ring R with unity. It will follows Theorem 4.5 below that this category is the free ∞-exact completion of the opposite of the category of injective R-modules (for details, see [8]).

4.4. Proposition. For any accessible category A with products, the restricted Yoneda embedding

y : A^{op} → ∏ Acc(A, Set)

has the following properties:

(i) y is full and faithful; (ii) y(A) is regularly projective in ∏ Acc(A, Set), for any A ∈ A; (iii) for any F ∈ ∏ Acc(A, Set), there are A ∈ A and a regular epimorphism A(A, −) → F in ∏ Acc(A, Set); (iv) y is ∞-flat.

Proof. (i) is trivial. (ii) and (iii) follows from Proposition 4.3. (iv) follows from (i), (ii) and (iii).

As proved in [1] and [9], an accessible category A has weak colimits iff it has products. Considering C = A^{op}, we therefore obtain the following theorem:

4.5. Theorem. Let C be a co-accessible category with weak limits. Then the free ∞-exact completion of C is equivalent to the category ∏ Acc(C^{op}, Set).

4.6. Corollary. Let C be a co-accessible category with weak limits, then the objects of the free ∞-exact completion of C are exactly the weakly representable functors from C^{op}.

References

[1] J.Adámek and J.Rosický, Locally Presentable and Accessible Categories (Cambridge Univ. Press, Cambridge 1994).

[2] M.Barr, P.A.Grillet and D.H.van Osdol, Exact categories, Lecture Notes in Math. 236 (Springer-Verlag, Berlin 1971).

[3] M.Barr, Representations of categories, J. Pure Appl. Algebra 41 (1986), 113-137

[4] A.Carboni and R.C.Magno, The free exact category on a left exact one, J. Austral. Math. Soc. (Series A) 33 (1982), 295-301

[5] A.Carboni and E.M.Vitale, Regular and exact Completions, J. Pure Appl. Algebra (to appear).

[6] P.Freyd, Representations in Abelian Categories, Proceedings of the conference on Categorical Algebra, La Jolla, 1965 (Springer-Verlag, Berlin 1966), pp 95-120.

[7] P.Freyd and A.Scedrov, Categories, Allegories (North-Holland, 1990).

[8] H.Hu, Dualities for accessible categories, Can. Math. Soc. Conference Proceedings 13 (AMS, Providence, 1992), pp 211-242.

[9] H.Hu, Flat functors and free exact completions, J. Austral. Math. Soc. (series A) 60(1996), 143-156

[10] H.Hu and W.Tholen, Limits in free coproduct completions, J. Pure Appl. Algebra 105 (1995), 277-291

[11] G.M. Kelly, Basic Concepts of Enriched Category Theory (Cambridge University Press, Cambridge 1982).

[12] S.Mac Lane, Categories for the Working Mathematician (Springer-Verlag, Berlin 1971).

[13] M.Makkai, A theorem on Barr-exact categories, with an infinitary generalization, Ann. Pure Appl. Logic 47(1990), 225-268.

[14] M.Makkai and R.Paré, Accessible Categories: the foundations of categorical model theory, Contemporary Math. 104 (AMS, Providence, 1990).

[15] J.Rosický and W.Tholen, Accessibility and the solution set condition, J. Pure Appl. Algebra 58 (1995), 189-208

[16] R.Street, Fibrations in bicategories, Cahiers Topologie Géom. Différentielle Catégoriques 21(1980), 111-160

Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395, U.S.A.

and

Department of Mathematics and Statistics, York University, North York, Ont., M3J 1P3, CANADA

Email: hongde@math.upenn.edu and tholen@mathstat.yorku.ca

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John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@triples.math.mcgill.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, University of Genoa: carboni@vmimat.mat.unimi.it P. T. Johnstone, University of Cambridge: ptj@pmms.cam.ac.uk G. Max Kelly, University of Sydney: kelly m@maths.su.oz.au Anders Kock, University of Aarhus: kock@mi.aau.dk F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.ruu.nl Susan Niefield, Union College: niefiels@gar.union.edu Robert Paré, Dalhousie University: pare@cs.dal.ca Andrew Pitts, University of Cambridge: ap@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@charlie.math.unc.edu Ross Street, Macquarie University: street@macadam.mpce.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Sydney: walters b@maths.su.oz.au R. J. Wood, Dalhousie University: rjwood@cs.da.ca