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

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

HONGDE HU AND WALTER THOLEN

Transmitted by Michael Barr

A^BSTRACT. 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 ^Awith nite limits isregular(cf. 2]) if every morphism factors into a regular epimorphism followed by a monomorphism, with the regular epimorphisms being stable under pullback it isexactif, in addition, equivalencerelations are e ective, that is, if every equivalence relation in ^A is a kernel pair. It was noted by Joyal that, in the de nition 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 comple- tion^Cexof a category^Cwith nite 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 nite limits (= weakly lex), and the free exact completion ^Aex=^reg

for a regular category ^A, so that ^Cex can be obtained as (^Creg)^ex=^reg for ^C with weak nite limits. The objects of ^Creg are given by nite sources (fi : X ^! Xi)i²I of arrows in ^C, and the morphisms are de ned to be equivalence classes of suitably compatible^C- morphisms between the domains of the given sources. Also for ^Aex=^reg the description of objects is simple, as^A-objects with a xed 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 di erent approach to ^Cex for ^C weakly lex and ^C small was given by Hu 9]

who generalized a result by Makkai 14]: the exact completion^Cexis given by the category

C

+ =^QFilt(^{C Set})

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

Partial nancial 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 Classi cation : 18A35, 18E10, 18G05.

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

c Hongde Hu and Walter Tholen 1996. Permission to copy for private use granted.

113

C

= Flat(^{C Set})

the category of at functors on ^C note that^C is a nitely 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 nite limits, ^Creg and ^Cex 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!(^{Cop Set})

when constructing ^Creg and ^Cex. Because of the correspondences X ^;!Yi

y(X)^;!y(Yi) y(X)^;!Y

i²Iy(Yi) y(X)^;!e A^;m!Y

i²Iy(Yi)

(with e being a regular epimorphism and m being a monomorphism) the Carboni-Vitale construction suggests to take as objects of ^Creg those F ² (^{Cop Set}) which appear at the same time as quotients of representables and as subobjects of nite products of rep- resentables.

This paper outlines Kelly's construction of^Creg and shows that ^Cex may be described conveniently within (^{Cop Set}) as well, as Kelly had anticipated in his talk: take those F ² (^{Cop 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 ^Cex constructed this way is indeed nitely complete remains a bit laborious, but we nd it convenient that most proofs reduce to checking closedness properties of ^Creg and ^Cex 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 e ort to prove all results for a category ^C with weak -limits, rather than just weak nite limits here is a regular in nite cardinal or the symbol¹. Recall that a (weak) -limit in^C is a (weak) limit of a diagram D : ^{J ! C} with #^J < , which in case = ¹ simply means that ^J must be small. Then the notion of regularity and exactness must be \- ed" as follows: ^C is -regular (-exact) if ^C is regular (exact), has -limits, and if -products of regular epimorphisms are regular epimorphisms. Note that for =^@0, the latter property comes for free:

f g = (f idD)(idAg) : AC ^!BD

is the composition of a pullback ofg with a pullback of f. Notice furthermore that -Barr- exactness in the sense of 13] and 9] implies-exactness ¹-regular is called \completely regular" in 5].

A simple description of ^Cex with ^C co-accessible is also given at the end of the pa- per. We show that for a co-accessible category ^C with weak limits, the objects of ^Cex are exactly all those F ² (^{Cop 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 be an in nite regular cardinal or the symbol¹. A category^Cis said to beweakly- completeif it has weak-limits, i.e., for any -diagram D : I ^!C (so that the morphism set of I has cardinality less than which, in case = ¹, just means that I must be small), a weak limit of D exists in ^C.

1.1. Definition. (9]) Let ^C be a weakly -complete category, and ^B a category with -limits. A functor F :^{C !B} is called -at, if for any -diagram G : I ^!C, and for each weak limit cone (fi : D ^! G(i))i²I on G, the morphism k : F(D)^! limF G with F(fi) =pi k for all i²I, is a regular epi here the morphisms pi are limit projections.

1.2. Remark. (i) For^C small, and ^B the category ^Set of sets, as pointed out in 9], F is -at i it is a - ltered colimit of representable functors.

(ii) For ^B having (regular epi, mono)-factorization, we notice that F is -at i there is a cone (fi :D ^!G(i))i²I on G with F(fi) =pik as in 1.1, so that k is a regular epi.

(iii)^@0-at functors were called left covering functors in 5].

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

Proof. Follows from 1.1 and the fact that a morphism a : M ^! N in (^{Cop Set}) is regular epi i aC :M(C)^!N(C) is surjective for each C ^2C.

1.4. Proposition. Let F :^C!B be any functor. If ^C and ^B have -limits, then F is - at i it preserves -limits.

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

1.5. Proposition. For any weakly -complete category ^C, the category ^C (= ^; Flat(^{C Set})) of set-valued - at functors has products. Moreover, for an innite reg- ular cardinal, ^C has -ltered colimits.

Proof. Let (Fi)i²I be a small family of functors of ^C. Since Fi is -at, for any - diagram G : J ^{! C}, the morphism ki : Fi(D) ^! limFi G is surjective for each i ² I.

Therefore, the morphism ^Qki : ^QFi(D) ^{! Q}limFi G is surjective. ^QlimFi G is isomorphic to lim^QFi G in ^Set, so we have that ^C is closed under products in (^C,

Set).

Suppose that is an in nite regular cardinal. Let M : I ^!C be a- ltered diagram.

Since M(i) is -at, the morphism ki : M(i)(D) ^! limM(i)G is surjective for each i ² I here G : J ^{! C} is a -diagram. For any u ² colimJlimI M(i) G, there are i ² I and ui ² limM(i)G such that u = fi(ui) with fi the colimit injection. Denote colimIlimJM(i)G by Q, and consider the following commutative diagram:

colimM(i)(D)

M(i)(D)

Q

limM(i)G k -

6

fi

6

fi(D)

ki-

where k is a morphism induced by the family ^hkiⁱi²I. Each ki is surjective, so there is x ² M(i)(D) such that ki(x) = ui we let y = fi(D)(x) and we have k(y) = u. Since - ltered colimits commute with -limits in^Set,Q is isomorphic to limJcolimIM(i)G.

Therefore,^C is closed under - ltered colimits in (^{C Set}).

2.

and

2.1. Definition. Let y : ^{C !} (^{Cop 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 dene extensions ^C;reg and ^C;ex of ^C as follows, for any weakly -complete category ^C:

(i) ^C;reg is the full subcategory of (^{Cop Set}) whose objects are these weakly repre- sentable functors which are subfunctors of -products of representable functors.

(ii) ^C;ex is the full subcategory of(^{Cop Set})of functorsF 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 xed, we shall write^Creg,^Cex for ^C;reg, ^C;ex, respectively.

2.2. ^Remark. Given B ^{2 Creg}, we have a mono m : B ^!Qi²Iy(Ci) with #I < . Let the pi's be the projections of the product ^Qy(Ci), and let (u v : A ^! B) be the kernel pair of a regular epi e : y(D) ^! B with D ^{2 C}. Then A is a limit of the family of (pime pime)i²I. By 1.3, A can be regularly covered by y(S), where S is a weak limit of the family (pi me pime). Thus, ^Creg is a full subcategory of ^Cex.

2.3. Theorem. If ^C is weakly -complete, then ^Creg is -regular, and ^Cex is ^-exact moreover, the inclusions from ^Creg and ^Cex into (^{Cop Set}) are-regular.

Proof. Step 1. ^Creg has -products.

Let (Fi)i²I be a family of objects of ^Creg with #I < . If, in (^{Cop Set}), Fi is a subobject of the product ^Qj²Iⁱy(Cj), then ^QFi is a subobject of the product of all

Qj²Iⁱy(Cj), with i ² I. If the morphisms pi : y(Di) ^! Fi are regular epis, also ^Qpi :

Qy(Di) ^{! Q}Fi is a regular epi. Let Q be a weak product of all Di in ^C by 1.1, the unique arrow y(Q)^!Qy(Di) is a regular epi. Consequently, we have a regular epi from y(Q) onto ^QFi. This shows that ^QFi is in ^Creg.

Step 2. ^Creg has equalizers.

Let (u v : M ^! N) be a pair of arrows in ^Creg, and m : P ^! M be an equalizer of u and v in (^{Cop Set}). P is a subobject of M, and M is a subobject of a -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. Letm⁰:P^{0 !}y(C) be an equalizer of us and vs. We then have a unique arrow w : P^0!P making the following diagram a pullback:

P⁰

P

y(C)

M m⁰ -

?

s

?

w

m -

wherew is a regular. Also, P⁰can be regularly covered by a representable functor. Indeed, let k : N ^{! Q}i²Iy(Bi) be mono, and qi : ^Qy(Bi) ^! y(Bi) be the product projections.

Then P⁰ is a joint equalizer of the family (qi us qivs : y(C) ^! y(Bi))i²I. Since C and Bi are in ^C and since y is full and faithful, we can write y(ui) = qius and y(vi) =qivs. By 1.2, P⁰is regularly covered byy(W) here W is a weak joint equalizer of the family (ui vi :C ^!Bi)i²I.

This completes the proof that ^Creg has -limits.

Step 3. ^Cex has -products.

Let (Fi)i²I be in ^Cex, with #I < . There are regular epis pi : y(Di) ^!Fi such that the kernel pair Mi of pi

Mi fi

;;!

;;!

gi y(Di) p^;;!i Fi

can be regularly covered by some y(Bi). Given the regular epis qi : y(Bi) ^! Mi (for all i), then also ^Qqi : ^Qy(Bi) ^{! Q}Mi is a regular epi. So ^QMi is regularly covered by y(W) here W is a weak product of the family of Bi. Mi is a subobject of the product y(Di)y(Di), so^QMi is a subobject of the product of ally(Di)y(Di). This shows that

QMi is in^Creg. Let s : y(Z)^!Qy(Di) be a regular epi hereZ is a weak product of Di.

Thus, we have the regular epi ^Qpis : y(Z)^{! Q}Fi. Forming pullbacks, we obtain the following diagram:

y(Z) G H

Qy(Di)

QMi

G⁰

QFi

Qy(Di) y(Z)

b⁰ - -

g

b - ^Qfi^-

s - ^Qpi^-

?

a⁰

?

f

?

a

? Qgi

?

s

? Qpi

Note that y(Z), ^QMi and ^Qy(Di) are in ^Creg. Hence, also H is in ^Creg and can be regularly covered by a representable functor. But H is the domain of the kernel pair of the regular epi ^Qpis. This shows that^QFi is in ^Cex.

Step 4. ^Cex has pullbacks.

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

M f

;;!

;;!

g y(D) p^;;!G

of a regular epi p : y(C) ^! G with G ^{2 Cex} such that M is regularly covered by a representable functor. Also, M is a subobject of the product y(C)y(C), so M ^2Creg. From Remark 2.2, we have that M is in ^Cex.

Case 2. For arrows s : y(A) ^! G and t : y(B) ^! G with G ^{2 Cex}, we can write s = ps⁰ and t = pt⁰ for the regular epimorphismp of Case 1. We form the following diagram of pullbacks:

H ^- M ^-y(B)

? ? ?

t⁰

N ^- G^{0 -}y(C)

? ? ?

p

y(A) s^{0 -}y(C) p ^- G

Since G⁰, y(A) and y(B) are in ^Creg, so is H. Thus, H is in ^Cex, i.e., the pullback of s and t is in ^Cex.

Case 3. For any arrows s : F ^! G and t : F^{0 !} G of ^Cex, let p : y(A) ^! F and q : y(B)^!F⁰ be regular epis. We form the following diagram of pullbacks:

H ^- M ^-y(B)

? ? ?

q

N ^- G^{0 -} F⁰

? ? ?

t

y(A) p ^- F s ^- G

As in Case 2,H is in ^Creg. Also, G⁰is regularly covered byH as p and q are regular epis.

Say that e : H ^!G⁰is the regular epi given by the diagram, and m : G^0!FF⁰ is the mono determined by the pullback of s and t. Note that F F⁰ is in ^Cex. We form the kernel pair of me:

H⁰ u

;;!

;;!

v H m;;;;;e!FF⁰

That H⁰ can be regularly covered by a representable functor follows from the same prop- erty for H. Also, H⁰ is a subobject of H H, so H^{0 2Creg}. This shows that the kernel pair ofe is in ^Creg. Using the same argument as in Step 3, we can show that G⁰ is in^Cex. This completes the proof that ^Cex has pullbacks.

Step 5. Every kernel pair of any morphism in ^Creg has a coequalizer.

In fact, let (u v : M ^! N) be the kernel pair of f : N ^! G in ^Creg. Take the coequalizer of u and v:

M u

;;!

;;!

v N g^;;!G⁰ Then G⁰ is a subobject of G. Thus, G⁰ is in ^Creg.

Step 6. Every equivalence relation in ^Cex has a coequalizer.

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

H ^- M^{00 -}y(A)

? ? ?

s

M^{0 -} M u ^- N

? ?

v

?

g

y(A) s ^- N g ^- G

Sincey(A), N and M are in ^Cex, so isH. Thus, the kernel pair H of the regular epi gs is regularly covered by a representable functor. This shows thatG is in ^Cex.

Finally, that regular epis in ^Creg and ^Cex are stable under pullback and under - products follows immediately from the corresponding properties of (^{Cop Set}).

2.4. Corollary. Let k : ^{C ! Creg} and l : ^{C ! Cex} be the restricted Yoneda embed- dings. If ^C is weakly -complete, then

(i) k and l are- at

(ii) for every C ^{2 C,} k(C) ^and l(C) are regularly projective in ^Creg and ^Cex, respec- tively

(iii) for each A ^2Cex, there areC ^2C and a regular epi l(C)^!A in^Cex. Moreover, A is in ^Creg i A is a subobject of a -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 ^Creg and ^Cex. Consequently, these are necessary and sucient conditions describing the free regular and exact completions of^C.

(ii) For any weakly-complete category, ^C of 1.5 has products. Let

Q(^{C Set})

be the category of set-valued functors preserving products. Then ^Q(^{C Set}) is ¹-exact, since products commute with regular epimorphisms and limits in ^Set. Consider the evaluation functor

e^C:^{C !Q}(^{C Set})

It is clear that e^C is full and faithful. From 2.3 and (i) above, ^Cex is equivalent to the full subcategory of ^Q(^{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 ^Q(^{C Set}).

(iii) For an in nite regular cardinal, let

C

+ =^QFilt(^{C Set})

be the category of set-valued functors preserving products and - ltered colimits. ^C+ is -exact, as - ltered colimits commute with regular epimorphisms and -limits in ^Set. Therefore,^Creg and ^Cex can be described as full subcategories of ^C+ as in (ii).

3. Universal properties of

and

3.1. Proposition. Let ^C be a weakly -complete category. With k and l as in 2.4 one has:(i) If ^B is-regular, then every - at functor F :^C!B has a left Kan extension F!

along k^{, and} F! preserves regular epimorphisms.

(ii) If ^B is -exact, then every - at 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 thatl 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 suces to show that the composite F P : l=C^{0 !C !B} has a colimit in ^B for each C^{02 Cex}, where P is the projection

hC C ^! C^{0i 7!}C. Since C^02Cex, we have a regular epimorphisme : A^!C⁰ withA in

C. Let

D u⁰

;;!

;;!

v⁰ A

be the kernel pair ofe so e is the coequalizer of (u⁰ v⁰), and there is a regular epimorphism d : S ^!D with S ^2C. Then e is a coequalizer of the morphisms (u⁰d v⁰d). Denote u⁰d by u, and v⁰d by v. De ne a category^E whose only non-trivial arrows are

e u

;;!

;;!

v eu

Leti :^{E !}l=C⁰ be the inclusion functor. One then has:

3.2. Lemma. i is nal.

Proof. Firstly, f=i is non-empty, for any f : C ^! C⁰ with C ^{2 C}. Indeed, by the projectivity of C, there is a morphism w : C ^!A such that f = ew.

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., em = en = f. Since ^hu⁰ v⁰ⁱ is the kernel pair of e, there is a unique morphism k⁰ : C ^! D such that m = u⁰k⁰ and n = v⁰k⁰. By the

projectivity ofC , we obtain a morphism k : C ^!S with k⁰=dk. Thus, m = uk and n = vk, i.e., u : k ^!m and v : k ^!n here k : f ^!eu is in f=i.

Case 2: m n : f ^! eu, i.e., eu⁰dm = ev⁰dn = f. Since ^hu⁰ v⁰ⁱ is the kernel pair of e, there is a unique morphism k⁰ : C ^! D such that u⁰dm = u⁰k⁰ and v⁰dn = v⁰k⁰. By the projectivity of C, we have a morphism k : C ^! S with k = dk⁰. We conclude thatum = uk and vn = vk. Thus, we have four morphisms u : m^!uk, u : k ^!uk, v : n^!vk, and v : k ^!vk joining m and n.

Case 3: m : f ^!e and n : f ^! eu. By the projectivity of C, there is a morphism m⁰:f ^!eu such that m = um⁰, because u is regular epi. Thus, we have morphisms m⁰ n : f ^! eu. 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. Sincei is nal, 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 : F(S)^!F(A)F(A) be the unique morphism so that F(u) = p and F(v) = q. Since ^B is{exact, has a factorization = yx with y : Q^!F(A)F(A) mono and x : F(S)^!Q regular epi, for someQ^2B.

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

Proof. (i) (Reexivity) The diagonal :F(A)^!F(A)F(A) factors through y.

Let k⁰:A^!D be the morphism so that idA =u⁰k⁰=v⁰k⁰. Since S is projective, one obtains a morphismk : A^!S with k⁰=dk, hence idA =uk = v k. It follows that

idF⁽A⁾ =F(u)F(k) = pyxF(k) = qyxF(k):

Consequently, = y(xF(k)):

(ii)(Symmetry) There exists a morphism t : Q ^! Q such that py = q yt and qy = pyt. Let (^{1 2}) be the product projections of AA, and let m : D ^!AA be the induced morphism of (u⁰ v⁰). Since the kernel pair of a morphism always yields an equivalence relation, there exists n : D ^! D such that ¹ m = ² m n and ²m = ¹ mn. Since d : S ^!D is regular epi, by the projectivity of S, there is a morphismn⁰:S^!S such that nd = dn⁰. Thus, we have¹mdn⁰ =²md and ²mdn⁰=¹md, i.e., un⁰ =v and vn⁰ =u. Applying F to the above equalities, we obtain F(u)F(n⁰) =F(v) and F(v)F(n⁰) =F(u). Hence

pyxF(n⁰) =qyx and qyxF(n⁰) =pyx:

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

pyxF(n⁰)f = qyxf = qyxg = pyxF(n⁰)g:

Similarly,

qyxF(n⁰)f = qyxF(n⁰)g:

Consequently,yxF(n⁰)f = yxF(n⁰)g. But y is monic, so xF(n⁰)f = xF(n⁰)g.

Since (f g) is the coequalizer of x, there is a unique morphism t : Q ^! Q such that tx = xF(n⁰). It is easily seen that t is the required morphism.

(iii) (Transitivity) For the pullback diagram of py and qy Q

P

F(A)

Q

-

py

6

qy

6

b

a -

the morphism =^h(py)a (qy)bⁱ: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 : U ^! P such that xz = b and xw = a. Then, is a regular epi. In fact, let b⁰ :U^{1 !}F(S) and x¹ :U^{1 !}P be the pullback of x and b, and let x² :U^{2 !} P and a⁰:U^{2 !}F(S) be the pullback of a and x. Since x is regular epi, so are x¹ and x². If x⁰² and x⁰¹ is the pullback of x¹ and x², then b⁰x⁰² and a⁰x⁰¹ is the pullback of F(u) and F(v). Therefore, = x¹x⁰² =x²x⁰¹. That is regular epi follows from the fact that x¹ and x⁰² are regular epis.

Since F is -at, there are two morphisms s t : V ^! S in ^C with us = vt such that F(s) = z and f(t) = w for some regular epi : F(V ) ^! U in ^B. Thus, : F(V )^! P is a regular epi in ^B.

Note that (u⁰ v⁰) is an equivalence relation on A. Let D

T

A

D u⁰ -

6

v⁰

6

c¹

c² -

be the pullback diagram ofu⁰andv⁰. Thus, the morphismc =^hu⁰c² v⁰c¹ⁱ:T ^!AA factors as c = mr⁰ for somer⁰:T ^!D here m is as in the proof for symmetry. Since us = vt, i.e., u⁰ds = v⁰dt, there exists a unique morphism d⁰ :V ^! T such that ds = c¹ d⁰ and dt = c²d⁰. By the projectivity of V , we have r : V ^! S with dr = r⁰d⁰. Since

u⁰dt = u⁰c²d⁰=¹cd⁰=¹mr⁰d⁰=u⁰dr

one obtains ut = ur. Similarly, vs = vr. 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) = pyx and F(v) = qyx, it follows that

pyxF(r) = pyxF(t) = pyxw = pya and qyxF(r) = qyb . Let (f⁰ g⁰) be the kernel pair of . Then

pyxF(r)f⁰=pyxF(r)g⁰ qyxF(r)f⁰=qyxF(r)g⁰

and consequentlyyxF(r)f⁰=yxF(r)g⁰, which impliesandxF(r)f⁰=xF(r)g⁰ as y is monic. Since is the coequalizer of f⁰ and g⁰, there is a unique morphism : P ^!Q such that = xF(r). Thus, pya = py and qyb = qy. Consequently, = y.

Finally we can complete the proof of 3.1. Since^Bis-exact, every equivalence relation is e ective, so (py qy) has a coequalizer. Since F(u) = pyx and F(v) = qyx, 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 epie : P ^!Q in^Cex, we take a regular epid : C ^!P with C ^2C. Since F(e)F(d) is a regular epi, so is F(e).

For -regular categories ^A and ^B, recall that a functor F :^{A !B} is -^regular if F preserves-limitsand regular epimorphisms. We denote the category of -regular functors from ^A into ^B by-Reg(^{A B}). For ^C and ^B as in 2.1, -Flat(^{C B}) is the category of -at functors from ^C into^B.

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

(a) ^Creg has the following universal property which characterizes it as the free-regular completion of ^C:

(i) For any ^{-regular B}, the functor

:^;Reg(^{Creg B})^!;Flat(^{C B}) M ^7!M k induced by k of 2.4 is an equivalence of categories.

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

(b) ^Cex has the following universal property which characterizes it as the free ^-exact completion of ^C:

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

:^;Reg(^{Cex B})^!;Flat(^{C B}) M ^7!M l induced by l is an equivalence of categories.

(ii) The quasi-inverse of the equivalence of (i) takes a - at 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 follow from the properties of ^Cex described in 2.5.

For details, see the proof of Proposition 5.8 in 8]. We now prove that is essentially surjective on objects. Since is full and faithful, by Corollary X.3.3 in 10], it suces to show that for any-at functor F :^{C !B},F has a left Kan extension F! of F along e^C, and F! is -regular. The existence of F! was shown in Proposition 3.1. Since F! preserves regular epimorphisms, it remains to be shown that F! preserves -limits, and by 1.4, we only need to show that F! is -at. We proceed in several steps.

Step 1. F! is at w.r.t. -products. Indeed, let (Bi)i²I be a family of objects in

C

ex with #I < , and pi : Ci ^! Bi be regular epis with Ci in ^C. By 3.1, F!(pi) is regular epi for every i ² I. Since regular epis are stable under -products in ^B,

QF!(pi) : ^QF!(Ci) ^{! Q}F!(Bi) is a regular epi. Let W be a weak product of Ci in

C. The induced arrow t : F(W) ^{! Q}F!(Ci) is a regular epi as F is -at. There is a canonical arrow s : F(^QBi)^!QF!(Bi), and the weak projections ei :W ^!Ci of W in

C, when composed with pi induce an m : W ^!QBi. SincesF!(m) = ^QF!(qi)t, with

QF!(qi)t regular epi, also s must be a regular epimorphism.

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

Case 1. Let e : A ^!B be a regular epi in ^Cex with A^2C. Consider the kernel pair of F!(e):

Q f

;;!

;;!

g F!(A) F!(e)^;;;;;!F!(B)

Let (u v : D ^!A) be the kernel pair of e in ^Cex, and a : S ^! D be a regular epi with S ^{2 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) = gb. Then x = bF!(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 ^2C. Since C is regular projective, f = eg for some g : C ^! A. We form the following diagram of pullbacks:

Q⁰ c^{0 -}F!(C) g⁰

? ?

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 ^Cex, and b : F!(D)^!Q and a : S ^!D be regular epis in Case 1. We form a weak pullbackS⁰ofg and ua in^Cex hereF!(u) = fb.

Thus, we have the following diagram of pullbacks:

Q⁰⁰ s ^- Q⁰ c^{0 -}F!(C) F!(d)

? ?

g⁰

?

F!(g) F!(S) ab^- Q c ^-F!(A)

where s is a regular epi as ab is a regular epi. Let S⁰ be a weak pullback of g and ua in ^C. Since F is -at, the unique arrow p : F!(S⁰) ^! Q⁰⁰ is a regular epi. Thus, we have a regular epi sp : F!(S⁰) ^! Q⁰. Let D⁰ be the pullback of e and f in ^Cex, and b⁰ : F!(D⁰) ^! Q⁰ be the unique arrow determined by the universal property of the pullbackQ⁰. Thensp = b⁰F!(s⁰) heres⁰ :S^0!D⁰ is the unique arrow determined by the universal property of the pullback D⁰. This shows that b⁰ is a regular epi.

Case 3. Now we consider arrows f : K ^!B and g : A ^! B of ^Cex with K A ^{2 C}. Lete : C ^!B be a regular epi with C ^2C. Then f = ef⁰ and g = eg⁰. We form the following pullback diagrams

Q⁰ m^{0 -}F!(K)

? ?

F!(g⁰)

Q m ^-F!(C)

e⁰

? ?

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 ^2C. Then bF!(a) : F!(D)^!Q is a regular epi. We form the consecutive pullback diagrams

Q⁰⁰ b^{0 -} Q⁰ m^{0 -}F!(K)

? ? ?

F!(g⁰) F!(D) bF!(a)^- Q m ^-F!(C)

LetD⁰be a weak pullback ofua and g⁰. SinceF is -at, the unique arrow b⁰:F!(D⁰)^! Q⁰⁰ is a regular epi. Thus, cb⁰ :F!(D⁰)^!Q⁰ is a regular epi. Consequently, the unique arrow F!(S⁰)^!Q⁰is a regular epi here S⁰ is the pullback of f and g.

Case 4. Next we consider arrows f : A ^! B and g : K ^! B with A ^{2 C}. Let e : C ^!K be a regular epi with C ^2C. We form the pullback diagrams

Q⁰ e^{0 -} Q g^{0 -}F!(A)

? ? ?

F!(f) F!(C) F!(e)^-F!(K) F!(g)^-F!(B)

LetS⁰ be the pullback off and ge. From the Case 3, the unique arrow b : F!(S⁰)^!Q⁰ is a regular epi. Thus, we have a regular epi e⁰b : F!(S⁰) ^! Q as e⁰ 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^Cex, we can reduce the proof to the case just mentioned.

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

Given f g : A ^!B in ^Cex, letm : Q^!F!(A) be the equalizer of F!(f), F!(g) in ^B. With a regular epi u : C ^!A in ^Cex with C ^2C one forms the pullback

P n ^-F!(C)

k

? ?

F!(u)

Q m ^-F!(A)

so that n is the equalizer of F!(fu), F!(gu). The pullback

R s ^-F!(C)

t

? ?

F!(gu) 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, gu, with projections x y : E ^!C in^C, and the induced arrow p : F(E)^!R is regular epi. For the pullback diagram

R⁰ s^{0 -} P t⁰

? ?

q

F(E) p ^- R

t⁰ is easily recognized as the equalizer of F(x), F(y). But since F is at, there is a weak equalizer Z of x and y such that the induced arrow i : F(Z) ^! R⁰ is regular epi. As pullbacks of regular epis, alsos⁰andk are regular epi. Consequently, ks⁰i : 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 at w.r.t. pullbacks.

But atness w.r.t. equalizers and preservation of monos makesF! preserve equalizers.