Homology, Homotopy and Applications, vol.3(3), 2001, pp.453–466

EXACT COMPLETION AND REPRESENTATIONS IN ABELIAN CATEGORIES

J. ROSICKÝ and E.M. VITALE

(communicated by Cristina Pedicchio)

Abstract When the exact completion of a category with weak finite limits is a Mal’cev category, it is possible to combine the universal property of the exact completion and the universal property of the coequalizer completion. We use this fact to explain Freyd’s representation theorems in abelian and Frobenius categories.

Introduction

In the first of his contributions to the La Jolla proceedings [7], P. Freyd shows that a small abelian category with enough projectives is the reflection of any of its projective covers (which are preadditive categories) in the 2-category of small abelian categories. Moreover, Freyd characterizes those small preadditive categories which occur as projective covers of small abelian categories.

There is a strong analogy between Freyd’s results and the properties of the exact completion established in [6]. The aim of this note is to make precise this analogy : we show that Freyd’s results follow using at the same time the exact completion of a category with weak finite limits and the reflexive coequalizer completion of a category [13, 3]. It is remarkable that the basic ingredient to combine the two completions is that abelian categories are Mal’cev categories : if the exact completion is a Mal’cev category, then it coincides with the reflexive coequalizer completion.

To complete our analysis, in the last section we consider the fact, proved in [8], that an additive functor from a projective-injective cover of a Frobenius category admits three different extensions to the whole category. This leads us to compare the exact completion with a third kind of completion, the pre-regular completion. We establish a necessary and sufficient condition on the base category for its exact and pre-regular completions to be equivalent. In the additive case, this condition is equivalent to be a projective-injective cover of a Frobenius category, and the universal property of a Frobenius category follows from the equivalence of the three quoted completions. Another corollary of our analysis is that the pre-regular completion of a triangulated category 𝕋 is the universal homological functor on 𝕋. This

First Author supported by the Grant Agency of the Czech Republic under the grant No. 201/99/0310. Received November 30, 2000; published on September 27, 2001. 2000 Mathematics Subject Classification: 18A35, 18B15, 18E10, 18G05. Key words and phrases: completions, Mal’cev categories, abelian categories, Frobenius categories. \begin{matrix}c \\ ◯ \end{matrix} 2001, J. Rosický and E.M. Vitale. Permission to copy for private use granted.

fact, first proved in [8] when 𝕋 is the stable homotopy category of spaces, has been generalized to arbitrary triangulated categories by M. Grandis [9] and A. Neeman [11].

A warning : the composition of two arrows \xymatrix{X \ar[r]^{f} & Y \ar[r]^{g} & Z} is written f · g . Aknowledgment : We warmly thank the editor and the referee for their help about

the correct formulation and proof of Lemma 8.

1. Review on free completions

In this section we recall construction and universal property of the various completions we need.

Recall that a regular category is a category with finite limits and (regular epi, mono) factorizations, in which regular epis are pullback stable. A category is exact if it is regular and moreover equivalence relations are effective (i.e., they have a coequalizer and they are the kernel pairs of their coequalizer) [1, 2]. A functor between regular or exact categories is exact if it preserves finite limits and regular epis. Recall also that a weak limit is defined as a limit, but deleting the uniqueness of the factorization. A functor F: ℂ → 𝔹, with ℂ having weak finite limits and 𝔹 exact, is left covering if for each finite diagram D in ℂ and for each weak limit W of D, the comparison between F (W) and the limit of F (D) is a regular epi.

In [6] the following results have been proved.

Proposition 1. Let ℂ be a category with weak finite limits. Then

1) there exists an exact category ℂ_{ex} and a full and faithful functor

Γ: ℂ → ℂ_{ex}

(the “exact completion” of ℂ) such that, for each exact category 𝔹, the composition with Γ induces an equivalence between the category of exact functors from ℂ_{ex} to 𝔹 and the category of left covering functors from ℂ to 𝔹

Γ · − : Ex[ℂ_{ex}, 𝔹] → Lco[ℂ, 𝔹] ;

2) Γ(ℂ) is a projective cover of ℂ_{ex}, i.e. each object of Γ(ℂ) is regular projective, and each object of ℂ_{ex} is a regular quotient of an object of Γ(ℂ);

3) if 𝔸 is an exact category and ℙ is a projective cover of 𝔸 then ℙ has weak finite limits, the full inclusion ℙ → 𝔸 is a left covering functor and its exact extension ℙ_{ex} → 𝔸 is an equivalence.

The exact category ℂ_{ex} can be described in the following way :

• an object of ℂ_{ex} is a pseudo-equivalence \xymatrix{X_{1} \ar[r]^{x_{0}} \ar[r]_{x_{1}} & X_{0}} in ℂ (that is, as an

equivalence relation, but without requiring that the pair is jointly monic); • a premorphism in ℂ_{ex} is a pair (f, \overline{f}) of arrows in ℂ as in the following diagram

\xymatrix{X_{1} \ar[r]^{\overline{f}} \ar[d]_{x_{0}} \ar[d]^{x_{1}} & Y_{1} \ar[d]_{y_{0}} \ar[d]^{y_{1}} \\ X_{0} \ar[r]_{f} & Y_{0}}

such that x_{0} · f = \overline{f} · y_{0} and x_{1} · f = \overline{f} · y_{1} ;

• a morphism in ℂ_{ex} is an equivalence class [f, \overline{f}] of premorphisms : we identify two pairs (f, \overline{f}) and (g, \overline{g}) of premorphisms if there exists l: X_{0} → Y_{1} such that l · y_{0} = f and l · y_{1} = g;

• composition and identities in ℂ_{ex}, and the full embedding Γ, are the obvious ones.

Let us now recall from [13, 3] the reflexive coequalizer completion of a category (a reflexive coequalizer is the coequalizer of a reflexive graph).

Proposition 2. Let ℂ be any category. Then

1) there exists a category ℂ_{rc} with reflexive coequalizers and a full and faithful

functor

Λ: ℂ → ℂ_{rc}

such that, for each category 𝔹 with reflexive coequalizers, the composition with Λ induces an equivalence between the category of reflexive coequalizer preserving functors from ℂ_{rc} to 𝔹 and the category of functors from ℂ to 𝔹

Λ · − : RC[ℂ_{rc}, 𝔹] → [ℂ, 𝔹] ;

2) if moreover ℂ has finite sums, then ℂ_{rc} has finite colimits and, for each category 𝔹 with finite colimits, the composition with Λ induces an equivalence between the category of right exact (= finite colimits preserving) functors from ℂ_{rc} to 𝔹 and the category of finite sums preserving functors from ℂ to 𝔹

Λ · − : Right[ℂ_{rc}, 𝔹] → F S[ℂ, 𝔹] .

The category ℂ_{rc} can be described as follows :

• an object in ℂ_{rc} is a reflexive graph \xymatrix{X_{1} \ar[r]^{x_{0}} \ar[r]_{x_{1}} & X_{0}} in ℂ

• consider the following diagram in ℂ, with \xymatrix{Y_{1} \ar[r]^{y_{0}} \ar[r]_{y_{1}} & Y_{0}} a reflexive graph

\xymatrix{ & Y_{1} \ar[d]_{y_{0}} \ar[d]^{y_{1}} \\ V \ar[r]_{g} \ar[r]^{f} & Y_{0}}

we say that f is preequivalent to g if there exists h: V → Y_{1} such that h·y_{0} = f and h · y_{1} = g; this is a reflexive relation in the hom-set ℂ(V, Y_{0}). We write f ≃ g if f and g are in the equivalence relation generated by the above reflexive relation ;

• a premorphism in ℂ_{rc} is an arrow f of ℂ as in the diagram

\xymatrix{X_{1} \ar[d]_{x_{0}} \ar[d]^{x_{1}} & Y_{1} \ar[d]_{y_{0}} \ar[d]^{y_{1}} \\ X_{0} \ar[r]_{f} & Y_{0}}

such that x_{0} · f ≃ x_{1} · f ;

• a morphism in ℂ_{rc} is an equivalence class [f] of premorphisms : we identify two premorphisms f and g if f ≃ g;

• composition and identities in ℂ_{rc}, and the full embedding Λ, are the obvious ones.

Recall that a factorization system (E, M) on a category 𝔹 (see [2] for the notion of factorization system) is proper if each E-arrow is an epi and each M-arrow is a mono ; it is stable if E is stable under pullbacks. A pre-regular category is a pair 𝔹 = (𝔹, (E, M)) with 𝔹 a category and (E, M) a proper and stable factorization system on 𝔹. Observe that any regular category is pre-regular in a canonical way : take as E regular epis and as M monos. The pre-regular completion of a category has been introduced, in the special case of the stable homotopy category, by Freyd in [8]. It has been studied in all its generality in [9] (see also [10]). We recall from [9] the following facts.

Proposition 3. Let ℂ be a category. Then

1) there exists a pre-regular category ℂ_{preg} and a full and faithful functor

Ψ: ℂ → ℂ_{preg}

such that, for each pre-regular category 𝔹, the composition with Ψ induces an equivalence between the category of functors from ℂ_{preg} to 𝔹 preserving the factorization system and the category of functors from ℂ to 𝔹

Ψ · − : fs[ℂ_{preg}, 𝔹] → [ℂ, 𝔹] ;

2) for each object C of ℂ, Ψ(C) is E-projective and M-injective in ℂ_{preg}. Moreover, for each object X of ℂ_{preg}, there are objects X_{0}, X′ in ℂ and arrows x_{0} : Ψ(X_{0}) → X , x′: X → Ψ(X′) with x_{0} in E and x′ in M. As a consequence, in ℂ_{preg} E-projectives coincide with M-injectives and they are exactly the retracts of the objects Ψ(C), for C ∈ ℂ.

The pre-regular category ℂ_{preg} can be described in the following way :

• an object of ℂ_{preg} is a morphism x: X_{0} → X′ in ℂ;

• a premorphism in ℂ_{preg} is a pair (f_{0}, f′) of arrows in ℂ making commutative the following diagram

\xymatrix{X_{0} \ar[r]^{f_{0}} \ar[d]_{x} & Y_{0} \ar[d]^{y} \\ X′ \ar[r]_{f′} & Y′}

• a morphism in ℂ_{preg} is an equivalence class [f_{0}, f′] of premorphisms : we identify two pairs (f_{0}, f′) and (g_{0}, g′) of premorphisms if x·f′ = x·g′ (equivalently, if f_{0} · y = g_{0} · y) ;

• composition and identities in ℂ_{preg}, and the full embedding Ψ, are the obvious ones.

2. Representations in abelian categories

Since many of the interesting examples are given by locally small, but not small, categories, in this section we give elementary proofs, i.e. without using embedding in presheaf categories; all the facts quoted in the first section can be proved in this way. This allows us to remove size conditions from Freyd’s results.

The second of the Freyd’s results quoted in the introduction immediately follows from Proposition 1. (For the notions of preadditive, additive and abelian category we follow [2]. In particular, we will use that a category is abelian iff it is exact and (pre)additive.)

Proposition 4. [7] A category ℂ is equivalent to a projective cover of an abelian category if and only if it is preadditive and it has weak finite products and weak kernels.

Proof: If ℂ is a projective cover of an abelian category 𝔸, then ℂ is preadditive because 𝔸 is preadditive. Moreover, given a finite diagram in ℂ, any ℂ-cover of the corresponding limit in 𝔸 is a weak limit in ℂ. Conversely, if ℂ has weak kernels, then it has weak equalizers and then, if it has also weak finite products, it has all weak finite limits [6]. Now consider its exact comletion ℂ_{ex} : since ℂ is preadditive, also ℂ_{ex} is preadditive (define [f, \overline{f}]+[g, \overline{g}] = [f + g, \overline{f} + \overline{g}]) and then it is abelian. By point 2 of Proposition 1, the proof is complete. □

To go further in our analysis of Freyd’s results, we now compare the exact completion and the reflexive coequalizer completion. The next lemma generalizes a result in [12] by deleting assumptions on sums.

Lemma 5. Let ℂ be a category with weak finite limits. Then ℂ_{ex} is equivalent to a full subcategory of ℂ_{rc}.

Proof: Consider the functor L: ℂ_{ex} → ℂ_{rc}

\xymatrix{X_{1} \ar[r]^{\overline{f}} \ar[d]_{x_{0}} \ar[d]^{x_{1}} & Y_{1} \ar[d]_{y_{0}} \ar[d]^{y_{1}} \\ X_{0} \ar[r]_{f} & Y_{0}} ↦ \xymatrix{X_{1} \ar[d]_{x_{0}} \ar[d]^{x_{1}} & Y_{1} \ar[d]_{y_{0}} \ar[d]^{y_{1}} \\ X_{0} \ar[r]_{f} & Y_{0}}

It is well-defined, in fact the arrow \overline{f} gives that f is a premorphism in ℂ_{rc}; moreover, if [f, \overline{f}] = [g, \overline{g}] in ℂ_{ex}, then the arrow l: X_{0} → Y_{1} gives that f and g are preequivalent, and then [f] = [g] in ℂ_{rc}.

Now observe that if \xymatrix{Y_{1} \ar[r]^{y_{0}} \ar[r]_{y_{1}} & Y_{0}} is a pseudo-equivalence in ℂ, then for each object

V the relation to be preequivalent in ℂ(V, Y_{0}) is already an equivalence relation, so that we can avoid the passage to the generated equivalence relation. This implies that [f, \overline{f}] = [g, \overline{g}] in ℂ_{ex} if and only if [f] = [g] in ℂ_{rc}, so that the functor L is faithful. It is also full because, given a premorphism in ℂ_{rc}

\xymatrix{X_{1} \ar[d]_{x_{0}} \ar[d]^{x_{1}} & Y_{1} \ar[d]_{y_{0}} \ar[d]^{y_{1}} \\ X_{0} \ar[r]_{f} & Y_{0}}

with (y_{0}, y_{1}) a pseudo-equivalence, x_{0} ·f ≃ x_{1} ·f means that they are preequivalent, so that there exists \overline{f}: X_{1} → Y_{1} such that (f, \overline{f}) is a premorphism in ℂ_{ex}. □

Recall that a category with finite limits is a Mal’cev category if any reflexive relation is an equivalence relation [4, 5]. Now we a little bit adapt this definition to the case of categories with weak finite limits, where it is more appropriate to work with graphs instead of relations.

Definition 6. Let ℂ be a category with weak finite limits. We say that ℂ is a G-Mal’cev category (G stays for graph) if any reflexive graph can be equipped with a structure of pseudo-equivalence.

Remark : Observe that, for a category ℂ with weak finite limits, to be G-Mal’cev is equivalent to the fact that any representable functor ℂ(C, −) sends reflexive graphs to pseudo-equivalences ; this is also equivalent to the fact that for each object C and for each reflexive graph \xymatrix{X_{1} \ar[r] \ar[r] & X_{0}} , the jointly monic part of its (epi, jointly mono) factorization in SET is an equivalence relation

\xymatrix{ℂ(C, X_{1}) \ar[rr] \ar[rr] \ar@{->>}[dr] &  & ℂ(C, X_{0}) \\  & I \ar[ur] \ar[ur]}

Observe also that any G-Mal’cev category having finite limits is Mal’cev.

Lemma 7. Let ℂ be a G-Mal’cev category with weak finite limits. Then ℂ_{ex} and ℂ_{rc} are equivalent.

Proof: If ℂ is G-Mal’cev, then the functor L: ℂ_{ex} → ℂ_{rc} is surjective on objects and then, by Lemma 5, it is an equivalence. □

Lemma 8. Let ℂ be a projective cover of a regular category 𝔸. Then 𝔸 is Mal’cev if and only if ℂ is G-Mal’cev. In particular, a category ℂ with weak finite limits is G-Mal’cev if and only if its exact completion ℂ_{ex} is Mal’cev.

Proof: (only if) : Consider a reflexive graph \xymatrix{X_{1} \ar[r]^{x_{0}} \ar[r]_{x_{1}} & X_{0}} in ℂ and its (regular

epi, jointly mono) factorization in 𝔸

\xymatrix{X_{1} \ar[rr]^{x_{0}} \ar[rr]_{x_{1}} \ar@{->>}[dr]_{p} &  & X_{0} \\  & I \ar[ur]^{i_{0}} \ar[ur]_{i_{1}}}

The relation (i_{0}, i_{1}) is reflexive so that, by assumption, it is an equivalence relation.

• Consider the symmetry s: I → I of (i_{0}, i_{1}); since X_{1} is regular projective and p: X_{1} → I is a regular epi, there exists a morphism σ: X_{1} → X_{1} such that σ·p = p·s; one checks that σ is a symmetry for the graph (x_{0}, x_{1}).

• Consider the following pullbacks in 𝔸

\xymatrix{P \ar[rr]^{^{\overline{x}^{1}}} \ar[dd]_{_{\overline{x}_{0}}} &  & X_{1} \ar[d]^{p} \\  & L \ar[r]^{_{\overline{i}_{1}}} \ar[d]_{_{\overline{i}_{0}}} & I \ar[d]^{i_{0}} \\ X_{1} \ar[r]_{p} & I \ar[r]_{i_{1}} & X_{0}}

together with the unique factorization π: P → L; consider now a regular epi q: X_{2} → P with X_{2} in ℂ. Then

\xymatrix{X_{1} & X_{2} \ar[l]_{q·\overline{x}_{0}} \ar[r]^{q·\overline{x}_{1}} & X_{1}}

is a weak pullback of x_{0} and x_{1} in ℂ. Let t: L → I be given by the transitivity of (i_{0}, i_{1}) ; since X_{2} is regular projective and p: X_{1} → I is a regular epi, there exists a morphism τ: X_{2} → X_{1} such that τ ·p = q ·π ·t; one checks that τ yields transitivity of (x_{0}, x_{1}).

(if) : Consider a reflexive relation \xymatrix{R \ar[r]^{r_{0}} \ar[r]_{r_{1}} & X} in 𝔸, with δ: X → R the common

section of r_{0} and r_{1}, an object \overline{X} in ℂ and a regular epi x: \overline{X} → X. Consider also the following pullback

\xymatrix{B \ar[r]^{⟨b_{0},b_{1}⟩} \ar[d]_{a} & \overline{X} × \overline{X} \ar[d]^{x×x} \\ R \ar[r]_{⟨r_{0},r_{1}⟩} & X × X}

and cover B with a regular epi β: P → B from an object P of ℂ. Since x·δ·⟨r_{0}, r_{1}⟩ = Δ · (x × x), where Δ: \overline{X} → \overline{X} × \overline{X} is the diagonal, there is a unique α: \overline{X} → B such that α · a = x · δ and α · ⟨b_{0}, b_{1}⟩ = Δ. Since β is a regular epi and \overline{X} is regular projective, there exists γ: \overline{X} → P such that γ · β = α. Now γ · β is a reflection for

the graph \xymatrix{P \ar[r]^{β·b_{0}} \ar[r]_{β·b_{1}} & \overline{X}} in ℂ. By assumption, this graph has a structure of pseudo-

equivalence. Let σ: P → P be a symmetry. Then σ is a symmetry also for the graph

\xymatrix{P \ar[r]^{β·b_{0}·x} \ar[r]_{β·b_{1}·x} & X} and then \xymatrix{R \ar[r]^{r_{0}} \ar[r]_{r_{1}} & X} , being the jointly monic part of its factorization,

is a symmetric relation. Since 𝔸 is regular, this is enough to prove that it is a Mal’cev category. □

Recall from [2] that a preadditive category is additive iff it has finite sums, and that a functor between additive categories is additive iff it preserves finite sums. Moreover, for a functor between abelian categories, to be right exact or exact in the sense of abelian categories is the same as to be right exact or exact in the sense used in section 1.

Proposition 9. Let ℂ be a preadditive category with weak finite products and weak kernels. Then there exists an abelian category ℂ_{ab} and a full and faithful functor

Σ: ℂ → ℂ_{ab}

such that, for each abelian category 𝔹, the composition with Σ induces an equivalence between the category of right exact functors from ℂ_{ab} to 𝔹 and the category of additive functors from ℂ to 𝔹

Σ · − : Right[ℂ_{ab}, 𝔹] → Add[ℂ, 𝔹]

Proof: We already know that such a category ℂ has weak finite limits. Moreover, up to replacing ℂ by its Cauchy-completion \overline{ℂ}, which is nothing else but the full subcategory of all the regular projective objects of the abelian category ℂ_{ex}, we can assume that ℂ has finite sums. (In fact \overline{ℂ} is still preadditive, and a functor ℂ → 𝔹 is additive iff its unique extension \overline{ℂ} → 𝔹 is additive.) Now we can take as ℂ_{ab} the exact completion ℂ_{ex}. It is abelian and then it is a Mal’cev category [4, 5]. By Lemma 7 and Lemma 8, it follows that L: ℂ_{ex} → ℂ_{rc} is an equivalence. Now take as Σ: ℂ → ℂ_{ab} the functor Λ · L^{−1} : ℂ → ℂ_{rc} → ℂ_{ex}; point 2 of Proposition 2 gives us exactly the requested universal property. □

Bearing in mind point 3 of Proposition 1, we have the following corollary, which is the first of Freyd’s results quoted in the introduction.

Corollary 10. [7] Let 𝔸 be an abelian category and ℙ a projective cover of 𝔸. Then, for each abelian category 𝔹, the composition with the full inclusion ℙ → 𝔸 induces an equivalence of categories Right[𝔸, 𝔹] → Add[ℙ, 𝔹] .

3. Frobenius categories

To explain the universal property of a Frobenius category (Theorem 3.1 in [8]), in this section we compare the exact and the pre-regular completion of a category.

Let ℂ be a category with weak finite limits. There is a canonical (pseudo) functor K: ℂ_{preg} → ℂ_{ex} induced by the universal property of ℂ_{preg}. Explicitely, K is defined in the following way

\xymatrix{X_{0} \ar[r]^{f_{0}} \ar[d]_{x} & Y_{0} \ar[d]^{y} \\ X′ \ar[r]_{f′} & Y′} ↦ \xymatrix{N(x) \ar[r]^{\overline{f}} \ar[d]_{x_{0}} \ar[d]^{x_{1}} & N(y) \ar[d]_{y_{0}} \ar[d]^{y_{1}} \\ X_{0} \ar[r]_{f_{0}} & Y_{0}}

where (x_{0}, x_{1}) is a weak kernel pair of x and \overline{f} is induced by the fact that (y_{0}, y_{1}) is a weak kernel pair of y. The functor K is well defined because two weak kernel pairs of the same arrow are objects canonically isomorphic in ℂ_{ex}. Moreover, K is faithful.

Lemma 11. Let ℂ be a category with weak finite limits. The functor K: ℂ_{preg} → ℂ_{ex} is an equivalence if and only if the following conditions hold in ℂ :

i - each arrow is a weak coequalizer ;

ii - each pseudo-equivalence is a weak kernel pair.

Proof: (if) : Consider two objects x: X_{0} → X′ and y: Y_{0} → Y′ in ℂ_{preg} and an arrow [f_{0}, \overline{f}]: K(x) → K(y) in ℂ_{ex}

\xymatrix{N(x) \ar[r]^{\overline{f}} \ar[d]_{x_{0}} \ar[d]^{x_{1}} & N(y) \ar[d]_{y_{0}} \ar[d]^{y_{1}} \\ X_{0} \ar[r]_{f_{0}} \ar[d]_{x} & Y_{0} \ar[d]^{y} \\ X′ \ar@{.>}[r]_{f′} & Y′}

Since x is, by condition (i), a weak coequalizer, it is a weak coequalizer of (x_{0}, x_{1}). Then there exists f′ such that x · f′ = f_{0} · y, so that K is full.

Now consider an object \xymatrix{C_{1} \ar[r]^{r_{0}} \ar[r]_{r_{1}} & C_{0}} in ℂ_{ex}. By condition (ii), (r_{0}, r_{1}) is a weak

kernel pair of some q: C_{0} → Q and then (r_{0}, r_{1}) and K(q) are isomorphic.

(only if) : Consider the following diagram

\xymatrix{ & N(y) \ar[d]_{y_{0}} \ar[d]^{y_{1}} \\ N(x) \ar@{.>}[ur]^{\overline{f}} \ar[r]_{x_{1}} \ar[r]^{x_{0}} & X_{0} \ar[r]^{x} \ar[d]_{y} & X′ \ar@{.>}[dl]^{f′} \\  & Y′}

where x is an arrow in ℂ, (x_{0}, x_{1}) is a weak kernel pair of x, y coequalizes x_{0} and x_{1} and (y_{0}, y_{1}) is a weak kernel pair of y. The condition x_{0} · y = x_{1} · y implies that there exists \overline{f} such that \overline{f} · y_{0} = x_{0} and \overline{f} · y_{1} = x_{1}. Now [1_{X′}, \overline{f}] is an arrow in ℂ_{ex} between K(x) and K(y). Since K is full, there is an arrow f′ such that x · f′ = y. This means that x is a weak coequalizer of (x_{0}, x_{1}).

To prove condition (ii), we identify ℂ with a full subcategory of ℂ_{ex}. Let \xymatrix{C_{1} \ar[r]^{r_{0}} \ar[r]_{r_{1}} & C_{0}}

be a pseudo-equivalence in ℂ, and consider the following diagram in ℂ_{ex}

\xymatrix{ &  & B \ar@{.>}[dll]_{l} \ar@{.>}[dl]_{h} \ar[d]_{f} \ar[d]^{g} \\ C_{1} \ar[r]_{p} & X \ar[r]^{i_{0}} \ar[r]_{i_{1}} & C_{0} \ar[r]^{q} & Q \ar[r]^{m} & E}

where (p, (i_{0}, i_{1})) is the regular epi-jointly monic factorization of (r_{0}, r_{1}) and q is the coequalizer of (i_{0}, i_{1}), which exists by Theorem 26 in [6]. Moreover, since K is an equivalence, there is an object E in ℂ and a mono m: Q → E. Now we prove that (r_{0}, r_{1}) is a weak kernel pair in ℂ of q · m. Assume that f · q · m = g · q · m. Since m is a mono and (i_{0}, i_{1}) is the kernel pair of q, there exists a unique h such that h · i_{0} = f and h · i_{1} = g. Since B is regular projective and p is a regular epi, there is l such that l · p = h. Finally, l · r_{0} = f and l · r_{1} = g. □

Remark : The previous lemma can be made more precise. The functor K: ℂ_{preg} → ℂ_{ex} factors through ℂ_{reg}, the regular completion of ℂ [6], and one can prove that

i - ℂ_{preg} → ℂ_{reg} is full iff in ℂ any arrow is a weak coequalizer ;

ii - ℂ_{reg} → ℂ_{ex} is an equivalence iff for any pseudo-equivalence \xymatrix{C_{1} \ar[r]^{r_{0}} \ar[r]_{r_{1}} & C_{0}} in

ℂ there exists a finite family of arrows (q_{i}: C_{0} → Q_{i})_{I} such that (r_{0}, r_{1}) is weak universal w.r.t. the condition r_{0} · q_{i} = r_{1} · q_{i} for all i in I.

Recall that a Frobenius category is an abelian category with enough projectives and injectives, and in which projectives and injectives coincide. In the additive case, the conditions of the previous lemma are equivalent to the fact that ℂ is a projective-injective cover of a Frobenius category. In fact, we have the following lemma (compare with Theorem 1.7 in [7]).

Lemma 12. Let ℂ be an additive category with weak kernels, and assume that in ℂ each arrow is a weak cokernel. The following conditions are equivalent :

1) each arrow has a weak cokernel, each arrow is a weak kernel ;

2) each pseudo-equivalence is a weak kernel pair.

Proof: 1) ⇒ 2) : Let \xymatrix{C_{1} \ar[r]^{r_{0}} \ar[r]_{r_{1}} & C_{0}} be a pseudo-equivalence in ℂ and consider a

weak cokernel q: C_{0} → Q of r_{0} −r_{1}. Now we prove that ( \begin{matrix}q \\ −q \end{matrix} ) : C_{0} ⊕C_{0} → Q is a

weak cokernel of ⟨r_{0}, r_{1}⟩: C_{1} → C_{0} ⊕ C_{0}. For this, consider an arrow ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) : C_{0} ⊕

C_{0} → Z such that ⟨r_{0}, r_{1}⟩ · ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) = 0 and let i: C_{0} → C_{1} be such that i · r_{0} =

1_{C_{0}} = i · r_{1}. We have 0 = i · 0 = i · (r_{0} · h_{1} + r_{1} · h_{2}) = i · r_{0} · h_{1} + i · r_{1} · h_{2} = h_{1} + h_{2}, so that h_{2} = −h_{1} and then 0 = r_{0} · h_{1} + r_{1} · h_{2} = (r_{0} − r_{1}) · h_{1}. But q is a weak cokernel of r_{0} − r_{1}, so that there exists z: Q → Z such that q · z = h_{1}. This

implies that ( \begin{matrix}q \\ −q \end{matrix} ) · z = ( \begin{matrix}h_{1} \\ h_{2} \end{matrix} ) , that is ( \begin{matrix}q \\ −q \end{matrix} ) is a weak cokernel of ⟨r_{0}, r_{1}⟩.

By assumption, ⟨r_{0}, r_{1}⟩ is a weak kernel, and then it is a weak kernel of its weak

cokernel ( \begin{matrix}q \\ −q \end{matrix} ) . This implies that \xymatrix{C_{1} \ar[r]^{r_{0}} \ar[r]_{r_{1}} & C_{0}} is a weak kernel pair of q. In

fact, if \xymatrix{T \ar[r]^{x} \ar[r]_{y} & C_{0}} are such that x · q = y · q, then ⟨x, y⟩ · ( \begin{matrix}q \\ −q \end{matrix} ) = 0 and then

there is t: T → C_{1} such that t · ⟨r_{0}, r_{1}⟩ = ⟨x, y⟩. This means t · r_{0} = x and t · r_{1} = y.

2) ⇒ 1) : Let f: X → Y be an arrow in ℂ (which we identify with a full subcategory of the abelian category ℂ_{ex}). Consider the (regular epi, mono) factorization of f in ℂ_{ex} and the cokernel q of its monic part i

\xymatrix{ & T \ar@{.>}[ddl]_{t′′} \ar@{.>}[d]_{t′} \ar[ddr]^{t} \\  & I \ar[dr]_{i} \\ X \ar[ur]_{e} \ar[rr]_{f} &  & Y \ar[r]^{q} & Q \ar[r]^{m} & E}

and embed Q in an object E of ℂ (this is possible because ℂ has weak finite limits and satisfies the conditions of the previous lemma, so that K: ℂ_{preg} → ℂ_{ex} is an equivalence). Let us show that f is a weak kernel of q · m in ℂ. Let t: T → Y be an arrow in ℂ such that t · q · m = 0. Since m is a mono, then t · q = 0. But i is a mono, so that it is the kernel of its cokernel q, and then there exists a unique t′ such that t′ · i = t. Since T is regular projective and e is a regular epi, there is t′′ such that t′′ · e = t′. Finally, t′′ · f = t, so that f is a weak kernel of q · m. Moreover, by assumption, q · m is a weak cokernel, so that it is a weak cokernel of its weak kernel f. □

Corollary 13. [8] Let ℂ be an additive category such that each arrow has a weak kernel and a weak cokernel, and it is a weak kernel and a weak cokernel. Then ℂ_{preg} is a Frobenius category. Moreover, for any abelian category 𝔹 and for any additive functor F: ℂ → 𝔹, there are three functors F_{i}: ℂ_{preg} → 𝔹 (i = 1, 2, 3) such that :

1) The following diagram commutes, up to isomorphism, for i = 1, 2, 3

\xymatrix{ℂ \ar[rr]^{Ψ} \ar[dr]_{F} &  & ℂ_{preg} \ar[dl]^{F_{i}} \\  & 𝔹}

2) F_{1} is right exact, F_{2} is left exact and F_{3} preserves images ;

3) Each F_{i} is essentially unique within these properties.

Proof: By Lemmas 7, 8, 11 and 12, we have two equivalences

K: ℂ_{preg} → ℂ_{ex} L: ℂ_{ex} → ℂ_{rc}

Since ℂ is additive, ℂ_{ex} ≃ ℂ_{preg} is abelian. It is Frobenius by point 2 of Proposition 3. Now, the existence of F_{3} follows from the universal property of ℂ_{preg} because, under the equivalence ℂ_{preg} ≃ ℂ_{ex}, the factorization system in ℂ_{preg} becomes the (regular epi, mono) factorization. As far as F_{1} is concerned, we simply apply Proposition 9 using the equivalence ℂ_{preg} ≃ ℂ_{rc}. Finally, since ℂ_{preg} is abelian, its dual category is exact. Moreover, by point 2 of Proposition 3, ℂ^{op} is a projective cover of (ℂ_{preg})^{op}. We obtain now F_{2} applying Proposition 9 to the dual categories.

□

To end this paper, we consider the case of triangulated categories. For definitions and basic facts, the reader can consult [11, 14]. The following lemma shows that the notion of homological functor on a triangulated category, which is defined in terms of the triangulated structure, is in fact an intrinsic notion.

Lemma 14. Let 𝕋 be a triangulated category and 𝔹 an abelian category. An additive functor H: 𝕋 → 𝔹 is a homological functor if and only if it is left covering.

Proof: Since H is additive, we have to prove that H is left covering w.r.t. weak kernels iff it sends triangles on exact sequences. This is the case because any arrow v: Y → Z fits into an triangle of the form (u, v, w) and, in any triangle

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX ,}

u: X → Y is a weak kernel of v. Now the exactness of

\xymatrix{H(X) \ar[r]^{H(u)} & H(Y) \ar[r]^{H(v)} & H(Z)}

means that the comparison between H(u) and the kernel of H(v) is a (regular) epi, i.e. that H is left covering w.r.t. weak kernels. □

Corollary 15. Let 𝕋 be a triangulated category. Then Γ: 𝕋 → 𝕋_{ex} is the universal homological functor on 𝕋.

Proof: Since 𝕋 is additive, 𝕋_{ex} is abelian. The statement follows now from Lemma 14 and point 1 of Proposition 1. □

Since any triangulated category satisfies the conditions of Lemma 12, in the previous corollary we can replace the exact completion by the pre-regular completion. We get in this way the theorem proved in [8] for the stable homotopy category, and in [9, 11] for an arbitrary triangulated category.

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2001/n3a1/v3n3a1.(dvi,ps,pdf)

J. Rosický rosicky@math.muni.cz

Department of Mathematics Masaryk University Janáčkovo nám. 2a, 662 95 Brno Czech Republic

E.M. Vitale vitale@agel.ucl.ac.be

Département de Mathématique Université catholique de Louvain Chemin du Cyclotron 2, 1348 Louvain-la-Neuve Belgium