Theory and Applications of Categories, Vol. 23, No. 4, 2010, pp. 76–91.

SNAKE LEMMA IN INCOMPLETE RELATIVE HOMOLOGICAL CATEGORIES

Dedicated to Dominique Bourn on the occasion of his 60th birthday

TAMAR JANELIDZE

Abstract. The purpose of this paper is to prove a new, incomplete-relative, version of Non-abelian Snake Lemma, where “relative” refers to a distinguished class of normal epimorphisms in the ground category, and “incomplete” refers to omitting all completeness/cocompleteness assumptions not involving that class.

1. Introduction

The classical Snake Lemma known for abelian categories (see e.g. [5]) has been extended to homological categories by D. Bourn; see F. Borceux and D. Bourn [1], and references there. In [3], we extended it to the context of relative homological categories: here “relative” refers to a distinguished class E of regular epimorphisms in a ground category C satisfying certain conditions which, in particular, make (C, E) relative homological when (a) C is a homological category and E is the class of all regular epimorphisms in C; (b) C is a pointed finitely complete category satisfying certain cocompleteness conditions, and E is the class of all isomorphisms in C. In this paper we drop the completeness/cocompleteness assumption and extend Snake Lemma further to the context of what was called an incomplete relative homological category in [4] (see, however, the correction below Definition 2.1).

Let us recall the formulation of Snake Lemma from [1] (see Theorem 4.4.2 of [1]). It says: “Let C be a homological category. Consider the diagram

Partially supported by the International Student Scholarship of the University of Cape Town, and Georgian National Science Foundation (GNSF/ST06/3-004).

Received by the editors 2009-1-17 and, in revised form, 2009-09-29.

Published on 2010-01-29 in the Bourn Festschrift.

2000 Mathematics Subject Classification: 18G50, 18G25, 18A20, 18B10.

Key words and phrases: incomplete relative homological category, homological category, normal epimorphism, snake lemma.

\begin{matrix}c \\ ◯ \end{matrix} Tamar Janelidze, 2010. Permission to copy for private use granted.

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\  & K_{u} \ar@{.>}[r]^{f_{K}} \ar[d]_{k_{u}} & K_{v} \ar@{.>}[r]^{g_{K}} \ar[d]_{k_{v}} & K_{w} \ar[d]^{k_{w}} \ar@{.>}[dddll]_{d} \\  & X \ar[r]^{f} \ar[d]_{u} & Y \ar[r]_{g} \ar[d]_{v} & Z \ar[r] \ar[d]^{w} & 0 \\ 0 \ar[r] & X′ \ar[r]^{f′} \ar[d]_{q_{u}} & Y′ \ar[r]^{g′} \ar[d]_{q_{v}} & Z′ \ar[d]^{q_{w}} \\  & Q_{u} \ar@{.>}[r]_{_{f′_{Q}}} \ar[d] & Q_{v} \ar@{.>}[r]_{_{g′_{Q}}} \ar[d] & Q_{w} \ar[d] \\  & 0 & 0 & 0}

where all squares of plain arrows are commutative and all sequences of plain arrows are exact. There exists an exact sequence of dotted arrows making all squares commutative”.

The proof given in [1] contains explicit constructions of the dotted arrows, which makes the result far more precise. This seems to suggest that these constructions should be included in the formulation of the lemma, but there is a problem: while the constructions of f_{K}, g_{K}, f′_{Q}, and g′_{Q} as induced morphisms are straightforward, the several-step construction of the “connecting morphism” d : K_{w} → Q_{u} is too long and technical. A natural solution of this problem, well-known in the abelian case, involves partial composition of internal relations in C and goes back at least to S. Mac Lane [6] (see also e.g. [2] for the so-called calculus of relations in regular categories): one should simply define d as the composite q_{u}f′^{∘}vg^{∘}k_{w}, where g^{∘} is the relation opposite to g, etc. We do not know if the non-abelian version of d = q_{u}f′^{∘}vg^{∘}k_{w} is mentioned anywhere in the literature, but we use its relative version in our relative Snake Lemma (Theorem 3.1), which is the main result of this paper.

For the reader’s convenience we are mostly using the same notation as in [1].

2. Incomplete relative homological categories

Throughout the paper we assume that C is a pointed category and E is a class of morphisms in C containing all isomorphisms. Let us recall from [4]:

2.1. Definition. A pair (C, E) is said to be an incomplete relative homological category if it satisfies the following conditions:

(a) Every morphism in E is a normal epimorphism; (b) The class E is closed under composition;

(c) If f ∈ E and gf ∈ E then g ∈ E;

(d) If f ∈ E then ker(f) exists in C;

(e) A diagram of the form

\xymatrix{ & A′ \ar[d]^{f′} \ar[ddr]^{g′} \\ A \ar[r]_{f} \ar[drr]_{g} & B \\  &  & B}

has a limit in C provided f and g are in E, and either (i) f = g and f′ = g′, or (ii) f′ and g′ are in E, (f, g) and (f′, g′) are reflexive pairs, and f and g are jointly monic;

(f) If

\xymatrix{A ×_{B} A′ \ar[d]_{π_{1}} \ar[r]^{π_{2}} & A′ \ar[d]^{f′} \\ A \ar[r]_{f} & B}

is a pullback and f is in E, then π_{2} is also in E;

(g) If h_{1} : H → A and h_{2} : H → B are jointly monic morphisms in C and if α : A → C and β : B → D are morphisms in E, then there exists a morphism h : H → X in E and jointly monic morphisms x_{1} : X → C and x_{2} : X → D in C making the diagram

\xymatrix{ &  & H \ar@{.>}[d]^{h} \ar[dl]_{h_{1}} \ar[dr]^{h_{2}} \\  & A \ar[dl]_{α} & X \ar@{.>}[dll]^{x_{1}} \ar@{.>}[drr]_{x_{2}} & B \ar[dr]^{β} \\ C &  &  &  & D}

commutative (it easily follows from the fact that every morphism in E is a normal epimorphism, that such factorization is unique up to an isomorphism); (h) The E-Short Five Lemma holds in C, i.e. in every commutative diagram of the

form

[figure: K; k; A; w; f; B; K; k′; A′; f′; B]

with f and f′ in E and with k = ker(f) and k′ = ker(f′), the morphism w is an isomorphism;

(i) If in a commutative diagram

[figure: K; u; k; A; w; f; B; K′; k′; A′; f′; B]

f, f′, and u are in E, k = ker(f) and k′ = ker(f′), then there exists a morphism e : A → M in E and a monomorphism m : M → A′ in C such that w = me.

Let us use this opportunity to make a correction to conditions 2.1(c) and 3.1(c) of [4]. They should be replaced, respectively, with:

(a) A diagram of the form

\xymatrix{ & A′ \ar[d]^{f′} \ar[ddr]^{g′} \\ A \ar[r]_{f} \ar[drr]_{g} & B \\  &  & B}

has a limit in C provided f, g, f′, and g′ are in E, and either (i) f = g and f′ = g′, or (ii) (f, g) and (f′, g′) are reflexive pairs (that is, fh = 1_{B} = gh and f′h′ = 1_{B} = g′h′ for some h and h′), and f and g are jointly monic.

(b) Condition 2.1(e) of the present paper.

Note that this replacement will not affect any results/arguments of [4], except that without it a pair (C, Iso(C)) would be an incomplete relative homological category only when C (is pointed and) admits equalizers of isomorphisms.

2.2. Remark. As easily follows from condition 2.1(g), if a morphism f : A → B in C factors as f = em in which e is in E and m is a monomorphism, then it also factors (essentially uniquely) as f = m′e′ in which m′ is a monomorphism and e′ is in E.

2.3. Lemma. If a pair (C, E) satisfies conditions 2.1(a)-2.1(d), then (C, E) satisfies conditions 2.1(h) and 2.1(i) if and only if in every commutative diagram of the form

[figure: K; u; k; A; w; f; B; K′; k′; A′; f′; B]  (2.1)

with k = ker(f), k′ = ker(f′), and with f, f′, and u in E, the morphism w is also in E.

Proof. Suppose (C, E) satisfies conditions 2.1(a)-2.1(d), 2.1(h) and 2.1(i). Consider the commutative diagram (2.1) with k = ker(f), k′ = ker(f′), and with f, f′, and u in E. By condition 2.1(i), w = me where e : A → M is in E and m : M → A′ is a monomorphism. Consider the commutative diagram:

\xymatrix{K \ar[d]_{ek} \ar[r]^{u} & K′ \ar@{.>}[dl]_{m̄} \ar[d]^{k′} \\ M \ar[r]_{m} & A′}

Since u is in E and m is a monomorphism, by condition 2.1(a) there exists a unique morphism m̄ : K′ → M with m̄u = ek and mm̄ = k′; m̄ is a monomorphism since so k′. Since f′mm̄ = 0, m̄ is a monomorphism, and k′ = ker(f′), we conclude that m̄ = ker(f′m). By condition 2.1(c), f′m is in E, therefore we can apply the E-Short Five Lemma to the diagram

[figure: K′; m̄; M; m; f′m; B; K′; k′; A′; f′; B]

and conclude that m is an isomorphism. Hence, by condition 2.1(b) w is in E, as desired.

Conversely, suppose for every commutative diagram (2.1) with k = ker(f), k′ = ker(f′), and with f and f′ in E, if u is in E then w is also in E. It is a well know fact that under the assumptions of condition 2.1(h), ker(w) = 0; moreover, since E contains all isomorphisms and f and f′ are in E, w : A → A′ is also in E. Since every morphism in E is a normal epimorphism, we conclude that w is an isomorphism, proving condition 2.1(h). The proof of condition 2.1(i) is trivial.

2.4. Lemma. Let (C, E) be a pair satisfying conditions 2.1(a)-2.1(d) and 2.1(g). Consider the commutative diagram:

\xymatrix{A \ar[r]^{f} \ar[d]_{α} & B \ar[d]^{β} \\ A′ \ar[r]_{f′} & B′}  (2.2)

(i) If α : A → A′ and β : B → B′ are in E and if f : A → B factors as f = me in which e is in E and m is a monomorphism, then f′ : A′ → B′ also factors as f′ = m′e′ in which e′ is in E and m′ is monomorphism.

(ii) If α : A → A′ and β : B → B′ are monomorphisms and if f′ : A′ → B′ factors as f′ = m′e′ in which e′ is in E and m′ is a monomorphism, then f : A → B also factors as f = me in which e is in E and m is a monomorphism.

Proof. (i): Consider the commutative diagram (2.2) and suppose α and β are in E and f = me in which e : A → C is in E and m : C → B is a monomorphism. Since β is in E and m is a monomorphism, by Remark 2.2 there exists a morphism γ : C → C′ in E and a monomorphism m′ : C′ → B′ such that βm = m′γ. Consider the commutative diagram:

\xymatrix{A \ar[r]^{α} \ar[d]_{γe} & A′ \ar@{.>}[dl]_{e′} \ar[d]^{f′} \\ C′ \ar[r]_{m′} & B′}

Since α is in E and m′ is a monomorphism, conditions 2.1(a) and 2.1(d) imply the existence of a unique morphism e′ : A′ → C′ with e′α = γe and m′e′ = f′. Since α, e, and γ are in E, conditions 2.1(b) and 2.1(c) imply that e′ is also in E. Hence, f′ = m′e′ in which e′ is in E and m′ is a monomorphism, as desired.

(ii) can be proved similarly.

Let (C, E) be an incomplete relative homological category. We will need to compose certain relations in C:

Let R = (R, r_{1}, r_{2}) : A → B be a relation from A to B, i.e. a pair of jointly monic morphisms r_{1} : R → A and r_{2} : R → B with the same domain, and let S = (S, s_{1}, s_{2}) : B → C be a relation from B to C. If the pullback (R ×_{B} S, π_{1}, π_{2}) of r_{2} and s_{1} exists in C, and if there exists a morphism e : R ×_{B} S → T in E and a jointly monic pair of morphisms t_{1} : T → A and t_{2} : T → C in C making the diagram

\xymatrix{ &  & R ×_{B} S \ar@{.>}[d]^{e} \ar[dl]_{π_{1}} \ar[dr]^{π_{2}} \\  & R \ar[dl]_{r_{1}} \ar[dr]_{r_{2}} & T \ar@{.>}[dll]^{t_{1}} \ar@{.>}[drr]_{t_{2}} & S \ar[dl]^{s_{1}} \ar[dr]^{s_{2}} \\ A &  & B &  & C}  (2.3)

commutative, then we will say that (T, t_{1}, t_{2}) : A → C is the composite of (R, r_{1}, r_{2}) : A → B and (S, s_{1}, s_{2}) : B → C. One can similarly define partial composition for three or more relations satisfying a suitable associativity condition. Omitting details, let us just mention that, say, a composite RR′R′′ might exist even if neither RR′ nor R′R′′ does.

2.5. Convention. We will say that a relation R = (R, r_{1}, r_{2}) : A → B is a morphism in C if r_{1} is an isomorphism.

2.6. Definition. Let (C, E) be an incomplete relative homological category. A sequence of morphisms

\xymatrix{… \ar[r] & A_{i−1} \ar[r]^{f_{i−1}} & A_{i} \ar[r]^{f_{i}} & A_{i+1} \ar[r] & …}

is said to be:

(i) E-exact at A_{i}, if the morphism f_{i−1} admits a factorization f_{i−1} = me, in which e ∈ E and m = ker(f_{i});

(ii) an E-exact sequence, if it is E-exact at A_{i} for each i (unless the sequence either begins with A_{i} or ends with A_{i}).

As easily follows from Definition 2.6, the sequence

\xymatrix{0 \ar[r] & A \ar[r]^{f} & B \ar[r]^{g} & C}

is E-exact if and only if f = ker(g); and, if the sequence

\xymatrix{A \ar[r]^{f} & B \ar[r]^{g} & C \ar[r] & 0}

is E-exact then g = coker(f) and g is in E.

In the next section we will often use the following simple fact:

2.7. Lemma. (Lemma 4.2.4(1) of [1]) If in a commutative diagram

\xymatrix{K′ \ar[r]^{k′} \ar[d]_{u} & A′ \ar[r]^{f′} \ar[d]^{v} & B′ \ar[d]^{w} \\ K \ar[r]_{k} & A \ar[r]_{f} & B}  (2.4)

in C, k = ker(f) and w is a monomorphism, then k′ = ker(f′) if and only if the left hand square of the diagram (2.4) is a pullback.

3. The Snake Lemma

This section is devoted to our main result which generalizes Theorem 4.4.2 of [1] and its relative version mentioned in [3]. Formulating it, we use the same notation as in [1].

3.1. Theorem. [Snake Lemma] Let (C, E) be an incomplete relative homological category. Consider the commutative diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\  & K_{u} \ar[r]^{f_{K}} \ar[d]_{k_{u}} & K_{v} \ar[r]^{g_{K}} \ar[d]_{k_{v}} & K_{w} \ar[d]^{k_{w}} \ar@{.>}[dddll]^{d} \\  & X \ar[r]^{f} \ar[d]_{u} & Y \ar[r]^{g} \ar[d]_{v} & Z \ar[r] \ar[d]^{w} & 0 \\ 0 \ar[r] & X′ \ar[r]^{f′} \ar[d]_{q_{u}} & Y′ \ar[r]^{g′} \ar[d]_{q_{v}} & Z′ \ar[d]^{q_{w}} \\  & Q_{u} \ar[r]_{_{f′_{Q}}} \ar[d] & Q_{v} \ar[r]_{_{g′_{Q}}} \ar[d] & Q_{w} \ar[d] \\  & 0 & 0 & 0}  (3.1)

in which all columns, the second and the third rows are E-exact sequences. If the morphism g′ factors as g′ = g′_{2}g′_{1} in which g′_{1} is in E and g′_{2} is a monomorphism, then:

(i) The composite d = q_{u}f′^{∘}vg^{∘}k_{w} : K_{w} → Q_{u} is a morphism in C.

(ii) The sequence

\xymatrix{K_{u} \ar[r] & K_{v} \ar[r] & K_{w} \ar[r]^{d} & Q_{u} \ar[r] & Q_{v} \ar[r] & Q_{w}}  (3.2)

where d = q_{u}f′^{∘}vg^{∘}k_{w}, is E-exact.

Proof. (i): Under the assumptions of the theorem, let (Y ×_{Z} K_{w}, π_{1}, π_{2}) be the pullback of g and k_{w} (by condition 2.1(e) this pullback does exist in C); since g is in E, by condition 2.1(f) the morphism π_{2} is also in E, and since k_{w} is a monomorphism so is π_{1}. Since f′ = ker(g′) and g′vπ_{1} = 0, there exists a unique morphism φ : Y ×_{Z} K_{w} → X′ with vπ_{1} = f′φ (see diagram (3.4) below). Using the fact that (Y ×_{Z} K_{w}, π_{1}, π_{2}) is the pullback of g and k_{w} and that k_{w} = ker(w), an easy diagram chase proves that (Y ×_{Z} K_{w}, π_{1}, φ) is the pullback of v and f′. Therefore, we obtain the commutative diagram

\xymatrix{ &  &  &  &  & P \ar[dl]_{1_{P}} \ar[dr]^{1_{P}} \\  &  &  &  & P \ar[dl]_{1_{P}} \ar[dr]^{1_{P}} &  & P \ar[dl]_{1_{P}} \ar[dr]^{1_{P}} \\  &  &  & P \ar[dl]_{1_{P}} \ar[dr]^{π_{1}} &  & P \ar[dl]_{π_{1}} \ar[dr]^{1_{P}} &  & P \ar[dl]_{1_{P}} \ar[dr]^{φ} \\  &  & P \ar[dl]_{π_{2}} \ar[dr]^{π_{1}} &  & Y \ar[dl]_{1_{Y}} \ar[dr]^{1_{Y}} &  & P \ar[dl]_{π_{1}} \ar[dr]^{φ} &  & X′ \ar[dl]_{1_{X′}} \ar[dr]^{1_{X′}} \\  & K_{w} \ar[dl]_{1_{Kw}} \ar[dr]^{k_{w}} &  & Y \ar[dl]_{g} \ar[dr]^{1_{Y}} &  & Y \ar[dl]_{1_{Y}} \ar[dr]^{v} &  & X′ \ar[dl]_{f′} \ar[dr]^{1_{X′}} &  & X′ \ar[dl]_{1_{X′}} \ar[dr]^{q_{u}} \\ K_{w} &  & Z &  & Y &  & Y′ &  & X′ &  & Q_{u}}

where P = Y ×_{Z} K_{w}, and all the diamond parts are pullbacks. Since π_{2} and q_{u} are in E, by condition 2.1(g) we have the factorization (unique up to an isomorphism)

\xymatrix{ &  & Y ×_{Z} K_{w} \ar@{.>}[d]^{r} \ar[dl]_{1_{Y ×_{Z} Kw}} \ar[dr]^{φ} \\  & Y ×_{Z} K_{w} \ar[dl]_{π_{2}} & R \ar@{.>}[dll]^{r_{1}} \ar@{.>}[drr]_{r_{2}} & X′ \ar[dr]^{q_{u}} \\ K_{w} &  &  &  & Q_{u}}  (3.3)

where r : Y ×_{Z} K_{w} → R is a morphism in E and r_{1} : R → K_{w} and r_{2} : R → Q_{u} are jointly monic morphisms in C. As follows from the definition of composition of relations, (R, r_{1}, r_{2}) is the composite relation q_{u}f′^{∘}vg^{∘}k_{w} from K_{w} to Q_{u} (Note, that since the pullback (Y ×_{Z} K_{w}, π_{1}, π_{2}) of k_{w} and g, and the pullback (Y ×_{Z} K_{w}, π_{1}, φ) of v and f′ exists in C, the composite relations g^{∘}k_{w} : K_{w} → Y and f^{′∘}v : Y → X′ also exist. Moreover, since π_{2} and q_{u} are in E, the composite q_{u}(f′^{∘}v)(g^{∘}k_{w}) of the three relations g^{∘}k_{w}, f^{′∘}v, and q_{u} also exists and we have q_{u}(f′^{∘}v)(g^{∘}k_{w}) = q_{u}f′^{∘}vg^{∘}k_{w}).

To prove that q_{u}f′^{∘}vg^{∘}k_{w} : K_{w} → Q_{u} is a morphism in C, consider the commutative diagram

\xymatrix{ & 0 \ar[d] &  & 0 \ar[d] &  & 0 \ar[d] \\  & K_{u} \ar[rr]^{f_{K}} \ar[dd]_{k_{u}} &  & K_{v} \ar[rr]^{g_{K}} \ar[dd]_{k_{v}} \ar@{.>}[dr]^{h} &  & K_{w} \ar[dd]^{k_{w}} \ar@{.>}[dddddllll]^{d} \\  &  &  &  & Y ×_{Z} K_{w} \ar[dddlll]^{φ} \ar[ur]^{π_{2}} \ar[dl]_{π_{1}} \\  & X \ar[rr]^{f} \ar[dd]_{u} \ar[dr]_{f_{1}} &  & Y \ar[rr]^{g} \ar[dd]^{v} &  & Z \ar[dd]^{w} \ar[r] & 0 \\  &  & X′′ \ar@{.>}[dl]_{u′} \ar[ur]_{f_{2}} \ar@{.>}[uurr]^{θ} \\ 0 \ar[r] & X′ \ar[rr]_{f′} \ar[d]_{q_{u}} &  & Y′ \ar[rr]_{g′} \ar[d]^{q_{v}} &  & Z′ \ar[d]^{q_{w}} \\  & Q_{u} \ar[rr]_{f′_{Q}} \ar[d] &  & Q_{v} \ar[rr]_{g′_{Q}} \ar[d] &  & Q_{w} \ar[d] \\  & 0 &  & 0 &  & 0}

(3.4)

in which:

- All the horizontal and vertical arrows are as in diagram (3.1).

- π_{1} : Y ×_{Z} K_{w} → Y, π_{2} : Y ×_{Z} K_{w} → K_{w}, and φ : Y ×_{Z} K_{w} → X′ are defined as above, i.e. (Y ×_{Z} K_{w}, π_{1}, π_{2}) is the pullback of g and k_{w}, and φ : Y ×_{Z} K_{w} → X′ is the unique morphism with vπ_{1} = f′φ.

- f = f_{2}f_{1} where f_{1} : X → X′′ is a morphism in E and f_{2} : X′′ → Y is the kernel of g (such factorization of f does exist in C since the second row of diagram (3.1) is E-exact).

- Since (Y ×_{Z} K_{w}, π_{1}, π_{2}) is the pullback of g and k_{w}, there exists a unique morphism θ : X′′ → Y ×_{Z} K_{w} with π_{1}θ = f_{2} and π_{2}θ = 0. Since π_{2} is in E and f_{2} = ker(g), we conclude that π_{2} = coker(θ).

- Since f′ = ker(g′) and g′vf_{2} = 0, there exists a unique morphism u′ : X′′ → X′ with f′u′ = vf_{2}. It easily follows that u′f_{1} = u, q_{u} = coker(u′), and φθ = u′.

- h : K_{v} → Y ×_{Z} K_{w} is the canonical morphism and an easy diagram chase proves that h = ker(φ) (we do not need this fact now, but we shall need it below when proving (ii)).

Since q_{u}φθ = q_{u}u′ = 0 and π_{2} = coker(θ), there exists a unique morphism d : K_{w} → Q_{u} with q_{u}φ = dπ_{2}. We obtain the following factorization:

\xymatrix{ &  & Y ×_{Z} K_{w} \ar@{.>}[d]^{π_{2}} \ar[dl]_{1_{Y ×_{Z} Kw}} \ar[dr]^{φ} \\  & Y ×_{Z} K_{w} \ar[dl]_{π_{2}} & K_{w} \ar@{.>}[dll]^{1_{Kw}} \ar@{.>}[drr]_{d} & X′ \ar[dr]^{q_{u}} \\ K_{w} &  &  &  & Q_{u}}  (3.5)

Comparing diagrams (3.3) and (3.5), we conclude that the relation (K_{w}, 1_{K_{w}}, d) can be identified with the relation (R, r_{1}, r_{2}). Therefore, r_{1} is an isomorphism, proving that q_{u}f′^{∘}vg^{∘}k_{w} is a morphism in C.

(ii): To prove that the sequence (3.2) is E-exact, we need to prove that it is E-exact at K_{v}, K_{w}, Q_{u}, and Q_{v}. E-exactness at K_{v}: It follows from the fact that the first column of the diagram (3.4) is E-exact at X′, that the kernel of u′ exists in C. Indeed, consider the commutative diagram

\xymatrix{ & X′′ \ar@{.>}[dd]|{u′_{1}} \ar[dr]^{u′} \\ X \ar[dr]_{u_{1}} \ar[ur]^{f_{1}} \ar[rr]_{u} &  & X′ \ar[r]_{q_{u}} & Q_{u} \\  & M \ar[ur]_{u_{2}}}  (3.6)

where u = u_{2}u_{1} is the factorization of u with u_{2} = ker(q_{u}) and u_{1} ∈ E (which does exists since the first column of the diagram (3.4) is E exact at X′), and u′_{1} is the induced morphism. Since f_{1} and u_{1} are in E, u′_{1} is also in E, and therefore the kernel of u′_{1} exists in C. Since u_{2} is a monomorphism we conclude that Ker(u′) ≈ Ker(u′_{1}).

Consider the following part of the diagram (3.4)

\xymatrix{ & 0 \ar[d] &  & 0 \ar[d] & 0 \ar[d] \\  & K_{u} \ar[rr]^{f_{K}} \ar[dd]_{k_{u}} \ar@{.>}[dr]_{e} &  & K_{v} \ar[dd]^{k_{v}} \ar[r]^{g_{K}} & K_{w} \ar[dd]^{k_{w}} \\  &  & K_{u′} \ar@{.>}[ur]_{m} \ar[dd]^{k_{u′}} \\  & X \ar[rr]^{f} \ar[dd]_{u} \ar[dr]_{f_{1}} &  & Y \ar[dd]^{v} \ar[r]^{g} & Z \ar[dd]^{w} \ar[r] & 0 \\  &  & X′′ \ar[ur]_{f_{2}} \ar[dl]^{u′} \\ 0 \ar[r] & X′ \ar[rr]_{f′} &  & Y′ \ar[r]_{g′} & Z′}  (3.7)

in which:

- k_{u′} = ker(u′);

- Since k_{v} = ker(v) and vf_{2}k_{u′} = f′u′k_{u′} = 0, there exists a unique morphism m :

K_{u′} → K_{v} with k_{v}m = f_{2}k_{u′};

- Since k_{u′} = ker(u′) and u′f_{1}k_{u} = uk_{u} = 0, there exists a unique morphism e : K_{u} →

K_{u′} with k_{u′}e = f_{1}k_{u}; since k_{v} is a monomorphism, we conclude that me = f_{K}.

The E-exactness at K_{v} will be proved if we show that e ∈ E and m = ker(g_{K}). The latter, however, easily follows from Lemma 2.7. Indeed: by Lemma 2.7, the squares f_{1}k_{u} = k_{u′}e and f_{2}k_{u′} = k_{v}m are pullbacks; therefore, since f_{1} is in E we conclude that e also is in E, and since k_{w} is a monomorphism, by the same lemma m = ker(g_{K}). E-exactness at K_{w}: Consider the commutative diagram (3.4), we have: dg_{K} = dπ_{2}h = q_{u}φh = 0. To prove that the sequence (3.2) is E-exact at K_{w}, it suffices to prove that the kernel of d exists in C and that the induced morphism from K_{v} to the kernel of d is in E.

It easily follows from Lemma 2.4 that there exists a factorization d = d_{2}d_{1} where d_{2} is a monomorphism and d_{1} is in E. Indeed: since the second column of the Diagram (3.4) is E-exact at Y′, there exists a factorization v = v_{2}v_{1}, where v_{2} = ker(q_{v}) is a monomorphism and v_{1} is in E. Applying Lemma 2.4(ii) to the diagram

\xymatrix{Y ×_{Z} K_{w} \ar[d]_{π_{1}} \ar[r]^{φ} & X′ \ar[d]^{f′} \\ Y \ar[r]_{v} & Y′}

we conclude that φ = φ_{2}φ_{1} where φ_{2} is a monomorphism and φ_{1} is in E, and then applying Lemma 2.4(i) to the diagram

\xymatrix{Y ×_{Z} K_{w} \ar[d]_{π_{2}} \ar[r]^{φ} & X′ \ar[d]^{q_{u}} \\ K_{w} \ar[r]_{d} & Q_{u}}

we obtain the desired factorization of d. Since d_{1} is in E and d_{2} is a monomorphism, we conclude that the kernel of d exists in C (precisely, Ker(d) ≈ Ker(d_{1})). Let k_{d} : K_{d} → K_{w} be the kernel of d and let e_{d} : K_{v} → K_{d} be the induced unique morphism with k_{d}e_{d} = g_{K}, it remains to prove that e_{d} is in E. For, consider the commutative diagram

\xymatrix{ & K_{v} \ar[d]_{h} \ar[rr]^{e_{d}} &  & K_{d} \ar@{.>}[ddl]_{h′} \ar[d]^{k_{d}} \\ X′′ \ar[r]^{θ} \ar[dd]_{u′_{1}} & Y ×_{Z}K_{w} \ar[dd]_{φ} \ar@{.>}[dr]_{s} \ar[rr]^{π_{2}} &  & K_{w} \ar[dd]^{d} \\  &  & _{X′×_{Q_{u}}K_{w}} \ar@{-->}[dl]^{π′_{1}} \ar@{-->}[ur]_{π′_{2}} \\ K_{q_{u}} \ar[r]_{u_{2}} \ar@{.>}[urr]^{θ′} & X′ \ar[rr]_{q_{u}} &  & Q_{u}}

in which:

- π′_{1}, π′_{2} are the pullback projections, and s = ⟨φ, π_{2}⟩, θ′ = ⟨u_{2}, 0⟩, and h′ = ⟨0, k_{d}⟩ are the induced morphisms (the pullback (X′×_{Q_{u}}K_{w}, π′_{1}, π′_{2}) of q_{u} and d does exist

in C since q_{u} is in E); since q_{u} is in E so is π′_{2}.
- u′_{1} : X′′ → K_{q_{u}} and u_{2} : K_{q_{u}} → X′ are as in diagram (3.6).

- Since u_{2} = ker(q_{u}) and k_{d} = ker(d), we conclude that θ′ = ker(π′_{2}) and h′ = ker(π′_{1}).

Since θ = ker(π_{2}), θ′ = ker(π′_{2}), and the morphisms π_{2}, π′_{2}, and u′_{1} are in E, by Lemma 2.3, s : Y ×_{Z}K_{w} → X′×_{Q_{u}}K_{w} is also in E. Therefore, since h = ker(φ) and h′ = ker(π′_{1}), by Lemma 2.7 and condition 2.1(f) the morphism e_{d} : K_{v} → K_{d} is also in E, as desired. E-exactness at Q_{u}: Consider the commutative diagram (3.4), we have: f′_{Q}dπ_{2} = f′_{Q}q_{u}φ = q_{v}f′φ = q_{v}vπ_{1} = 0, and since π_{2} is an epimorphism we conclude that f′_{Q}d = 0. To prove that the sequence (3.2) is E-exact at Q_{u}, it suffices to prove that the kernel of f′_{Q} exists in C and that the induced morphism from K_{w} to the kernel of f′_{Q} is in E.

It easily follows from Lemma 2.4(i) that there exists a factorization f′_{Q} = f′_{Q_{2}}f′_{Q_{1}} where f′_{Q_{2}} is a monomorphism and f′_{Q_{1}} is in E. Indeed: since f′ is a monomorphism and E contains all isomorphisms, applying Lemma 2.4(i) to the diagram

\xymatrix{X′ \ar[r]^{f′} \ar[d]_{q_{u}} & Y′ \ar[d]^{q_{v}} \\ Q_{u} \ar[r]_{f′_{Q}} & Q_{v}}

we obtain the desired factorization of f′_{Q}; since f′_{Q_{1}} is in E and f′_{Q_{2}} is a monomorphism, we conclude that the kernel of f′_{Q} exists in C. Let k_{f′_{Q}} : K_{f′_{Q}} → Q_{u} be the kernel of f′_{Q} and let e_{f′_{Q}} : K_{w} → K_{f′_{Q}} be the induced unique morphism with e_{f′_{Q}}k_{f′_{Q}} = d, it remains to prove that e_{f′_{Q}} is in E. Since q_{u} is in E, the pullback (K ×_{Q_{u}} X′, p_{1}, p_{2}) of k_{f′_{Q}} and q_{u} exists in C and p_{1} is in E; therefore, we have the commutative diagram

\xymatrix{ &  & Y′′ \ar[d]^{v_{2}} \\ K ×_{Q_{u}} X′ \ar@{.>}[urr]^{p} \ar[d]_{p_{1}} \ar[r]^{p_{2}} & X′ \ar[r]^{f′} \ar[d]_{q_{u}} & Y′ \ar[d]^{q_{v}} \\ K_{f′_{Q}} \ar[r]_{k_{f′_{Q}}} & Q_{u} \ar[r]_{f′_{Q}} & Q_{v}}

in which v_{2} = ker(q_{v}) (recall, that since the second column of the diagram (3.4) is Eexact at Y′, we have v = v_{2}v_{1} where v_{1} ∈ E and v_{2} = ker(q_{v})) and p : K ×_{Q_{u}} X′ → Y′′ is the induced morphism. Using the fact that v_{2} and p_{2} are monomorphisms and that k_{f′_{Q}} = ker(f′_{Q}), an easy diagram chase proves that the square f′p_{2} = v_{2}p is the pullback of f′ and v_{2}.

Next, consider the commutative diagram

\xymatrix{Y \ar[d]_{v_{1}} \ar[dd]_{v} & Y ×_{Z}K_{w} \ar@{.>}[d]_{ψ} \ar[dd]_{φ} \ar[l]_{π_{1}} \ar[r]^{π_{2}} & K_{w} \ar[d]_{e_{f′_{Q}}} \ar[dd]^{d} \\ Y′′ \ar[d]_{v_{2}} & K ×_{Q_{u}} X′ \ar[l]^{p} \ar[r]_{p_{1}} \ar[d]_{p_{2}} & K_{f′_{Q}} \ar[d]_{k_{f′_{Q}}} \\ Y′ & X′ \ar[l]^{f′} \ar[r]_{q_{u}} & Q_{u}}

were ψ = ⟨φ, v_{1}π_{1}⟩; since k_{f′_{Q}} is a monomorphism and the equalities

k_{f′_{Q}}p_{1}ψ = q_{u}p_{2}ψ = q_{u}φ = dπ_{2} = k_{f′_{Q}}e_{f′_{Q}}π_{2}

hold, we conclude that p_{1}ψ = e_{f′_{Q}}π_{2}. Recall, that the square f′φ = vπ_{1} is a pullback (see the proof of (i)), therefore, (Y ×_{Z}K_{w}, π_{1}, ψ) is the pullback of p and v_{1}, yielding that ψ is in E. Since p_{1}ψ = e_{f′_{Q}}π_{2}, and ψ, p_{1}, and π_{2} are in E, we conclude that e_{f′_{Q}} is also in E, as desired. E-exactness at Q_{v}: Consider the commutative diagram (3.4). According to the assumptions of the theorem, we have g′ = g′_{2}g′_{1} were g′_{1} is a morphism in E and g′_{2} is a monomorphism. Then, by Lemma 2.4(i) there exists a factorization g′_{Q} = g′_{Q_{2}}g′_{Q_{1}} were g′_{Q_{1}} is a morphism in E and g′_{Q_{2}} is a monomorphism, hence, the kernel of g′_{Q} exists in C. Since g′_{Q}f′_{Q}q_{u} = q_{w}g′f′ = 0 and q_{u} is an epimorphism, we conclude that g′_{Q}f′_{Q} = 0, therefore, to prove that the sequence (3.2) is E-exact at Q_{v} it suffices to prove that the induced morphism from Q_{u} to the kernel of g′_{Q} is in E. For, consider the commutative diagram

\xymatrix{X \ar[rr]^{f} \ar[dd]_{u} &  & Y \ar[dd]_{v} \ar[rrrr]^{g} \ar[dr]^{v_{1}} &  &  &  & Z \ar[dd]^{w} \ar[dl]_{w_{1}} \\  &  &  & Ȳ \ar[dl]^{v_{2}} \ar@{.>}[rr]_{ȳ} &  & Z̄ \ar[dr]_{w_{2}} \ar@{.>}[ddl]_{z̄} \\ X′ \ar[dd]_{q_{u}} \ar@{.>}[dr]_{e_{1}} \ar[rr]^{f′} &  & Y′ \ar@{.>}[drr]_{e_{2}} \ar[dd]_{q_{v}} \ar[rrrr]^{g′} &  &  &  & Z′ \ar[dd]^{q_{w}} \\  & K_{g′_{Q}} \ar[dr]_{k_{g′_{Q}}} \ar@{.>}[rrr]_{f′′} &  &  & Q_{v}×_{Q_{w}}Z′ \ar[dll]^{p′_{1}} \ar@{-->}[urr]_{p′_{2}} \\ Q_{u} \ar[ur]_{e_{g′_{Q}}} \ar[rr]_{f′_{Q}} &  & Q_{v} \ar[rrrr]_{g′_{Q}} &  &  &  & Q_{w}}  (3.8)

in which:

- k_{g′_{Q}} = ker(g′_{Q}) and e_{g′_{Q}} : Q_{u} → K_{g′_{Q}} is the induced unique morphism with k_{g′_{Q}}e_{g′_{Q}} = f′_{Q}.

- (Q_{v}×_{Q_{w}}Z′, p′_{1}, p′_{2}) is the pullback of g′_{Q} and q_{w} (this pullback does exist since q_{w} is in E), and e_{2} = ⟨q_{v}, g′⟩ and f′′ = ⟨k_{g′_{Q}}, 0⟩ are the canonical morphisms; since k_{g′_{Q}} = ker(g′_{Q}) we conclude that f′′ = ker(p′_{2}).

- Since f′′ = ker(p′_{2}) and p′_{2}e_{2}f′ = 0 there exists a unique morphism e_{1} : X′′ → K_{g′_{Q}} with f′′e_{1} = e_{2}f′; it easily follows that e_{g′_{Q}}q_{u} = e_{1}.

- Since the second and the third columns of the diagram (3.8) are E-exact at Y′ and Z′ respectively, we have the factorizations v = v_{2}v_{1} and w = w_{2}w_{1}, where v_{1}, w_{1} ∈ E and v_{2} = ker(q_{v}) and w_{2} = ker(q_{w}).

- z̄ = ⟨0, w_{2}⟩, and since w_{2} = ker(q_{w}) we conclude that z̄ = ker(p′_{1}).

- Since z̄ = ker(p′_{1}) and p′_{1}e_{2}v_{2} = 0, there exists a unique morphism ȳ : Ȳ → Z̄ with

e_{2}v_{2} = z̄ȳ.

Since w_{1}, g, and v_{1} are in E, we conclude that ȳ is in E; therefore, by Lemma 2.3, e_{2} is also in E. Then, Lemma 2.7 implies that e_{1} is in E, and since e_{g′_{Q}}q_{u} = e_{1} we conclude that e_{g′_{Q}} is also in E, as desired.

References

[1] F. Borceux, D. Bourn, Mal’cev, protomodular, homological and semi-abelian categories, Mathematics and its Applications, Kluwer, 2004.

[2] A. Carboni, G. M. Kelly, and M. C. Pedicchio, Some remarks on Mal’cev and Goursat categories, Applied Categorical Structures 1, 1993, 385-421.

[3] T. Janelidze, Relative homological categories, Journal of Homotopy and Related Structures 1, 2006, 185-194.

[4] T. Janelidze, Incomplete relative semi-abelian categories, Applied Categorical Structures, available online from March 2009.

[5] S. Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, 1971.

[6] S. Mac Lane, An algebra of additive relations, Proc. Nat. Acad. Sci. USA 47, 1961, 1043-1051.

Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa

Email: tamar.janelidze@uct.ac.za

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