Theory and Applications of Categories, Vol. 30, No. 12, 2015, pp. 410–432.

NORMALIZERS, CENTRALIZERS AND ACTION ACCESSIBILITY

J. R. A. GRAY

Abstract. We give several reformulations of action accessibility in the sense of D. Bourn and G. Janelidze. In particular we prove that a pointed exact protomodular category is action accessible if and only if for each normal monomorphism κ : X → A the normalizer of ⟨κ, κ⟩ : X → A × A exists. This clarifies the connection between normalizers and action accessible categories established in a joint paper of D. Bourn and the author, in which it is proved that for pointed exact protomodular categories the existence of normalizers implies action accessibility. In addition we prove a pointed exact protomodular category with coequalizers is action accessible if centralizers of normal monomorphisms exist, and the normality of unions holds.

1. Introduction

Recall for a pointed category ℂ a split extension is a diagram

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

where κ is the kernel of α and αβ = 1_{B}. A morphism of split extensions is a diagram

\xymatrix{X \ar[r]^{κ} \ar[d]_{f} & A \ar[r]^{α} \ar[d]^{g} & B \ar[l]^{β} \ar[d]^{h} \\ X′ \ar[r]^{κ′} & A′ \ar[r]^{α′} & B′ \ar[l]^{β′}}

where the top and bottom are split extensions (the domain and codomain respectively), κ′f = gκ, α′g = hα, and β′h = gβ. We will denote by SPLEXT(ℂ) the category of split extensions, and by SPLEXT_{X}(ℂ) the category with objects those split extensions with kernel X, and with morphisms those morphisms of split extensions where the morphism between their kernels is 1_{X}. An extension

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

is said to be faithful if it is a sub-terminal object in SPLEXT_{X}(ℂ), that is, there is at most one morphism from any object to it in this category. A pointed protomodular

Received by the editors 2014-05-27 and, in revised form, 2015-03-26. Transmitted by Walter Tholen. Published on 2015-04-03. 2010 Mathematics Subject Classification: 18D35, 18A35, 18A05, 18A99. Key words and phrases: action accessible, protomodular, Barr exact, normality, centrality, normal-

izer, centralizer.

\begin{matrix}c \\ ◯ \end{matrix} J. R. A. Gray, 2015. Permission to copy for private use granted.

category ℂ is a called action accessible [7] if for each X the category SPLEXT_{X}(ℂ) has enough sub-terminal objects, that is, each object admits a morphism into a sub-terminal object.

It is well known that each split extension of groups

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

is determined by a morphism

B → Aut(X).

This can be understood categorically as the fact that when ℂ is the category of groups, the category SPLEXT_{X}(ℂ) has a terminal object. Semi-abelian categories which satisfy this property are called action representative and were introduced and studied in [2] (see also [3]) by F. Borceux, G. Janelidze and G. M. Kelly. Action accessible categories were introduced by D. Bourn and G. Janelidze in [7] as a weakening of action representable categories so as to include the category of rings as an example but still to allow, amongst other things, centralizers of equivalence relations to be constructed in a similar way as in the category of groups.

In [4] D. Bourn studied further how the existence of centralizers of equivalence relations is related to the concept of action accessibility (as well as groupoid accessibility). In particular it was shown that a split extension

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

in an action accessible category is faithful if and only if the inverse image of the centralizer of the kernel pair of α along β is indiscrete.

A. S. Cigoli and S. Mantovani showed in [8] that a pointed exact protomodular category ℂ is action accessible if and only if what they called non-symmetric centralizers exist in ℂ. In addition in a talk by A. S. Cigoli on their joint work, it was explained that under certain conditions a category ℂ is action accessible provided that certain centralizers of subobjects exist and satisfy certain properties (see Theorem 2.15 and the paragraph before it).

In Section 2 we recall many of these results relating to the existence of centralizers of subobjects and add other equivalent formulations (see Theorem 2.15). In addition we show that a pointed exact protomodular category with coequalizers is action accessible if it has centralizers of normal monomorphisms which are normal and the normality of unions holds in ℂ. Recall from [3] that the normality of unions holds in ℂ if for any subobjects A, B and C of an object D, if A is normal in both B and C, then it is normal in the join of B and C (in D).

Let us denote by K the functor sending a split extension in SPLEXT(ℂ) to its kernel. Recall that the normalizer [9] of a monomorphism is defined as the terminal object in the category of factorizations as a normal monomorphism (i.e. a kernel) followed by a monomorphism i.e. the normalizer of S ≤ X is the largest subobject of X in which S is normal. D. Bourn together with the author introduced in [6] another notion of normalizer (using a different notion of normal) which can be defined for a finitely complete category and which coincides with the definition given above in a pointed exact protomodular category. In the same paper it was shown that ℂ has normalizers in the sense of [6], and hence in the sense defined above (when ℂ is pointed exact protomodular), if and only if for each split extension

\xymatrix{X′ \ar[r]^{κ′} & A′ \ar[r]^{α′} & B′ \ar[l]^{β′}}  (1)

and for each monomorphism f : X → X′ there exists a K-precartesian lifting of f to (1) . Such a K-precartesian lifting can be seen to be the terminal object in the following subcategory of the category of morphisms of SPLEXT(ℂ). The objects are morphisms of split extensions of the form

\xymatrix{X \ar[r]^{κ} \ar[d]_{f} & A \ar[r]^{α} \ar[d]^{g} & B \ar[l]^{β} \ar[d]^{h} \\ X′ \ar[r]^{κ′} & A′ \ar[r]^{α′} & B′; \ar[l]^{β′}}

and morphisms are morphisms of the form

[figure: X; κ_{1}; f; A_{1}; α_{1}; g_{1}; θ; B_{1}; β_{1}; h_{1}; ϕ; X; κ_{2}; f; A_{2}; α_{2}; g_{2}; B_{2}; β_{2}; h_{2}; X′; κ′; A′; α′; B′; β′; X′; κ′; A′; α′; B′.; β′]

As mentioned above, in the paper [8], A. S. Cigoli and S. Mantovani showed that a pointed exact protomodular category is action accessible if and only if the category has what they called non-symmetric centralizers. In the context of a pointed exact protomodular action accessible category ℂ it can be seen that an equivalence relation r_{1}, r_{2} : R → A with common section s : A → R has a non-symmetric centralizer if and only if the category of relations on

\xymatrix{X \ar[r]^{k} & R \ar[r]^{r_{1}} & A \ar[l]^{s}}

in SPLEXT_{X}(ℂ) has a terminal object (this is essentially Proposition 4.1 of [8]). Using the isomorphism of categories between subobjects of products and relations (considered as parallel pairs) it can be seen that the existence of such a terminal object is equivalent to the existence of a K-precartesian lifting of ⟨1, 1⟩ : X → X × X to

\xymatrix{X × X \ar[r]^{k×k} & R × R \ar[r]^{r_{1}×r_{1}} & A × A. \ar[l]^{s×s}}

From this observation and what was recalled previously it follows that a pointed exact protomodular category with normalizers is action accessible [6]. In Section 3 of this paper we show that a pointed exact protomodular category is action accessible if and only if each normal monomorphism composed with the diagonal morphism has a normalizer (see Theorem 3.1). The proof of the fact that the existence of normalizers implies action accessibility given in [6] was different from what was described above and made use of a simple property studied by D. Bourn in [5] which is equivalent to the existence of centralizers for Mal’tsev categories.

2. Action accessibility, centralizers and the normality of unions

In this section we recall various results from the papers [8] and [4] in order to give reformulations of action accessibility. Note that a diagram

\xymatrix{C \ar[r]^{γ} \ar[d]_{p} & D \ar[l]^{δ} \ar[d]^{q} \\ A \ar[r]^{α} & B \ar[l]^{β}}

where γδ = 1_{D}, αβ = 1_{B}, qγ = αp and βq = pδ will be called a split pullback if the diagram consisting of rightward and downward directed arrows is a pullback. Recall that a pair of morphisms f : A → C and g : B → C commute in a (weakly) unital category ℂ [12, 1] if there exists a (necessarily unique) morphism φ : A×B → C, where (A×B, π_{1}, π_{2}) is the product of A and B, making the diagram

\xymatrix{A \ar[r]^{⟨1,0⟩} \ar[dr]_{f} & A × B \ar[d]^{φ} & B \ar[l]_{⟨0,1⟩} \ar[dl]^{g} \\  & C,}

in which ⟨1, 0⟩ and ⟨0, 1⟩ are the unique morphisms such that π_{1}⟨1, 0⟩ = 1_{A}, π_{2}⟨1, 0⟩ = 0, π_{1}⟨0, 1⟩ = 0 and π_{2}⟨0, 1⟩ = 1, commute.

2.1. Lemma. [8], Lemma 2.3. Let ℂ be a pointed protomodular category and let

\xymatrix{X \ar[r]^{κ} \ar[d]_{1_{X}} & A \ar[r]^{α} \ar[d]^{g} & B \ar[l]^{β} \ar[d]^{h} \\ X \ar[r]^{κ′} & A′ \ar[r]^{α′} & B′, \ar[l]^{β′}}

be a morphism of split extensions. For each morphism s′ : S′ → B′ such that β′s′ commutes with κ′ the morphism s : S → B obtained from the pullback

\xymatrix{S \ar[d]_{i} \ar[r]^{s} & B \ar[d]^{h} \\ S′ \ar[r]_{s′} & B′}

is such that βs commutes with κ.

2.2. Lemma. [8], Lemma 2.4, Proposition 2.5. Let ℂ be a pointed protomodular category. For each morphism of split extensions

\xymatrix{X \ar[r]^{κ} \ar[d]_{1_{X}} & A \ar[r]^{α} \ar[d]^{g} & B \ar[l]^{β} \ar[d]^{h} \\ X \ar[r]^{κ′} & A′ \ar[r]^{α′} & B′ \ar[l]^{β′}}

the kernel of h, ker(h) : Ker(h) → B has the properties:

(a) βker(h) commutes with κ;

(b) βker(h) is a normal monomorphism.

When the codomain of the morphism is faithful then ker(h) has the additional property:

(c) if s : S → B is a morphism such that βs commutes with κ, then there exists a unique morphism \overline{s} such that s = ker(h)\overline{s}.

2.3. Remark. The above proofs hold for morphisms of split extensions in an arbitrary pointed category with finite limits provided the square on the right is a split pullback.

2.4. Definition. For any morphisms f : A → C and g : B → C the centralizer of f relative to g is the morphism z_{f,g} : Z_{f,g} → B with the properties:

(a) gz_{f,g} commutes with f;

(b) for each morphism s : S → B such that gs commutes with f there exists a unique morphism \overline{s} such that z_{f,g}\overline{s} = s.

The centralizer of f relative to 1_{C} will be denoted by z_{f} (rather than z_{f,1_{C}}) and called the centralizer of f.

2.5. Corollary. [8], Proposition 2.5. Let ℂ be a pointed protomodular action accessible category. For each split extension

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

the centralizer z_{κ,β} of κ relative to β exists, and βz_{κ,β} is normal.

Recall that a monomorphism is called protosplit if it is normal and its cokernel is a split epimorphism [3]. Using the same construction as in [8] we obtain:

2.6. Proposition. Let ℂ be a unital category with cokernels. The following are equivalent:

(a) for each split extension

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

the centralizer of κ relative to β exists;

(b) ℂ has centralizers of normal monomorphisms;

(c) ℂ has centralizers of protosplit monomorphisms.

Proof. The implication (b) ⇒ (c) follows from the fact that each protosplit monomorphism is a normal monomorphism.

(a) ⇒ (b) : Let n : N → X be a normal monomorphism and consider the morphism of split extensions

\xymatrix{N \ar[r]^{k} \ar[d]_{n} & R \ar[r]^{r_{1}} \ar[d]^{⟨r_{1},r_{2}⟩} & X \ar[l]^{s} \ar[d]^{1_{X}} \\ X \ar[r]^{⟨0,1⟩} & X × X \ar[r]^{π_{1}} & X \ar[l]^{⟨1,1⟩}}

determined by its denormalization (the kernel pair of the cokernel of n). Since 0 commutes with any morphism and ⟨r_{1}, r_{2}⟩ is a monomorphism it follows (see e.g. [1]) that for each morphism u : U → X, su commutes with k if and only if n commutes with u. It follows that the morphism z_{k,s} : Z_{k,s} → X is the centralizer of n.

(c) ⇒ (a) : For each split extension

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

it is easy to check that t obtained by the pullback

\xymatrix{T \ar[d]_{t} \ar[r]^{\overline{β}} & Z_{κ} \ar[d]^{z_{κ}} \\ B \ar[r]_{β} & A}

is the centralizer of κ relative to β.

2.7. Corollary. Let ℂ be a unital category with cokernels. The following are equivalent:

(a) for each split extension

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

the centralizer of κ relative to β exists and the composite βz_{κ,β} is normal; (b) ℂ has centralizers of normal monomorphisms which are normal and have the property

that if their cokernel splits, then the intersection with each splitting is normal.

(c) ℂ has centralizers of protosplit monomorphisms which are normal and have the property that their intersection with each splitting of their cokernel is normal.

Proof. The proof of Proposition 2.6 can easily be extended to prove that the above statements are equivalent.

The existence of centralizers of normal monomorphisms which are normal was proved for a homological action accessible categories in [7].

2.8. Corollary. Let ℂ be a homological action accessible category with cokernels. Centralizers of normal monomorphisms exist, are normal, and have the property that if their cokernel splits, then the intersection with each splitting is normal.

Proof. The proof follows from Corollary 2.5 and Corollary 2.7

Recall that in a category with finite limits, for each diagram

\xymatrix{D \ar[r]^{δ} \ar[d]_{p} & E \ar[l]^{ϵ} \ar[d]^{q} \\ A \ar[r]^{α} & B \ar[l]^{β}}

in which p and q are regular epimorphisms, αβ = 1_{B}, δϵ = 1_{E}, pϵ = βq and qδ = αp, the diagram

\xymatrix{D \ar[r]^{δ} \ar[d]_{p} & E \ar[d]^{q} \\ A \ar[r]^{α} & B}

is a pushout (since the induced morphism between the kernel pair of p and the kernel pair of q is a (split) epimorphism.)

2.9. Lemma. Let ℂ be a regular subtractive category [10]. Each morphism

\xymatrix{X′ \ar[r]^{κ′} \ar[d]_{f} & A′ \ar[r]^{α′} \ar[d]^{g} & B′ \ar[l]^{β′} \ar[d]^{h} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

can be decomposed as a composite

\xymatrix{X′ \ar[r]^{κ′} \ar[d]_{p} & A′ \ar[r]^{α′} \ar[d]^{q} & B′ \ar[l]^{β′} \ar[d]^{r} \\ Y \ar[r]^{σ} \ar[d]_{l} & C \ar[r]^{γ} \ar[d]^{m} & D \ar[l]^{δ} \ar[d]^{n} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

where p, q and r are regular epimorphisms, and l, m and n are monomorphisms.

Proof. Let

\xymatrix{X \ar[r]^{κ′} \ar[d]_{f} & A′ \ar[r]^{α′} \ar[d]^{g} & B′ \ar[l]^{β′} \ar[d]^{h} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

be a morphism of split extensions. Consider the diagram

\xymatrix{X′ \ar[r]^{κ′} \ar[d]_{p} & A′ \ar[r]^{α′} \ar[d]^{q} & B′ \ar[l]^{β′} \ar[d]^{r} \\ Y \ar[r]^{σ} \ar[d]_{l} & C \ar[r]^{γ} \ar[d]^{m} & D \ar[l]^{δ} \ar[d]^{n} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}  (*)

where each of the vertical morphisms from left to right is the factorization as a regular epimorphism followed by a monomorphism of f, g and h respectively, and σ ,γ and δ are the induced morphisms between them. The top part of the diagram can be extended to a 3x3 lemma diagram (with zeros omitted)

\xymatrix{Ker(p) \ar[r]^{\overline{κ}} \ar[d]_{ker(p)} & Ker(q) \ar[r]^{\overline{α}} \ar[d]^{ker(q)} & Ker(r) \ar[l]^{\overline{β}} \ar[d]^{ker(r)} \\ X′ \ar[r]^{κ′} \ar[d]_{p} & A′ \ar[r]^{α′} \ar[d]^{q} & B′ \ar[l]^{β′} \ar[d]^{r} \\ Y \ar[r]^{σ} & C \ar[r]^{γ} & D \ar[l]^{δ}}

where the top two lines and all columns are exact (in the sense of Z. Janelidze in [11]). It follows [11] that the bottom row is exact and since σ is a monomorphism that the bottom row is a split extension. Therefore, the diagram (*) gives the desired decomposition.

Following Z. Janelidze in [11] we will call a category normal if it is pointed, regular, and each regular epimorphism is normal. The following result is a generalization for normal subtractive categories of Corollary 2.8 of [8], we omit the proof as it essentially the same.

2.10. Corollary. Let ℂ be a normal subtractive action accessible category (where we have dropped the pointed protomodularity requirement in the definition of action accessible category). For each split extension

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

There exists a morphism of split extensions

\xymatrix{X \ar[r]^{κ} \ar[d]_{1_{X}} & A \ar[r]^{α} \ar[d]^{q} & B \ar[l]^{β} \ar[d]^{r} \\ X \ar[r]^{σ} & C \ar[r]^{γ} & D \ar[l]^{δ}}

with codomain faithful, and with q and r normal epimorphisms.

It easily follows from the definition of an eccentric extension in [5] that a split extension

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

in a pointed protomodular category is eccentric if and only if z_{κ,β} = 0. In this paper we will use this as the definition of an eccentric extension. The following proposition is essentially the same as [4] Corollary 4.1 we omit the proof.

2.11. Proposition. Let ℂ be a homological action accessible category. Eccentric extensions are faithful.

Recall from [1]

2.12. Proposition. Let ℂ be a pointed protomodular category and let f : A → C and g : B → C be normal monomorphisms in ℂ. If the intersection A ∩ B is trivial then f and g commute.

2.13. Proposition. Let ℂ be a pointed protomodular category. For each split pullback

\xymatrix{A ×_{B} B′ \ar[d]_{π_{1}} \ar[r]^{π_{2}} & B′ \ar[l]^{⟨βh,1⟩} \ar[d]^{h} \\ A \ar[r]^{α} & B \ar[l]^{β}}

where h is a normal monomorphism, ⟨βh, 1⟩ is normal if and only if βh commutes with the kernel of α Proof. Let κ : X → A be the kernel of α. It follows that ⟨κ, 0⟩ : X → A ×_{B} B′ is the kernel of π_{2} : A ×_{B} B′ → B′. Suppose that ⟨βh, 1⟩ is a normal monomorphism. Since by Proposition 2.12 the morphisms ⟨κ, 0⟩ and ⟨βh, 1⟩ commute, it follows that κ = π_{1}⟨κ, 0⟩ and βh = π_{1}⟨βh, 1⟩ commute. Conversely suppose that βh commutes with κ and that φ is the morphism showing they commute. Since the diagram

\xymatrix{X \ar[r]^{⟨1,0⟩} \ar[d]_{1_{X}} & X × B′ \ar[r]^{π_{2}} \ar[d]^{φ} & B′ \ar[l]^{⟨0,1⟩} \ar[d]^{h} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

is a morphism of split extensions and ℂ is protomodular it follows that the square on the right is a split pullback. It easily follows that ⟨βh, 1⟩ is a normal monomorphism.

Next we show that a weak form of normality of unions holds in any homological action accessible category.

2.14. Proposition. Let ℂ be a homological action accessible category. For each split pullback

\xymatrix{A ×_{B} B′ \ar[d]_{π_{1}} \ar[r]^{π_{2}} & B′ \ar[l]^{⟨βh,1⟩} \ar[d]^{h} \\ A \ar[r]^{α} & B \ar[l]^{β}}

if h and ⟨βh, 1⟩ are normal, then βh is normal.

Proof. Let κ : X → A be the kernel of α. Since by Proposition 2.13 it follows that κ and βh commute and since by Corollary 2.5 z_{κ,β} exists, it follows that there exists a unique morphism h′ : B′ → Z_{κ,β} such that h = z_{κ,β}h′. By Corollary 2.10 and Corollary 2.5 it follows that there exists a morphism

\xymatrix{X \ar[r]^{κ} \ar[d]_{1_{X}} & A \ar[r]^{α} \ar[d]^{q} & B \ar[l]^{β} \ar[d]^{r} \\ X \ar[r]^{σ} & C \ar[r]^{γ} & D \ar[l]^{δ}}

with codomain faithful, and with q and r normal epimorphisms, such that the kernel of r is z_{κ,β}. Consider the diagram

\xymatrix{A \ar[r]^{α} \ar[dd]_{q} \ar[d]^{⟨q,eα⟩} & B \ar[l]^{β} \ar[dd]^{r} \ar[d]_{e} \\ C ×_{D} E \ar[r]^{π_{2}} \ar[d]^{π_{1}} & E \ar[l]^{⟨δf,1⟩} \ar[d]_{f} \\ C \ar[r]^{γ} & D \ar[l]^{δ}}

in which e is the cokernel of h, and f is the unique morphism such that fe = r (which exists since rh = rz_{κ,β}h′ = 0h′ = 0) . Since the outer arrows of the diagram above form a split pullback and the bottom square is a split pullback it follows that the top square is also a split pullback, this means that βh is the kernel of ⟨q, eα⟩ and is therefore normal as required.

As pointed out to me by D. Bourn, the fact that the conditions (a) and (b) of the following theorem are equivalent follows from the results in [4], the same fact appeared in a talk by A. S. Cigoli on joint work with S. Mantovani entitled Action accessibility and centralizers at the CT conference in 2010.

2.15. Theorem. Let ℂ be a homological category. The following are equivalent:

(a) ℂ is action accessible;

(b) (i) for each split extension

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

the centralizer of κ relative to β exists and βz_{κ,β} is a normal monomorphism; (ii) Eccentric extensions are faithful;

(c) (i) ℂ has centralizers of normal monomorphisms;

(ii) the intersection of the centralizer of a protosplit monomorphism and any splitting

of its cokernel is normal; (iii) Eccentric extensions are faithful;

(d) (i) ℂ has centralizers of normal monomorphisms which are normal; (ii) for each split pullback

\xymatrix{A ×_{B} B′ \ar[d]_{π_{1}} \ar[r]^{π_{2}} & B′ \ar[l]^{⟨βh,1⟩} \ar[d]^{h} \\ A \ar[r]^{α} & B \ar[l]^{β}}

if h and ⟨βh, 1⟩ are normal, then βh is normal;

(iii) Eccentric extensions are faithful.

Proof. The equivalence of (b) and (c) follows from Corollary 2.7. The implication (a) ⇒ (d) follows from Corollary 2.8 and Propositions 2.11 and 2.14. The implication (d) ⇒ (b) follows from Corollary 2.7 and Proposition 2.13. To complete the proof we will show that (b) ⇒ (a). Let

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

be a split extension. Consider the diagram

\xymatrix{Z_{κ,β} \ar[r]^{1_{Z_{κ,β}}} \ar[d]_{βz_{κ,β}} & Z_{κ,β} \ar[l]^{1_{Z_{κ,β}}} \ar[d]^{z_{κ,β}} \\ A \ar[r]^{α} \ar[d]_{q} & B \ar[l]^{β} \ar[d]^{r} \\ C \ar[r]^{γ} & D \ar[l]^{δ}}

in which r and q are the cokernels of z_{κ,β} and βz_{κ,β} respectively and γ and δ are the induced morphisms between them. Since βz_{κ,β} is a normal monomorphism it follows that the square at the bottom is a split pullback and so can be completed as a morphism of split extensions

\xymatrix{X \ar[r]^{κ} \ar[d]_{1_{X}} & A \ar[r]^{α} \ar[d]^{q} & B \ar[l]^{β} \ar[d]^{r} \\ X \ar[r]^{σ} & C \ar[r]^{γ} & D. \ar[l]^{δ}}

Since in the diagram

\xymatrix{Z_{σ,δ} ×_{D} B \ar[d]_{π_{1}} \ar[r]^{π_{2}} & B \ar[d]_{r} \\ Z_{σ,δ} \ar[r]_{z_{σ,δ}} & D}

by Lemma 2.1, βπ_{2} commutes with κ, it follows by the universal property of z_{κ,β} that there exists a unique morphism t : Z_{σ,δ} ×_{D} B → Z_{κ,β} such that π_{2} = z_{κ,β}t. Since z_{σ,δ}π_{1} = rπ_{2} = rz_{κ,β}t = 0t = 0 = 0π_{1} and since π_{1} is a (regular) epimorphism (being the pullback of a regular epimorphism), it follows that z_{σ,δ} = 0 and so by assumption the split extension

\xymatrix{X \ar[r]^{σ} & C \ar[r]^{γ} & D \ar[l]^{δ}}

is faithful, and every split extension admits a morphism into a faithful split extension as required.

The following result is a generalization for subtractive categories of Lemma 3.1 of [8] we omit the proof:

2.16. Lemma. Let ℂ be a regular subtractive category. Each parallel pair of morphisms

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{κ′} & A′ \ar[d]_{g} \ar[d]^{g′} \ar[r]^{α′} & B′ \ar[d]_{h} \ar[d]^{h′} \ar[l]^{β′} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

can be decomposed as a composite

\xymatrix{X \ar[r]^{κ′} \ar[d]_{1_{X}} & A′ \ar[r]^{α′} \ar[d]^{e} & B′ \ar[l]^{β′} \ar[d]^{f} \\ X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{\overline{κ}} & \overline{A} \ar[d]_{m} \ar[d]^{m′} \ar[r]^{\overline{α}} & \overline{B} \ar[d]_{n} \ar[d]^{n′} \ar[l]^{\overline{β}} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

where e and f are regular epimorphisms, and the pairs m and m′, and n and n′ are jointly monomorphic.

2.17. Corollary. Let ℂ be a regular subtractive category. A split extension

\xymatrix{K \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

is faithful if and only if for each parallel pair of morphisms

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{κ′} & A′ \ar[d]_{g} \ar[d]^{g′} \ar[r]^{α′} & B′ \ar[d]_{h} \ar[d]^{h′} \ar[l]^{β′} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

where the pairs g and g′, and h and h′ are jointly monomorphic, g = g′ and h = h′.

2.18. Proposition. Let ℂ be a pointed exact protomodular category with coequalizers in which the normality of unions holds. Eccentric extensions are faithful.

Proof. Let

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

be an eccentric extension. It follows from Corollary 2.17 and protomodularity that it is sufficient to show for each pair of jointly monomorphic morphisms of split extensions

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{κ′} & A′ \ar[d]_{g} \ar[d]^{g′} \ar[r]^{α′} & B′ \ar[d]_{h} \ar[d]^{h′} \ar[l]^{β′} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B. \ar[l]^{β}}

that h = h′. Consider the diagram

\xymatrix{X \ar[r]^{κ′} \ar[d]_{κ} & A′ \ar[ddr]^{⟨g,g′⟩} \ar[d]^{u} \\ A \ar[r]^{e} \ar[drr]_{⟨1,1⟩} & R \ar[dr]_{⟨r_{1},r_{2}⟩} \\  &  & A × A}

in which r_{1}, r_{2} : R → A is the kernel pair of the coequalizer of g, g′ : A′ → A, and e and u are the unique morphisms such that r_{1}e = 1_{A} and r_{2}e = 1_{A}, and r_{1}u = g and r_{2}u = g′. Since e and u are jointly extremal-epimorphic (indeed if they factor through some monomorphism m : S → R, then it will make S a reflexive relation and hence an effective equivalence relation contained in R and containing A′), it follows by the normality of unions k = eκ = uκ′ is a normal monomorphism. Since in the pullback

\xymatrix{Y \ar[r]^{λ} \ar[d]_{σ} & R \ar[d]^{⟨r_{1},r_{2}⟩} \\ A \ar[r]_{⟨0,1⟩} & A × A}

λ is a normal monomorphism with intersection with k equal to 0, it follows by Proposition 2.12 that k and λ commute. Therefore since (β × β)⟨0, 1⟩ = ⟨0, 1⟩β it follows that in the diagram

\xymatrix{Z \ar[r]^{μ} \ar[d]_{η} & S \ar[d]^{⟨s_{1},s_{2}⟩} \ar[r]^{\overline{β}} & R \ar[d]^{⟨r_{1},r_{2}⟩} \\ B \ar[r]_{⟨0,1⟩} & B × B \ar[r]_{β×β} & A × A}

where both the left and right hand squares are pullbacks, \overline{β}μ commutes with k. It follows that βη = r_{2}\overline{β}μ commutes with κ = r_{2}k and therefore by assumption η = 0. Since ⟨s_{1}, s_{2}⟩ : S → B × B is the preimage of an equivalence relation it is also an equivalence relation, and since η = 0 is the normalization of ⟨s_{1}, s_{2}⟩ : S → B × B it follows that s_{1} = s_{2}. Therefore the unique morphism v making the diagram

\xymatrix{B′ \ar[dd]_{⟨h,h′⟩} \ar[r]^{β′} \ar@{-->}[d]^{v} & A′ \ar[d]_{u} \ar[dd]^{⟨g,g′⟩} \\ S \ar[d]^{⟨s_{1},s_{2}⟩} \ar[r]^{\overline{β}} & R \ar[d]_{⟨r_{1},r_{2}⟩} \\ B × B \ar[r]_{β×β} & A × A}

commute, forces h = s_{1}v = s_{2}v = h′ as required.

2.19. Theorem. Let ℂ be a pointed exact protomodular category with coequalizers. If ℂ satisfies the normality of unions and has centralizers of normal monomorphisms which are normal, then ℂ is action accessible.

Proof. The proof follows from Theorem 2.15 and Proposition 2.18 since the diagram consisting of leftward and downward directed arrows in Condition (d)(ii) of Theorem 2.15 is a union by protomodularity.

3. Action accessibility and normalizers

It was shown in [6] that for a pointed exact protomodular category action accessibility follows from the existence of normalizers. In this section we show that for a pointed exact protomodular category action accessibility is equivalent to the existence of certain normalizers.

3.1. Theorem. Let ℂ be a pointed exact protomodular category. The following are equivalent:

(a) ℂ is action accessible;

(b) for each normal monomorphism κ : X → A the normalizer of ⟨κ, κ⟩ exists;

(c) for each protosplit monomorphism κ : X → A the normalizer of ⟨κ, κ⟩ exists;

(d) for each split extension

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

the category of relations in SPLEXT_{X}(ℂ) on

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

has a terminal object;

NORMALIZERS, CENTRALIZERS AND ACTION ACCESSIBILITY  425

(e) for each equivalence relation

\xymatrix{R \ar[r]^{r_{1}} \ar[r]_{r_{2}} & A \ar[l]|{s}}

the category of relations in SPLEXT_{X}(ℂ) on

\xymatrix{X \ar[r]^{k} & R \ar[r]^{r_{1}} & A, \ar[l]^{s}}

where k is the kernel of r_{1}, has a terminal object;

(f) for each split extension

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

the category of parallel morphisms in SPLEXT_{X}(ℂ) with codomain

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

has a terminal object;

(g) for each equivalence relation

\xymatrix{R \ar[r]^{r_{1}} \ar[r]_{r_{2}} & A \ar[l]|{s}}

the category of parallel morphisms in SPLEXT_{X}(ℂ) with codomain

\xymatrix{X \ar[r]^{k} & R \ar[r]^{r_{1}} & A, \ar[l]^{s}}

where k is the kernel of r_{1}, has a terminal object;

(h) for each split extension

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

there is a K-precartesian lifting of ⟨1, 1⟩ : X → X × X to

\xymatrix{X × X \ar[r]^{κ×κ} & A × A \ar[r]^{α×α} & B × B. \ar[l]^{β×β}}

(i) for each equivalence relation

\xymatrix{R \ar[r]^{r_{1}} \ar[r]_{r_{2}} & A \ar[l]|{s}}

there is a K-precartesian lifting of ⟨1, 1⟩ : X → X × X to

\xymatrix{X × X \ar[r]^{k×k} & R × R \ar[r]^{r_{1}×r_{1}} & A × A, \ar[l]^{s×s}}

where k is the kernel of r_{1}.

Proof. The implications (b) ⇒ (c), (d) ⇒ (e), (f) ⇒ (g), and (h) ⇒ (i) follow trivially, and the implication (c) ⇒ (h) follows from Proposition 2.4 and Lemma 2.7 in [6]. The implications (d) ⇔ (f) and (e) ⇔ (g) follow from Lemma 2.16, and the implications (f) ⇔ (h) and (g) ⇔ (i) follow from the fact that for any category with binary products there is an isomorphism of categories between relations on an object and monomorphisms into the product of that object with itself. Therefore the proof will be completed if we show that (a) ⇒ (f), (d) ⇒ (a), and (i) ⇒ (b).

(a) ⇒ (f): Let

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

be a split extension. Since ℂ is action accessible there exists a morphism

\xymatrix{X \ar[r]^{κ} \ar[d]_{1_{X}} & A \ar[r]^{α} \ar[d]^{g} & B \ar[l]^{β} \ar[d]^{h} \\ X \ar[r]^{\overline{κ}} & \overline{A} \ar[r]^{\overline{α}} & \overline{B} \ar[l]^{\overline{β}}}

with codomain a faithful extension. Consider the diagram

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{κ̃} & R \ar[d]_{r_{1}} \ar[d]^{r_{2}} \ar[r]^{α̃} & S \ar[d]_{s_{1}} \ar[d]^{s_{2}} \ar[l]^{β̃} \\ X \ar[r]^{κ} \ar[d]_{1_{X}} & A \ar[r]^{α} \ar[d]^{g} & B \ar[l]^{β} \ar[d]^{h} \\ X \ar[r]^{\overline{κ}} & \overline{A} \ar[r]^{\overline{α}} & \overline{B} \ar[l]^{\overline{β}}}

in which (S, s_{1}, s_{2}) and (R, r_{1}, r_{2}) are the kernel pair of h and g respectively, α̃ and β̃ are the induced morphisms between the kernel pairs, and κ̃ is the kernel of α̃. Let

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{κ′} & A′ \ar[d]_{g_{1}} \ar[d]^{g_{2}} \ar[r]^{α′} & B′ \ar[d]_{h_{1}} \ar[d]^{h_{2}} \ar[l]^{β′} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

be a parallel pair of morphism. Since the extension

\xymatrix{X \ar[r]^{\overline{κ}} & \overline{A} \ar[r]^{\overline{α}} & \overline{B} \ar[l]^{\overline{β}}}

is faithful it follows that in the diagram

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{κ′} & A′ \ar[d]_{g_{1}} \ar[d]^{g_{2}} \ar[r]^{α′} & B′ \ar[d]_{h_{1}} \ar[d]^{h_{2}} \ar[l]^{β′} \\ X \ar[r]^{κ} \ar[d]_{1_{X}} & A \ar[r]^{α} \ar[d]^{g} & B \ar[l]^{β} \ar[d]^{h} \\ X \ar[r]^{\overline{κ}} & \overline{A} \ar[r]^{\overline{α}} & \overline{B} \ar[l]^{\overline{β}}}

gg_{1} = gg_{2} and hh_{1} = hh_{2} so by the universal properties of the kernel pairs (R, r_{1}, r_{2}) and (S, s_{1}, s_{2}), there exists a unique morphism

\xymatrix{X \ar[r]^{κ′} \ar[d]_{1_{X}} & A′ \ar[r]^{α′} \ar[d]^{p} & B′ \ar[l]^{β′} \ar[d]^{q} \\ X \ar[r]^{κ̃} & R \ar[r]^{α̃} & S \ar[l]^{β̃}}

such that g_{1} = r_{1}p, g_{2} = r_{2}p, h_{1} = s_{1}q and h_{2} = s_{2}q. This proves that

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{κ̃} & R \ar[d]_{r_{1}} \ar[d]^{r_{2}} \ar[r]^{α̃} & S \ar[d]_{s_{1}} \ar[d]^{s_{2}} \ar[l]^{β̃} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

is the terminal object in the category of parallel morphisms in SPLEXT_{X}(ℂ) with codomain

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B. \ar[l]^{β}}

(d) ⇒ (a): Let

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

be a split extension and let

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{κ̃} & R \ar[d]_{r_{1}} \ar[d]^{r_{2}} \ar[r]^{α̃} & S \ar[d]_{s_{1}} \ar[d]^{s_{2}} \ar[l]^{β̃} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

be the terminal object in the category of relations in SPLEXT_{X}(ℂ) with codomain

\xymatrix{X \ar[r]^{κ} & A \ar[r]^{α} & B. \ar[l]^{β}}

Note that since

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{κ} & A \ar[d]_{1_{A}} \ar[d]^{1_{A}} \ar[r]^{α} & B \ar[d]_{1_{B}} \ar[d]^{1_{B}} \ar[l]^{β} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

is an object in the same category it follows that (R, r_{1}, r_{2}) and (S, s_{1}, s_{2}) are effective equivalence relations and hence the pairs r_{1}, r_{2} and s_{1}, s_{2} admit coequalizers. Consider the diagram

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{κ̃} & R \ar[d]_{r_{1}} \ar[d]^{r_{2}} \ar[r]^{α̃} & S \ar[d]_{s_{1}} \ar[d]^{s_{2}} \ar[l]^{β̃} \\ X \ar[r]^{κ} \ar[d]_{1_{X}} & A \ar[r]^{α} \ar[d]^{g} & B \ar[l]^{β} \ar[d]^{h} \\ X \ar[r]^{\overline{κ}} & \overline{A} \ar[r]^{\overline{α}} & \overline{B} \ar[l]^{\overline{β}}}

in which g and h are the coequalizers of r_{1} and r_{2} and s_{1} and s_{2} respectively, and \overline{κ} is the kernel of \overline{α} (which has domain X since in an exact category, every equivalence relation is effective and every regular epimorphism is an effective descent morphism, the lower right hand side square is a split pullback since so are upper ones). We will show that the split extension

\xymatrix{X \ar[r]^{\overline{κ}} & \overline{A} \ar[r]^{\overline{α}} & \overline{B} \ar[l]^{\overline{β}}}

is faithful. It follows from Corollary 2.17 that it is sufficient to show that, for each relation

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{\overline{σ}} & \overline{U} \ar[d]_{_{\overline{u}_{1}}} \ar[d]^{_{\overline{u}_{2}}} \ar[r]^{\overline{θ}} & \overline{V} \ar[d]_{_{\overline{v}_{1}}} \ar[d]^{_{\overline{v}_{2}}} \ar[l]^{\overline{ϕ}} \\ X \ar[r]^{\overline{κ}} & \overline{A} \ar[r]^{\overline{α}} & \overline{B}, \ar[l]^{\overline{β}}}

in SPLEXT_{X}(ℂ), \overline{v}_{1} = \overline{v}_{2}. By forming the pullback

\xymatrix{ & X \ar[rr]^{σ} \ar[dd]^{1_{X}} \ar[dl]_{⟨1,1⟩} &  & U \ar[rr]^{θ} \ar[dd]^{g̃} \ar[dl]_{⟨u_{1},u_{2}⟩} &  & V \ar[ll]^{ϕ} \ar[dd]^{h̃} \ar[dl]_{⟨v_{1},v_{2}⟩} \\ X × X \ar[rr]_{κ×κ} \ar[dd]_{1_{X×X}} &  & A × A \ar[rr]^{α×α} \ar[dd]^{g×g} &  & B × B \ar[ll]^{β×β} \ar[dd]^{h×h} \\  & X \ar[rr]^{\overline{σ}} \ar[dl]_{⟨1,1⟩} &  & \overline{U} \ar[rr]^{\overline{θ}} \ar[dl]^{⟨\overline{u}_{1},\overline{u}_{2}⟩} &  & \overline{V} \ar[ll]^{\overline{ϕ}} \ar[dl]^{⟨\overline{v}_{1},\overline{v}_{2}⟩} \\ X × X \ar[rr]_{\overline{κ}×\overline{κ}} &  & \overline{A} × \overline{A} \ar[rr]^{\overline{α}×\overline{α}} &  & \overline{B} × \overline{B} \ar[ll]^{\overline{β}×\overline{β}}}

in SPLEXT(ℂ) we obtain a new relation

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{σ} & U \ar[d]_{u_{1}} \ar[d]^{u_{2}} \ar[r]^{θ} & V \ar[d]_{v_{1}} \ar[d]^{v_{2}} \ar[l]^{ϕ} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

which therefore factors through the relation

\xymatrix{X \ar[d]_{1_{X}} \ar[d]^{1_{X}} \ar[r]^{κ̃} & R \ar[d]_{r_{1}} \ar[d]^{r_{2}} \ar[r]^{α̃} & S \ar[d]_{s_{1}} \ar[d]^{s_{2}} \ar[l]^{β̃} \\ X \ar[r]^{κ} & A \ar[r]^{α} & B \ar[l]^{β}}

via a unique morphism

\xymatrix{X \ar[r]^{σ} \ar[d]_{1_{X}} & U \ar[r]^{θ} \ar[d]^{u} & V \ar[l]^{ϕ} \ar[d]^{v} \\ X \ar[r]^{κ̃} & R \ar[r]^{α̃} & S. \ar[l]^{β̃}}

It follows that \overline{v}_{1}h̃ = hv_{1} = hs_{1}v = hs_{2}v = hv_{2} = \overline{v}_{2}h̃ and similarly \overline{u}_{1}g̃ = \overline{u}_{2}g̃ and therefore, since g̃ and h̃ being pullbacks of regular epimorphism are (regular) epimorphisms, that \overline{u}_{1} = \overline{u}_{2} and \overline{v}_{1} = \overline{v}_{2}.

(i) ⇒ (b) : Let κ : X → A be a normal monomorphism and let γ : A → C be a morphism which it is the kernel of. By forming the pullback

\xymatrix{A ×_{C} A \ar[d]_{π_{1}} \ar[r]^{π_{2}} & A \ar[d]^{γ} \\ A \ar[r]_{γ} & C}

we obtain the split extension

\xymatrix{X \ar[r]^{⟨κ,0⟩} & A ×_{C} A \ar[r]^{π_{2}} & A. \ar[l]^{⟨1,1⟩}}

Let

\xymatrix{X \ar[r]^{σ} \ar[d]_{⟨1,1⟩} & T \ar[r]^{ϵ} \ar[d]^{⟨⟨t_{1,1},t_{1,2}⟩,⟨t_{2,1},t_{2,2}⟩⟩} & N \ar[l]^{δ} \ar[d]^{⟨m_{1},m_{2}⟩} \\ X × X \ar[r]^{⟨κ,0⟩×⟨κ,0⟩} & (A ×_{C} A) × (A ×_{C} A) \ar[r]^{π_{2}×π_{2}} & A × A \ar[l]^{⟨1,1⟩×⟨1,1⟩}}

be a K-precartesian lifting. Since the diagram

\xymatrix{X \ar[r]^{⟨1,0⟩} \ar[d]_{⟨1,1⟩} & X × X \ar[r]^{π_{2}} \ar[d]^{⟨κ×κ,κ×κ⟩} & X \ar[l]^{⟨1,1⟩} \ar[d]^{⟨κ,κ⟩} \\ X × X \ar[r]^{⟨κ,0⟩×⟨κ,0⟩} & (A ×_{C} A) × (A ×_{C} A) \ar[r]^{π_{2}×π_{2}} & A × A \ar[l]^{⟨1,1⟩×⟨1,1⟩}}

is a lifting of the same morphism to the same extension there exists a unique morphism

\xymatrix{X \ar[r]^{⟨1,0⟩} \ar[d]_{1_{X}} & X × X \ar[r]^{π_{2}} \ar[d]^{φ} & X \ar[l]^{⟨1,1⟩} \ar[d]^{n} \\ X \ar[r]^{σ} & T \ar[r]^{ϵ} & N \ar[l]^{δ}}

such that ⟨κ × κ, κ × κ⟩ = ⟨⟨t_{1,1}, t_{1,2}⟩, ⟨t_{2,1}, t_{2,2}⟩⟩φ and ⟨κ, κ⟩ = ⟨m_{1}, m_{2}⟩n. We will show that (N, n, ⟨m_{1}, m_{2}⟩) is the normalizer of ⟨κ, κ⟩. It is easy to check that the diagram

\xymatrix{X \ar[ddr]_{n} \ar[r]^{σ} & T \ar[d]|{⟨t_{1,1},t_{2,1}⟩} & N \ar[ddl]^{1_{N}} \ar[l]_{δ} \\  & A × A \\  & N \ar[u]|{⟨m_{1},m_{2}⟩}}

commutes, and therefore since ⟨m_{1}, m_{2}⟩ is a monomorphism and σ and δ are jointly strongly epimorphic, there exist a unique morphism ϵ′ : T → N such that ⟨m_{1}, m_{2}⟩ϵ′ = ⟨t_{1,1}, t_{2,1}⟩, and hence ϵ′δ = 1_{N} and ϵ′σ = n. Since ⟨⟨t_{1,1}, t_{1,2}⟩, ⟨t_{2,1}, t_{2,2}⟩⟩ is a monomorphism and π_{1} × π_{1}, π_{2} × π_{2} : (A ×_{C} A) × (A ×_{C} A) → A × A are jointly monomorphic it follows that ϵ, ϵ′ : T → N are jointly monomorphic. Therefore since ϵδ = 1_{N} = ϵ′δ and since ℂ is a Mal’tsev category, it follows that

\xymatrix{T \ar[r]^{ϵ} \ar[r]_{ϵ′} & N \ar[l]|{δ}}

is an equivalence relation. It follows that since the diagram

\xymatrix{X \ar[r]^{σ} \ar[d]_{n} & T \ar[r]^{ϵ} \ar[d]^{⟨ϵ′,ϵ⟩} & N \ar[l]^{δ} \ar[d]^{1_{N}} \\ N \ar[r]^{⟨1,0⟩} & N × N \ar[r]^{π_{2}} & N \ar[l]^{⟨1,1⟩}}

is a morphism of split extensions, n is normal. Let (N′, n′, ⟨m′_{1}, m′_{2}⟩) be a factorization of ⟨κ, κ⟩ as a normal monomorphism followed by a monomorphism, and let

\xymatrix{X \ar[r]^{λ} \ar[d]_{n′} & R \ar[r]^{r_{2}} \ar[d]^{⟨r_{1},r_{2}⟩} & N′ \ar[l]^{s} \ar[d]^{1_{N′}} \\ N′ \ar[r]^{⟨1,0⟩} & N′ × N′ \ar[r]^{π_{2}} & N′ \ar[l]^{⟨1,1⟩}}

be the morphism of split extensions determined by the denormalization of n′. Since λ and s are jointly epimorphic, γm′_{i}r_{1}λ = γm′_{i}n′ = γκ = 0 = γ0 = γm′_{i}0 = γm′_{i}r_{2}λ, and γm′_{i}r_{1}s = γm′_{i} = γm′_{i}r_{2}s it follows that γm′_{i}r_{1} = γm′_{i}r_{2}. Therefore for each i in {1, 2} there exists a unique morphism ⟨m′_{i}r_{1}, m′_{i}r_{2}⟩ making the diagram

\xymatrix{X \ar[ddr]_{m′_{i}r_{1}} \ar[drr]^{m′_{i}r_{2}} \ar[dr]^{⟨m′_{i}r_{1},m′_{i}r_{2}⟩} \\  & A ×_{C} A \ar[r]^{π_{2}} \ar[d]_{π_{1}} & A \ar[d]^{γ} \\  & A \ar[r]_{γ} & C}

commute. Since the diagram

\xymatrix{X \ar[d]_{⟨1,1⟩} \ar[r]^{λ} & R \ar[d]|{⟨⟨m′_{1}r_{1},m′_{1}r_{2}⟩,⟨m′_{2}r_{1},m′_{2}r_{2}⟩⟩} \ar[r]^{r_{2}} & N′ \ar[d]^{⟨m′_{1},m′_{2}⟩} \ar[l]^{s} \\ X × X \ar[r]^{⟨κ,0⟩×⟨κ,0⟩} & (A ×_{C} A) × (A ×_{C} A) \ar[r]^{π_{2}×π_{2}} & A × A \ar[l]^{⟨1,1⟩×⟨1,1⟩}}

is a morphism of split extensions there exists a unique morphism

\xymatrix{X \ar[r]^{λ} \ar[d]_{1_{X}} & R \ar[r]^{r_{2}} \ar[d]^{g′} & N′ \ar[l]^{s} \ar[d]^{h′} \\ X \ar[r]^{σ} & T \ar[r]^{ϵ} & N \ar[l]^{δ}}

such that ⟨⟨m′_{1}r_{1}, m′_{1}r_{2}⟩, ⟨m′_{2}r_{1}, m′_{2}r_{2}⟩⟩ = ⟨⟨t_{1,1}, t_{1,2}⟩, ⟨t_{2,1}, t_{2,2}⟩⟩g′ and ⟨m′_{1}, m′_{2}⟩ = ⟨m_{1}, m_{2}⟩h′. This proves that (N, n, ⟨m_{1}, m_{2}⟩) is the normalizer of ⟨κ, κ⟩.

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[12] N. Martins-Ferreira, Low-dimensional internal categorial structures in weakly mal’cev sesquicategories, Ph.D. thesis, University of Cape Town, 2008.

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