Theory and Applications of Categories, Vol. 16, No. 2, 2006, pp. 46–58.

ACTION GROUPOID IN PROTOMODULAR CATEGORIES

DOMINIQUE BOURN

Abstract. We give here some examples of non pointed protomodular categories ℂ satisfying a property similar to the property of representation of actions which holds for the pointed protomodular category Gp of groups: any slice category of Gp, any category of groupoids with a fixed set of objects, any essentially affine category. This property gives rise to an internal construction of the center of any object X, and consequently to a specific characterization of the abelian objects in ℂ.

1. Introduction.

It is well known that, a group X being given, the group Aut X of automorphisms of X has the following property: given any other group G, the set Gp(G, Aut X) of group homomorphisms between G and Aut X is in bijection with the set of actions of the group G on the group X, which is itself, via the semidirect product, in bijection with the set of isomorphisms classes SExt(G, X) of split exact sequences:

[figure: 1; X; k; H; g; G; s; 1]

This is a universal property which can be described in a very general way: consider any category ℂ which is pointed (i.e. finitely complete with a zero object) and protomodular (see the precise definition below). An object X of ℂ is said to have a split extension classifier when there is a split extension:

[figure: X; γ; D_{1}(X); d_{0}; D(X); s_{0}]

which is universal in the sense that any other split extension with kernel X as above determines a unique morphism χ such that, in the following diagram, the right hand side squares are pullbacks and the left hand side square commutes:

[figure: X; k; 1_{X}; H; χ_{1}; g; G; χ; s; X; γ; D_{1}(X); d_{0}; D(X); s_{0}]

Received by the editors 2005-11-03 and, in revised form, 2006-01-11.

Transmitted by R. Rosebrugh. Published on 2006-01-15.

2000 Mathematics Subject Classification: 25A05,18E05.

Key words and phrases: Protomodular categories; representation of actions; internal groupoids; abelian objects; central relations and center.

\begin{matrix}c \\ ◯ \end{matrix} Dominique Bourn, 2006. Permission to copy for private use granted.

This implies that the map χ_{1} is uniquely determined by the map χ, and this is the reason of the assumption of protomodularity. Indeed a category ℂ is protomodular [5] when it is finitely complete [12] and when, given any split epimorphism (g, s) and any pullback diagram:

\xymatrix{U \ar[r]^{h̄} \ar[d]_{ḡ} & X \ar[d]_{g} \\ V \ar[r]_{h} & Y \ar[u]_{s}}

the pair (s, h̄) is jointly strongly epic (of course, the category Gp is the leading and guiding example of a pointed protomodular category). This insures that, in the diagram defining the universal property of the split extension classifier, the pair (k, s) is jointly epimorphic and the map χ_{1} uniquely determined by the map χ and the equations χ_{1}.k = γ and χ_{1}.s = s_{0}.χ. In the category Gp, the split extension classifier associated with the group X is the split exact sequence determined by the group homomorphism Id : Aut X → Aut X, namely:

[figure: 1; X; γ; Aut X ⋉ X; d_{0}; Aut X; s_{0}; 1]

with γ(x) = (Id_{X}, x), d_{0}(f, x) = f and s_{0}(f) = (f, 1).

A pointed protomodular category ℂ is said to be action representative when such a split extension classifier does exist for any object X [2]. We just observed that this is the case for the category Gp. This is also classically true for the category R-Lie of Lie algebras on a ring R, with D(X) = Der(X) the Lie algebra of derivations of X. Many other examples and counterexamples are given in [3] in the more restricted context of semi-abelian categories [10].

Let X be an object with split extension classifier. Then it is shown in [2] that this classifier is underlying a groupoid structure D_{•}(X), endowed with a canonical discrete fibration j_{•} : ∇X → D_{•}(X) (where ∇X is the indiscrete equivalence relation associated with X). For that, first denote by j_{X} : X → D(X) the classifying map of the upper split extension which makes the following diagram commute:

[figure: X; r_{X}; 1_{X}; X × X; j̃_{X}; p_{0}; X; j_{X}; s_{0}; X; γ; D_{1}(X); d_{0}; D(X); s_{0}]

Then consider the following split extension, with R[d_{0}] the kernel equivalence relation of d_{0} : D_{1}(X) → D(X), and s_{1} (according to the simplicial notations) the unique map such that p_{0}.s_{1} = s_{0}.d_{0} and p_{1}.s_{1} = 1_{D_{1}(X)}:

[figure: X; s_{1}.γ; R[d_{0}]; d_{0}; D_{1}(X); s_{0}]

It determines a unique pair (d_{1}, δ_{2}) of arrows making the following commutative square a pullback:

\xymatrix{R[d_{0}] \ar[d]_{d_{0}} \ar[r]^{δ_{2}} & D_{1}(X) \ar[d]_{d_{0}} \\ D_{1}(X) \ar[r]_{d_{1}} \ar[u]_{s_{0}} & D(X) \ar[u]_{s_{0}}}

Since any protomodular category ℂ is Mal’cev, this is sufficient (see [9]) to produce the following internal groupoid D_{•}(X):

\xymatrix{R[d_{0}] \ar[r]^{δ_{2}} \ar[r]_{d_{0}} \ar[r]^{d_{1}} & D_{1}(X) \ar[r]^{d_{1}} \ar[r]_{d_{0}} & D(X) \ar[l]_{s_{0}}}

We call this D_{•}(X) the action groupoid of the object X. Moreover we have d_{1}.j̃_{X} = j_{X}.p_{1} since the two maps clearly classify the same split extension. This produces an internal functor j_{•} : ∇X → D_{•}(X) which is actually a discrete fibration. When ℂ = Gp, this action groupoid is just the internal groupoid associated with the canonical crossed module X → Aut X, where the map d_{1} : Aut X ⋉X → X is defined by d_{1}(f, x) = ι_{x}∘f, and where ι_{x} is the inner automorphism associated with x. In any pointed protomodular category ℂ, this discrete fibration j_{•} has still a universal property, and the existence of split extension classifiers is equivalent to the existence of action groupoids (again see [2]).

Actually this universal property still makes sense even if the category ℂ is no longer pointed, and even no longer protomodular. Consequently the notion of action groupoid (and consequently of action representative category) is still valid in any finitely complete context. The point of this work is mainly to show that there are actual examples of such objects in the non pointed case, more precisely: 1) when the pointed protomodular category ℂ is action representative, the non pointed protomodular slice category ℂ/X is still action representative in this new sense, 2) the fibration ()_{0} : Grd → Set which associates with any groupoid its object of objects (and whose fibre above 1 is precisely Gp) has any of its (non pointed and protomodular, see [5]) fibres action representative,

3) when ℂ is essentially affine [5], it is not only naturally Mal’cev [11], but also action representative.

On the model of what happens in the category Gp of groups, the existence of action groupoids in ℂ gives rise to an internal construction of the center of any object X, and consequently to a specific characterization of the abelian objects in ℂ. This article gives the details of the abstract published for the announcement of the Charles Ehresmann’s centennial birthday Meeting [7].

2. Action groupoid.

From now on we shall consider a protomodular category ℂ [5], see also [1] where the fundamentals on this notion are collected. An internal groupoid Z_{•} in ℂ will be presented (see [4]) as a reflexive graph endowed with an operation ζ_{2}:

[figure: R^{2}[z_{0}]; R(ζ_{2}); z_{2}; z_{0}; z_{1}; R[z_{0}]; ζ_{2}; z_{0}; z_{1}; Z_{1}; z_{1}; z_{0}; Z_{0}; s_{0}]

making the diagram above satisfy all the simplicial identities (including the ones involving the degeneracies), where R[z_{0}] is the kernel equivalence relation of the map z_{0}. In the set theoretical context, this operation ζ_{2} associates the composite g.f^{−1} with any pair (f, g) of arrows with same domain. Actually, when the category ℂ is protomodular, and thus Mal’cev, we can even truncate this diagram at level 2, see [9].

2.1. Definition. An object X in ℂ is said to be action representative, or to have an action groupoid, when there is an internal groupoid D_{•}(X):

\xymatrix{R[d_{0}] \ar[r]^{δ_{2}} \ar[r]_{d_{0}} \ar[r]^{d_{1}} & D_{1}(X) \ar[r]^{d_{1}} \ar[r]_{d_{0}} & D(X) \ar[l]_{s_{0}}}

endowed with a discrete fibration j : ∇X → D_{•}(X):

\xymatrix{R[p_{0}] \ar[d]_{R(j̃)} \ar[r]^{p_{2}} \ar[r]_{p_{0}} \ar[r]^{p_{1}} & X × X \ar[d]^{j̃} \ar[r]^{p_{1}} \ar[r]_{p_{0}} & X \ar[d]^{j} \ar[l]_{s_{0}} \\ R[d_{0}] \ar[r]^{δ_{2}} \ar[r]_{d_{0}} \ar[r]^{d_{1}} & D_{1}(X) \ar[r]^{d_{1}} \ar[r]_{d_{0}} & D(X) \ar[l]_{s_{0}}}

where ∇X is the indiscrete equivalence relation associated with X (i.e. the upper internal groupoid in the diagram above), satisfying the following property: given any other internal groupoid Z_{•} endowed with a discrete fibration k_{•} = (k_{0}, k_{1}) : ∇X → Z_{•}, there is a unique internal functor ǩ_{•} = (ǩ_{0}, ǩ_{1}) : Z_{•} → D_{•} such that ǩ_{•}.k_{•} = j_{•}. The category ℂ is said to be action representative when any object X has an action groupoid.

If ℂ is also protomodular, this implies that ǩ_{•} is itself a discrete fibration. When ℂ = Gp, this action groupoid is just the internal groupoid associated with the canonical crossed module X → Aut X. In the pointed protomodular case, thanks to the results of [2], many examples are available in [3]. We have moreover:

2.2. Proposition. Let ℂ be an action representative protomodular category. Then any coslice category Y \ℂ is still action representative.

Proof. The category Y \ℂ has the same underlying products and pullbacks as ℂ. So it is straitforward that if (X, e), e : Y → X is an object in Y \ℂ, we have D(X, e) = (D(X), j.e) and D_{1}(X, e) = (D_{1}(X), s_{0}.j.e).

2.3. Remark. In particular the category ℂ_{∗} of pointed objects in ℂ, which is nothing but the coslice category 1\ℂ, is still action representative, and then any object in ℂ_{∗} (which is clearly a pointed category) has a split extension classifier, as in [2].

3. Slice categories in pointed case.

Suppose now ℂ is an action representative pointed protomodular category. The slice categories ℂ/Y are no longer pointed.

3.1. Proposition. When ℂ is an action representative pointed protomodular category, the slice categories ℂ/Y are still action representative. For any h : Y′ → Y, the change of base functor h^{∗} : ℂ/Y → ℂ/Y′ preserves the action groupoids.

Proof. Let f : X → Y be an object of ℂ/Y. Consider the following diagram where K is the kernel of f:

[figure: K × K; R(k); p_{1}; p_{0}; R[f]; ǩ_{1}; f_{1}; f_{0}; D_{1}(K); d_{1}; d_{0}; K; k; X; f; ǩ; D(K); 0; α_{Y}; Y]

Then the upper left hand side part of this diagram is underlying a discrete fibration, and thus produces the internal functorial factorization ǩ_{•} (actually a discrete fibration, since ℂ is protomodular). Whence the following diagram in ℂ/Y:

\xymatrix{R[f] \ar[rr]^{(ǩ_{1},f.f_{0})} \ar[d]^{f_{1}} \ar[d]_{f_{0}} &  & D_{1}(K) × Y \ar[d]^{d_{1}×Y} \ar[d]_{d_{0}×Y} \\ X \ar[dr]_{f} \ar[rr]_{(ǩ,f)} &  & D(K) × Y \ar@{-->}[dl]^{p_{Y}} \\  & Y}

The upper part of this diagram is a discrete fibration since the composite with the projection towards D_{•}(K) is the discrete fibration ǩ_{•}. We claim that this discrete fibration in ℂ/Y makes the internal groupoid on the right be the action groupoid of the object f in ℂ/Y, the equivalence relation R[f] being, in the slice category ℂ/Y, the indiscrete equivalence relation associated with this object f. So, consider an internal discrete fibration h_{•} : R[f] → Z_{•} in ℂ/Y:

\xymatrix{R[f] \ar[r]^{h_{1}} \ar[d]^{f_{1}} \ar[d]_{f_{0}} & Z_{1} \ar[d]^{z_{1}} \ar[d]_{z_{0}} \ar@{.>}[r] & D_{1}(K) × Y \ar[d]^{d_{1}×Y} \ar[d]_{d_{0}×Y} \\ X \ar[dr]_{f} \ar[r]_{h_{0}} & Z_{0} \ar@{.>}[r] \ar[d]^{z} & D(K) × Y \ar[dl]^{p_{Y}} \\  & Y}

We must find the dotted factorization. The projection of this factorization towards Y is forced by the diagram. The projection towards D_{•}(K) is given by the following diagram:

[figure: K × K; R(k); p_{1}; p_{0}; R[f]; h_{1}; f_{1}; f_{0}; Z_{1}; l_{1}; z_{1}; z_{0}; D_{1}(K); d_{1}; d_{0}; K; k; X; f; h_{0}; Z_{0}; l_{0}; D(K); 0; α_{Y}; Y]

where the functor l_{•} : Z_{•} → D_{•}(K) is the factorization produced by the discrete fibration h_{•}.k_{•} : ∇K → Z_{•} in ℂ.

The second point of the statement comes from the fact that the image h^{∗}(f) in ℂ/Y′ of the object f in ℂ/Y by the change of base functor h^{∗} has the same kernel K as f in ℂ.

We thus get the following:

3.2. Corollary. Let ℂ be an action representative pointed protomodular category; and π : Ptℂ → ℂ [5] the associated fibration of pointed objects. Then its fibres are action representative and its change of base functors preserve the action groupoids and the split extension classifiers.

Proof. The fibre Pt_{Y} ℂ above Y is nothing but the category (ℂ/Y)_{∗} of points of ℂ/Y (in other words: split epimorphisms with codomain Y). Then by Propositions 2.1 and 1.1, it is action representative. The change of base functors are given by pullbacks, so, by Proposition 2.1, they preserve the action groupoids, and consequently the split extension classifiers which are part of this structure, see [2].

4. The fibres of the fibration ()_{0} : Grd → Set.

Let Set and Grd be respectively the categories of sets and groupoids, and ()_{0} : Grd → Set the forgetful functor associating the object of objects Z_{0} with any groupoid Z_{•}. This functor is a fibration whose fibre above the singleton 1 is nothing but the category Gp of groups which is action representative. On the other hand, any fibre Grd_{X} above a set X is protomodular [5] and clearly non pointed unless X ≃ 1. We are going to show that it is still action representative. This fibre has an initial object ΔX, namely the discrete equivalence relation on X, and a final object ∇X the indiscrete equivalence relation on X. For any groupoid Z_{•} in Grd_{X}, we shall need the subgroupoid defined by the following pullback in Grd_{X} (namely the subgroupoid of endomaps of Z_{•}):

[figure: Aut Z_{•}; Z_{•}; ΔX; ∇X]

Let Z_{•} be a groupoid such that Z_{0} = X. We shall denote by Z(x, x′) the set of arrows from x to x′. Its action groupoid in Grd_{X}, if ever it exists, must be an internal groupoid in Grd_{X}, which is nothing but a 2-groupoid with X as object of objects. Let us denote by D(Z_{•}) the groupoid whose object of objects is X, and arrows ϕ : x → x′ are the group isomorphisms ϕ : Z(x, x) → Z(x′, x′). There is a canonical bijective on objects functor j_{•} : Z_{•} → D(Z_{•}): given a map f : x → x′ in Z_{•}, its image j(f) : x → x′ in D(Z_{•}) is the group isomorphism j(f) : Z(x, x) → Z(x′, x′) given by j(f)(α) = f.α.f^{−1} , ∀α ∈ Z(x, x), i.e. such that the following diagram commutes in the groupoid Z_{•}:

\xymatrix{x \ar[r]^{f} \ar[d]_{α} & x′ \ar[d]^{j(f)(α)} \\ x \ar[r]_{f} & x′}

This groupoid D(Z_{•}) is actually underlying a 2-groupoid. A 2-cell ν : ϕ ⇒ ψ is given by a map ν ∈ Z(x′, x′) such that ∀α ∈ Z(x, x) we have ψ(α) = ν.ϕ(α).ν^{−1}. The ”vertical” composition is induced by the composition in Z_{•}, the ”horizontal” one:

\xymatrix{x \ar[rr]^{ϕ} \ar[rr]_{ψ} & ⇓ ν & x′ \ar[rr]^{ϕ′} \ar[rr]_{ψ′} & ⇓ ν′ & x′′ ⟼ x \ar[rr]^{ϕ′.ϕ} \ar[rr]_{ψ′.ψ} & ⇓ ν′ ∘ ν & x′′}

is defined by ν′ ∘ ν = ν′.ϕ′(ν)(= ψ′(ν).ν′).

Accordingly, we define the groupoid D_{1}(Z_{•}) as the groupoid whose object of objects is X, and arrows x → x′ are the pairs (ϕ, ν), with ϕ : x → x′ an arrow of D(Z_{•}) and ν ∈ Z(x′, x′). The composition is defined by (ϕ′, ν′)∘(ϕ, ν) = (ϕ′.ϕ, ν′.ϕ′(ν)). The bijective on objects functors d_{i} : D_{1}(Z_{•}) → D(Z_{•}) are defined by d_{0}(ϕ, ν) = ϕ and d_{1}(ϕ, ν) = ψ with ψ(α) = ν.ϕ(α).ν^{−1}. We have also a bijective on objects functor:

j̃_{•} : Z_{•} ×_{X} Z_{•} → D_{1}(Z_{•})

(where Z_{•} ×_{X} Z_{•} denotes the product in the fibre Grd_{X}), which is defined by j̃(f, g) = (j(f), g.f^{−1}) for any arrow in Z_{•} ×_{X} Z_{•}, i.e. for any parallel pair of arrows (f, g) : x ⇉ x′ in Z_{•}. Whence the following commutative diagram in Grd_{X} which is actually underlying a discrete fibration:

\xymatrix{R[p_{0}] \ar[d]_{R(j̃_{•})} \ar[r]^{p_{2}} \ar[r]_{p_{0}} \ar[r]^{p_{1}} & Z_{•} ×_{X} Z_{•} \ar[d]^{j̃_{•}} \ar[r]^{p_{1}} \ar[r]_{p_{0}} & Z_{•} \ar[d]^{j_{•}} \ar[l]_{s_{0}} \\ R[d_{0}] \ar[r]^{δ_{2}} \ar[r]_{d_{0}} \ar[r]^{d_{1}} & D_{1}(Z_{•}) \ar[r]^{d_{1}} \ar[r]_{d_{0}} & D(Z_{•}) \ar[l]_{s_{0}}}

4.1. Remark. Actually the definition of the lower groupoid depends uniquely on Aut Z_{•}. Only the comparison j_{•} depends on Z_{•}. This observation will be essential in the proof of the uniqueness in the following proposition:

4.2. Proposition. The diagram above determines the lower groupoid as the action 2groupoid D_{•}(Z_{•}) associated with the groupoid Z_{•} in the protomodular fibre Grd_{X}. Accordingly the fibres Grd_{X} are action representative.

Proof. Suppose we are given a discrete fibration in Grd_{X}:

\xymatrix{R[p_{0}] \ar[d]_{R(k_{•,1})} \ar[r]^{p_{2}} \ar[r]_{p_{0}} \ar[r]^{p_{1}} & Z_{•} ×_{X} Z_{•} \ar[d]^{k_{•,1}} \ar[r]^{p_{1}} \ar[r]_{p_{0}} & Z_{•} \ar[d]^{k_{•,0}} \ar[l]_{s_{0}} \\ R[w_{0}] \ar[r]^{w_{2}} \ar[r]_{w_{0}} \ar[r]^{w_{1}} & W_{•,1} \ar[r]^{w_{1}} \ar[r]_{w_{0}} & W_{•,0} \ar[l]_{s_{0}}}

The fact that this is a discrete fibration means that any 2-cell in the 2-groupoid W_{•}:

\xymatrix{x \ar[rr]^{k(f)} \ar[rr]_{h} & ⇓ ν & x′}

determines a unique arrow g : x → x′ in the groupoid Z_{•} such that k(g) = h. In particular any 2-cell in W_{•}:

\xymatrix{x \ar[rr]^{1_{x}} \ar[rr]_{h} & ⇓ ν & x}

produces a unique arrow g : x → x in the groupoid Z_{•} such that k(g) = h. We must now define a 2-functor:

\xymatrix{R[w_{0}] \ar[d]_{R(ǩ_{•,1})} \ar[r]^{w_{2}} \ar[r]_{w_{0}} \ar[r]^{w_{1}} & W_{•,1} \ar[d]^{ǩ_{•,1}} \ar[r]^{w_{1}} \ar[r]_{w_{0}} & W_{•,0} \ar[d]^{ǩ_{•,0}} \ar[l]_{s_{0}} \\ R[d_{0}] \ar[r]^{δ_{2}} \ar[r]_{d_{0}} \ar[r]^{d_{1}} & D_{1}(Z_{•}) \ar[r]^{d_{1}} \ar[r]_{d_{0}} & D(Z_{•}) \ar[l]_{s_{0}}}

Let h : x → x′ be an arrow in W_{•}; let us define ǩ(h) : Z(x, x) → Z(x′, x′). So consider α : x → x in Z_{•}. We have a 2-cell k(1_{x}, α) in W_{•}. Consider the following diagram:

\xymatrix{x \ar[rr]^{1_{x}} \ar[rr]_{k(α)} \ar[d]_{h} & ⇓ k(1_{x}, α) & x \ar[d]^{h} \\ x′ \ar[rr]^{1_{x′}} \ar[rr]_{h.k(α).h^{−1}} & ⇓ h.k(1_{x}, α).h^{−1} & x′}

Then the lower 2-cell in the diagram above produces a unique arrow ǩ(h)(α) : x′ → x′ in Z_{•} such that k(ǩ(h)(α)) = h.k(α).h^{−1}. The unicity of this map assures that ǩ(h) is a group homomorphism, and the last equation that we have ǩ_{•,0}.k_{•,0} = j_{•}. It is easy to check that the construction ǩ_{•} : W_{•,0} → D(Z_{•}) is functorial.

We must now extend ǩ to the 2-cells of the 2-groupoid W_{•}. So let ν : h ⇒ h′ be a 2-cell in W_{•}. Then ν.h^{−1} : 1_{x′} ⇒ h′.h^{−1} is a 2-cell in W_{•} which determines a unique map ν̌ : x′ → x′ in Z_{•} such that k(ν̌) = h′.h^{−1}. We define ǩ(ν) as the 2-cell (ǩ(h), ν̌) in D_{•}(Z_{•}). This completes the 2-functor ǩ_{•,•} we were looking for.

To prove the unicity of this factorization, let us look at the following diagram:

\xymatrix{R[p_{0}] \ar[d] \ar[r]^{p_{2}} \ar[r]_{p_{0}} \ar[r]^{p_{1}} & Aut Z_{•} ×_{X} Aut Z_{•} \ar[d] \ar[r]^{p_{1}} \ar[r]_{p_{0}} & Aut Z_{•} \ar[d] \ar[l]_{s_{0}} \\ R[p_{0}] \ar[d]_{R(k_{•,1})} \ar[r]^{p_{2}} \ar[r]_{p_{0}} \ar[r]^{p_{1}} & Z_{•} ×_{X} Z_{•} \ar[d]^{k_{•,1}} \ar[r]^{p_{1}} \ar[r]_{p_{0}} & Z_{•} \ar[d]^{k_{•,0}} \ar[l]_{s_{0}} \\ R[w_{0}] \ar[d]_{R(ǩ_{•,1})} \ar[r]^{w_{2}} \ar[r]_{w_{0}} \ar[r]^{w_{1}} & W_{•,1} \ar[d]^{ǩ_{•,1}} \ar[r]^{w_{1}} \ar[r]_{w_{0}} & W_{•,0} \ar[d]^{ǩ_{•,0}} \ar[l]_{s_{0}} \\ R[d_{0}] \ar[r]^{δ_{2}} \ar[r]_{d_{0}} \ar[r]^{d_{1}} & D_{1}(Z_{•}) \ar[r]^{d_{1}} \ar[r]_{d_{0}} & D(Z_{•}) \ar[l]_{s_{0}}}

The upper part of this diagram, induced by the inclusion Aut Z_{•} ↣ Z_{•}, is actually a discrete fibration. Moreover, as we noticed in the previous remark the lower 2-groupoid is also the action 2-groupoid associated with Aut Z_{•}. So the unicity of the factorization ǩ_{•} can be equally checked from Aut Z_{•}. This is relatively easy by using the similar result holding in the category Gp of groups, since the groupoid Aut Z_{•} is nothing but a family of ordinary groups.

4.3. Remark. As in any (even not pointed) protomodular category, there is, in the fibres Grd_{X}, an intrinsic notion of normal subobject which is explicited in [6] (Theorem 3). The normal subobjects are closely related to the action groupoids, see [2]. Let us quickly mention here, that a subgroupoid V_{•} ↣ Z_{•} in Grd_{X} is normal if and only if, given any map f : x → x′ in Z_{•}, the restriction of the isomorphism j(f) : Z(x, x) → Z(x′, x′) to V (x, x) takes its values in V (x′, x′).

5. Action groupoid and centrality.

In any protomodular category ℂ, there is an intrinsic notion of abelian objects, see for instance [8] or [1]. The existence of action groupoids allows us to measure the obstruction to abelianness:

5.1. Proposition. Suppose the object X in ℂ admits an action groupoid. The kernel relation R[j] of the map j : X → D(X) is the centre of X, i.e. the greatest central equivalence relation on X.

Proof. Recall that an equivalence relation (r_{0}, r_{1}) : R ⇉ X on X is central when there is a ”connector” between the equivalence relations R and ∇X, which is a map p : R × X → X, satisfying internally the Mal’cev equations p(x, y, y) = x and p(x, x, y) = y, see [8]. Now let us consider the following diagram:

\xymatrix{R[j̃] \ar[d]_{R(p_{0})} \ar@{.>}[d]^{R(p_{1})} \ar[r]_{p_{0}} \ar[r]^{p_{1}} & X × X \ar[d]_{p_{0}} \ar[r]^{j̃} \ar@{.>}[d]^{p_{1}} & D_{1}(X) \ar[d]_{d_{0}} \ar@{.>}[d]^{d_{1}} \\ R[j] \ar[r]_{p_{0}} \ar[r]^{p_{1}} & X \ar[r]_{j} & D(X)}

Since the downward right hand side square is a pullback (as a part of a discrete fibration), the downward left hand side squares are pullbacks and R[j̃] is then isomorphic to R × X. Consequently the map p_{0}.R(p_{1}) : R[j̃] → X is a connector which makes R[j] central.

If R is another central relation, the connector p between R and ∇X allows us to complete the following diagram in a way which makes the central squares commute and determine internal functors, see [8]:

\xymatrix{X × X \ar@{.>}[r]^{s_{0}×X} \ar[d]_{p_{0}} \ar[d]^{p_{1}} & R × X \ar[d]_{p_{R}} \ar[d]^{(p,p_{X})} \ar[r]_{(r_{0}.p_{R},p)} \ar[r]^{r_{1}×X} & X × X \ar[d]_{p_{0}} \ar[d]^{p_{1}} \ar@{.>}[r]^{j̃} & D_{1}(X) \ar[d]_{d_{0}} \ar[d]^{d_{1}} \\ X \ar@{.>}[r]_{s_{0}} & R \ar[r]_{r_{0}} \ar[r]^{r_{1}} & X \ar@{.>}[r]_{j} & D(X)}

On the other hand, the dotted arrows on the left hand side determine a discrete fibration, while the composition by s_{0} (resp. by s_{0}×X) equalizes the horizontal arrows. Accordingly both maps j.r_{0} and j.r_{1} are the classifier of the left hand side dotted discrete fibration, and are consequently equal. Accordingly we get R ≤ R[j].

In this way we obtain a characterization of abelian objects in ℂ, where, classically an object X is called abelian when the indiscrete equivalence relation ∇X is central:

5.2. Corollary. Let the object X have an action groupoid. Then the following conditions are equivalent:

1) the object X is abelian in ℂ

2) d_{0} = d_{1} (in other words the action groupoid D_{•}(X) is absolutely disconnected)

Proof. Suppose X abelian. Then its centre is the indiscrete relation ∇X and, according to the previous proposition, R[j] = ∇X. It is the case if and only if j.p_{0} = j.p_{1} : X ×X ⇉ X → D(X). Now, considering the following diagram:

\xymatrix{X × X \ar[d]_{p_{0}} \ar[r]^{j̃} \ar[d]^{p_{1}} & D_{1}(X) \ar[d]_{d_{0}} \ar[d]^{d_{1}} \\ X \ar[r]_{j} & D(X)}

we have always d_{i}.j̃ = p_{i}.j. So that X is abelian if and only if d_{0}.j̃ = d_{1}.j̃. But the pair (s_{0}, j̃) is jointly strongly epic. Since d_{0} and d_{1} are clearly equalized by s_{0}, this last equality holds if and only if d_{0} = d_{1}.

6. Essentially affine categories.

A category ℂ is essentially affine [5] when it admits pullbacks of split epimorphisms, pushouts of split monomorphisms and is such that, given any commutative square of split epimorphisms:

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

the downward square is a pullback if and only if the upward square is a pushout. A pointed finitely complete category ℂ is essentially affine if and only if it is additive. The slice categories of any finitely complete additive category are essentially affine. On the other hand, any finitely complete essentially affine category is necessarily protomodular and naturally Mal’cev in the sense of [11]. Inside the protomodular context, this naturally Mal’cev condition exactly means that any object is abelian.

We showed in [2] that when the category ℂ is additive, then, for each object X, the action groupoid structure is nothing but the canonical (internal) abelian group structure on X, namely:

[figure: X × X; d; p_{0}; p_{1}; X; τ_{X}; 0; α_{X}]

where d = p_{1} − p_{0}. In the same order of ideas, we have:

6.1. Proposition. Any finitely complete essentially affine category ℂ is action representative.

Proof. Given any object X let us consider the following pushout of the split monomorphism s_{0} along the terminal map:

[figure: X × X; d; p_{0}; p_{1}; X̄; X; s_{0}; 1; e]

The object X̄, being pointed by e and abelian, is canonically endowed with an internal (abelian) group structure, and the map d determines an internal functor d_{•} : ∇X → X̄. Moreover the downward squares are pullbacks and this functor is a dicrete fibration. Let us show that this dicrete fibration makes the group structure on X̄ be the action groupoid of X. So consider any other discrete fibration k_{•} : ∇X → W_{•}:

\xymatrix{X × X \ar[rr]^{d} \ar[r]_{k_{1}} \ar[d]_{p_{0}} \ar[d]^{p_{1}} & W_{1} \ar@{.>}[r]_{ǩ_{1}} \ar[d]_{w_{0}} \ar[d]^{w_{1}} & X̄ \ar[d] \\ X \ar[r]_{k_{0}} \ar[u] & W_{0} \ar@{.>}[r]_{τ} \ar[u] & 1 \ar[u]_{e}}

We must explicit a unique dotted factorization. Clearly ǩ_{0} is the terminal map τ. Since k_{•} is a discrete fibration, the downward left hand side squares are pullbacks, and consequently the upward left hand side square is a pushout. Accordingly there is a unique map ǩ_{1} such that ǩ_{1}.k_{1} = d and ǩ_{1}.s_{0} = e.τ. It is easy to check that this diagram is actually underlying an internal functor ǩ_{•} : W_{•} → X̄.

6.2. Remark. As we emphasized it in the introduction, the definition of action groupoid is still valid in any finitely complete category. Would there be examples of action groupoids in a non protomodular context?

References

[1] F. Borceux and D. Bourn, Mal’cev, protomodular, homological and semi-abelian categories, Kluwer, Mathematics and its applications, vol. 566, 2004.

[2] F. Borceux and D. Bourn, Split extension classifier and centrality, Preprint 2005.

[3] F. Borceux, G. Janelidze and G.M. Kelly, On the representability of actions in a semi-abelian category, Theory and Applications of Categories, 14, 2005, 244-286.

[4] D. Bourn, The shift functor and the comprehensive factorization for internal groupoids, Cahiers de Top. et Géom. Diff. Catégoriques, 28, 1987, 197-226.

[5] D. Bourn, Normalization equivalence, kernel equivalence and affine categories, Springer Lecture Notes Math. 1488, 1991, 43-62.

[6] D. Bourn, Aspherical abelian groupoids and their directions, J. Pure Appl. Algebra, 168, 2002, 133-146.

[7] D. Bourn, Action groupoid in protomodular categories, Cahiers de Top. Géom. Diff. Catégoriques, 46, 2005, 180-182.

[8] D. Bourn and M. Gran, Centrality and normality in protomodular categories, Th. and Applications of Categories, 9, 2002, 151-165.

[9] A. Carboni, M.C. Pedicchio and N. Pirovano, Internal graphs and internal groupoids in Mal’cev categories, CMS Conference Proceedings, 13, 1992, 97-109.

[10] G. Janelidze, L. Marki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra, 168, 2002, 367-386.

[11] P.T. Johnstone, Affine categories and naturally Mal’cev categories, J. Pure Appl. Algebra, 61, 1989, 251-256.

[12] S. Mac Lane, Categories for the working mathematician, 2nd edition, Springer Verlag, 1998.

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