Theory and Applications of Categories, Vol. 28, No. 5, 2013, pp. 150–165.

ON THE MONAD OF INTERNAL GROUPOIDS

DOMINIQUE BOURN

Abstract. We deeply analyse the structural organisation of the fibration of points and of the monad of internal groupoids. From that we derive: 1) a new characterization of internal groupoids among reflexive graphs in the Mal’cev context; 2) a setting in which a Mal’cev category is necessarily a protomodular category.

Introduction

This article is the result of the effort to understand the universal property of the split epimorphism (p_{0}^{Y} , s_{0}^{Y}) : Y × Y ⇆ Y among all the split epimorphisms, a question that arises from many aspects of Mal’cev and protomodular categories, where it plays a pivotal role. For that, we had to analyse the remarkable structural organisation of the fibration of points ¶_{𝔼} : P t𝔼 → 𝔼 (the codomain functor). It was already known that there is on P t𝔼 a monad (T, λ, μ) whose category of algebras Alg^{T} is nothing but the category Grd𝔼 of internal groupoids in 𝔼, see [4]. It appears, here, that the functor J : 𝔼 → P t𝔼 defined by J(Y) = (p_{0}^{Y} , s_{0}^{Y}) is precisely the right adjoint of the functor ¶_{𝔼}.T : P t𝔼 → 𝔼 which is nothing but the domain functor. What is unexpected is that this functor J is monadic (where is originated the conceptual reason why the functor ( )_{0} = ¶_{𝔼}.U^{T} : Alg^{T} = Grd𝔼 → 𝔼 is still a fibration) and that it allows a localization of the monad (T, λ, μ) into a monad (T_{Y} , λ_{Y} , μ_{Y}) on the fibre P t_{Y} 𝔼 which is described by the following diagram:

[figure: X; (f,1); f; Y × X; p_{Y}; Y × (Y × X); p_{Y}; Y ×p_{X}; Y; s; (1,s); (s_{0},s)]

Of this last monad we could get out three considerations of the highest interest in the context of a Mal’cev category ℂ:

1) any reflexive graph (d_{0}, d_{1}) : X_{1} ⇉ X_{0} in ℂ is a groupoid if and only if the following two subobjects commute in the unital fibre P t_{X_{0}}ℂ:

Received by the editors 2012-02-22 and, in revised form, 2013-03-05.

Transmitted by Stephen Lack. Published on 2013-03-11.

2010 Mathematics Subject Classification: 18G50, 18D35, 20J15, 08C05.

Key words and phrases: Fibration of points, monad of internal groupoids, Mal’cev and protomodular categories, split exact sequence, algebraic exponentiation.

\begin{matrix}c \\ ◯ \end{matrix} Dominique Bourn, 2013. Permission to copy for private use granted.

[figure: X_{1}; (d_{0},1); d_{0}; X_{0} × X_{1}; p_{X_{0}}; X_{1}; (d_{1},1); d_{1}; X_{0}; s_{0}; (1,s_{0}); s_{0}]

which is a localization of the classical characterization [R[d_{0}], R[d_{1}]] = 0,

2) when a pointed Mal’cev category ℂ satisfies the property of algebraic exponentiation (see [13] and [8]) for any map, it is necessarily protomodular,

3) when the Mal’cev category ℂ is not pointed and regular, if it satisfies the property of algebraic exponentiation for any map, the full subcategory ℂ_{♯} whose objects have global support is protomodular. Section 1 is devoted to the fibration of points and to the monad of internal groupoids, Section 2 to the “localization” (T_{Y} , λ_{Y} , μ_{Y}) and to the T_{Y} -algebras, and Section 3 to the applications in the Mal’cev context. We should like to thank S. Lack for a very useful bibliographical suggestion.

1. The monad of internal groupoids

1.1. The first observation. Let (T, λ, μ) be any monad on a category 𝔼 and let us consider the following commutative diagram:

\xymatrix{𝔼 \ar[rr]^{T} \ar[dr]_{W} &  & 𝔼 \ar[dl]^{V} \\  & 𝔽}

1.2. Proposition. Suppose that the functor W = V.T has a right adjoint W̃, then the functor \overline{W} in the following commutative diagram:

\xymatrix{Alg^{T} \ar[rr]^{U^{T}} \ar@{-->}[dr]_{\overline{W}} &  & 𝔼 \ar[dl]^{V} \\  & 𝔽}

has a right adjoint Ŵ, and it is such that: U^{T} .Ŵ ≃ W̃.

Proof. This is a mere corollary of Theorem 3.7.2 in [2]. Consider the following commutative diagram:

\xymatrix{Alg^{T} \ar[rr]^{\overline{W} =V.U^{T}} &  & 𝔽 \\  & 𝔼 \ar[ur]_{V.T} \ar[ul]^{F^{T}}}

It satisfies the conditions of Theorem 3.7.2 in [2]: since F^{T} has a right adjoint U^{T} which is monadic and V.T has a right adjoint W̃, it remains to show that \overline{W} = V.U^{T} preserves coequalizers of U^{T} -contractible pairs; it is the case for U^{T}, and such a coequalizer becomes a contractible coequalizer in 𝔼, and thus it is preserved by any functor. The right adjoint functor Ŵ can be described in the following way; let us denote by w : W W̃ ⇒ 1_{𝔽} the natural transformation associated with the right adjoint W̃, then the natural transformation w.V μ : V T^{2}W̃ ⇒ 1_{𝔽} induces a unique natural transformation: β : T W̃ ⇒ W̃ such that: w.V T β = w.V μW̃. It is easy to check that β satisfies the axioms of T -algebras. We have Ŵ (Y) = (W̃ (Y), β_{Y}), the natural map ϵ̂: V.U^{T} .Ŵ ⇒ 1_{𝔽} being given by w_{Y} .V (λ_{W̃ (Y)}) : V.W̃ (Y) → Y.

When the functor V is the identity on 𝔼, then we get the well known result of Eilenberg-Moore [11] according to which when the endofunctor of a monad (T, λ, μ) admits a right adjoint G, this functor G is underlying a comonad (G, ϵ, δ) such that the categories Alg^{T} and Coal^{C} coincide. Actually we shall need the following precisions:

1.3. Proposition. Assume the assumptions of the previous proposition, and denote by (Θ, η, W̃ (w_{V T})) the monad on 𝔼 induced by the right adjoint W̃. There is a unique natural transformation m : T ⇒ Θ = W̃ .V.T such that w_{V T} .V T (m) = V (μ) and which is actually a morphism of monads:

(T, λ, μ) ⇒ (Θ, η, W̃ (w_{V T}))

namely such that the following diagram commutes:

\xymatrix{X \ar[r]^{λ_{X}} \ar[dr]_{η_{X}} & T (X) \ar[d]^{m_{X}} & T^{2}(X) \ar[l]_{μ_{X}} \ar[d]^{T (m_{X})} \\  & W̃ V T (X) & T W̃ V T (X) \ar[d]^{m_{W̃ V T X}} \\  &  & W̃ V T W̃ V T (X) \ar@{-->}[ul]^{W̃ (w_{V T (X)})}}

If we denote by M : Alg^{Θ} → Alg^{T} the induced comparison functor, the right adjoint Ŵ of V.U^{T} is nothing but the functor M.Φ_{Θ} in the following diagram:

\xymatrix{ & Alg^{T} \ar[dr]^{U^{T}} \ar[d]^{U^{T}} \\ Alg^{Θ} \ar[r]^{U^{Θ}} \ar@{-->}[ur]^{M} & 𝔼 & 𝔼 \ar[dl]^{V} \\  & 𝔽 \ar[ul]^{Φ_{Θ}} \ar[u]_{W̃}}

where Φ_{Θ} is the natural comparison functor to the category of Θ-algebras and any triangle on the left hand side commutes.

Proof. By adjunction, the map V (μ_{X}) : V.T^{2}(X) → V.T (X) determines a unique map m_{X} : T (X) → W̃ .V.T (X) such that w_{V T (X)}.V T (m_{X}) = V (μ_{X}). Let us show it is underlying a morphism of monads. First we have m_{X}.λ_{X} = η_{X} just checking: w_{V T (X)}.V T (m_{X}).V T (λ_{X}) = V (μ_{X}.V T (λ_{X})) = 1_{V T (X)} = w_{V T (X)}.V T (η_{X}). Then we have m_{X}.μ_{X} = W̃ (w_{V T (X)}).m_{W̃ V T (X)}.T (m_{X}) just checking: w_{V T (X)}.V T W̃ (w_{V T (X)}).V T (m_{W̃ V T (X)}).V T^{2}(m_{X}) = w_{V T (X)}.w_{V T W̃ V T (X)}.V T (m_{W̃ V T (X)}).V T^{2}(m_{X}) = w_{V T (X)}.V (μ_{W̃ V T (X)}).V T^{2}(m_{X}) = w_{V T (X)}.V T (m_{X}).V (μ_{T (X)}) = V (μ_{X}).V (μ_{T (X)}) = V (μ_{X}).V T (μ_{X}) = w_{V T (X)}.V T (m_{X}).V T (μ_{X}) Now we have M.Φ_{Θ}(Y) = (W̃ (Y), W̃ (w_{Y}).m_{W̃ (Y)}), and it is easy to check that the map W̃ (w_{Y}).m_{W̃ (Y)} is the map β_{Y} of the end of the proof of the previous proposition. The natural transformation η̂ : 1_{Alg^{T}} ⇒ M.Φ_{Θ}.V.U^{T} is given, for any T -algebra (X, ξ) by the unique map η̂(X, ξ) : X → W̃ V (X) satisfying: w_{V (X)}.V T (η̂(X, ξ)) = V (ξ).

Let us end this very general section by a relatively straightforward result we shall need later on:

1.4. Proposition. Let U : 𝔼 → 𝔽 be a functor having a right adjoint G. Then U preserves the jointly strongly epic families.

Proof. Let W_{i} →^{f_{i}} X; i ∈ I be a jointly strongly epic family in 𝔼; now consider a map h : U(X) → Y and a monomorphism t : T ↣ Y through which any map h.U(f_{i}) factorizes by a map g_{i} : U(W_{i}) → T. The image G(t) by the right adjoint G is a monomorphism G(T) ↣ G(Y) in 𝔼. On the other hand, the map h produces a map h̄ : X → G(Y) in the same way as the maps g_{i} produce maps ḡ_{i} : W_{i} → G(T) such that G(t).ḡ_{i} = h̄.f_{i}. Since the family (f_{i})_{i∈I} is jointly strongly epic, then necessarily, we get a factorization l:

[figure: W_{i}; f_{i}; ḡ_{i}; G(T); G(t); X; h̄; l; G(Y)]

The adjunction gives rise to a map l̃: U(X) → T such that t.l̃= h.

1.5. The fibration of points. We shall study here a specific example of the situation introduced above. Recall that, 𝔼 being any category, we denote by P t𝔼 the category whose objects are the split epimorphisms in 𝔼 with a given splitting and morphisms the commutative squares between these data, and by ¶_{𝔼} : P t𝔼 → 𝔼 the functor associating its codomain with any split epimorphism. As soon as the category 𝔼 has pullbacks, the left exact functor ¶_{𝔼} is a fibration whose cartesian maps are the pullbacks between split epimorphisms. More precisely it is a fibered reflection [4], in the sense that it admits a fully faithful right adjoint I defined by I(Y) = (1_{Y} , 1_{Y}). The fibre above Y will be denoted P t_{Y} 𝔼. From now on we shall suppose that any category has finite limits. There is a left exact monad (T, λ, μ) on P t𝔼 defined by the following diagram:

[figure: X; s_{1}; f; R[f]; p_{0}; R^{2}[f]; p_{0}; p_{2}; Y; s; s; X; s_{0}; R[f]; s_{0}; p_{1}]

where T (f, s) = (p_{0}, s_{0}), and the other arrows in the diagram above are given by the iterated kernel equivalence relations:

\xymatrix{R^{2}[d_{0}] \ar[r]^{d_{2}} \ar[r]^{d_{1}} \ar[r]_{d_{0}} & R[f] \ar[r]^{d_{1}} \ar[r]_{d_{0}} & X \ar[r]_{f} \ar[l] & Y \ar[l]_{s}}

Now consider the diagram:

\xymatrix{P t𝔼 \ar[rr]^{T} \ar[dr]_{¶_{𝔼}.T} &  & P t𝔼 \ar[dl]^{¶_{𝔼}} \\  & 𝔼}

Actually, the functor ¶_{𝔼}.T : P t𝔼 → E is nothing but the domain functor. Finally let us recall that the category Alg^{T} is nothing but the category Grd𝔼 of internal groupoids in 𝔼, see [4].

1.6. Proposition. The functor ¶_{𝔼}.T : P t𝔼 → E has a right adjoint J defined by J(Y) = (p_{0}, s_{0}) : Y × Y ⇆ Y. In other words the split epimorphism J(Y) is the universal split epimorphism with respect to the domain functor. Moreover this functor J is monadic and consequently conservative.

Proof. Let us denote by w : ¶_{𝔼}.T.J ⇒ 1_{𝔼} the natural transformation defined by w_{Y} = p_{1}^{Y} : Y × Y → Y. Given any split epimorphism (f′, s′) and any map g : ¶_{𝔼}.T (f′, s′) = X′ → Y, the unique factorization (m, n) : (f′, s′) → J(Y) in P t𝔼 such that we get w_{Y} .¶_{𝔼}.T (m, n) = g is given by the following diagram:

\xymatrix{X′ \ar[r]^{(g.s′.f′,g)} \ar[d]_{f′} & Y × Y \ar[d]_{p_{0}} \\ Y′ \ar[r]_{g.s′} \ar[u]_{s′} & Y \ar[u]_{s_{0}}}

Finally, it is straightforward that the functor J creates coequalizers of J-contractible pairs.

Accordingly, we are here in the situation of our first observation where moreover any functor is left exact, the endofunctor T as well, and with two very special features. We already noticed the first one: the functor W̃ = J is monadic. On the other hand, given any groupoid \underline{X}_{1}, the natural map η̂_{\underline{X}_{1}} (see Proposition 1.3) is the following one:

[figure: X_{1}; d_{0}; d_{1}; (d_{0},d_{1}); X_{0} × X_{0}; p_{0}; p_{1}; X_{0}; X_{0}]

Accordingly, and this is the second special feature, the map η̂_{Ŵ (Y)} is an isomorphism:

[figure: Y × Y; p_{0}; p_{1}; (p_{0},p_{1}); Y × Y; p_{0}; p_{1}; Y; Y]

It is the conceptual reason why the forgetful left exact functor ( )_{0} : Alg^{T} = Grd𝔼 → 𝔼 is a fibered reflection, namely is such that the natural transformation ϵ̂ (see the end of the proof of Proposition 1.2) is an isomorphism: indeed, since the map η̂_{Ŵ (Y)} is an isomorphism, it is the case for any Ŵ (ϵ̂_{X}); but, since the functor W̃ = U^{T} .Ŵ is monadic, the functor Ŵ is conservative, and consequently ϵ̂_{X} is an isomorphism. Rather unexpectedly, the Proposition 1.2 seems to show that the fibered reflection aspect of ( )_{0} is independent from the existence of the right adjoint I of ¶_{𝔼}, namely from the fact that ¶_{𝔼} is itself a fibered reflection.

1.7. Strongly split epimorphisms. We shall need also the following, when the category 𝔼 is pointed. Any split epimorphism (f, s) determines now a split sequence:

[figure: Kerf; k; X; f; Y; s]

1.8. Definition. We shall say that the split epimorphism (resp. the split sequence) is a strongly split epimorphism (resp. a strongly split exact sequence), when the pair (k, s) is jointly strongly epic.

This terminology is justified by the following:

1.9. Proposition. Let 𝔼 be a pointed category with finite limits. Any strongly split epimorphism (f, s) is a normal epimorphism. In other words, any split sequence associated with a strongly split epimorphism is a strongly split exact sequence.

Proof. Consider the associated split sequence:

[figure: Kerf; k; X; f; Y; s]

We have to show that f is the cokernel of k. Let g : X → T be a map such that: g.k = 0. The only possible factorization Y → T is g.s. So we need to show that g.s.f = g. For that let us introduce the equalizer i of the pair (g, g.s.f). It is clear that both k and s factor through i. Consequently i is an isomorphism, and the two maps in question are equal.

This result was noticed independently in [15]. According to this proposition, a pointed category ℂ with finite limits is protomodular [3] if and only if any split epimorphism is a strongly split epimorphism.

2. The induced monad on the fibres P t_{Y} 𝔼

The monad (T, λ, μ) on P t𝔼 has other very strong properties: the endofunctor T preserves the ¶_{𝔼}-cartesian maps, while the natural transformations λ and μ are themselves ¶_{𝔼}-cartesian, see [4]. Finally the natural transformation m : T ⇒ W̃ V T defined in Proposition 1.3 is given, for any split epimorphism (f, s) : X ⇆ Y, by the following map, and consequently, lies inside the fibre P t_{X}𝔼:

[figure: R[f]; d_{0}; (d_{0},d_{1}); X × X; p_{0}; X; s_{0}; X; s_{0}]

In other words, we have a third special feature, namely: V (m_{X}) = 1_{V T (X)}.

2.1. The monad (T_{Y} , λ_{Y} , μ_{Y}). With all this information, we can derive a monad (T_{Y} , λ_{Y} , μ_{Y}) on the fibre P t_{Y} 𝔼 from the monad (T, λ, μ) on P t𝔼 by means of the following fibered construction, where, first, the map λ̄_{X} is the ¶_{𝔼}-cartesian map above V (λ_{X}), and λ_{Y} X is the factorization inside the fibre P t_{Y} 𝔼 induced by m_{X}; this makes the upper square a pullback:

\xymatrix{X \ar[r]^{λ_{X}} \ar[d]_{λ_{Y} X} & T (X) \ar[d]^{m_{X}} \\ T_{Y} (X) \ar[r]^{λ̄_{X}} & W̃ V T (X) \\ T_{Y}^{2}(X) \ar[r]_{λ̄_{T_{Y} X}} \ar[u]^{μ_{Y} X} & W̃ V T (T_{Y} (X)) \ar[u]_{W̃ (n_{X})}}

On the other hand, by adjunction, the map λ̄_{X} : T_{Y} (X) → W̃ V T (X) in P t𝔼 corresponds to a map n_{X} : V T (T_{Y} (X)) → V T (X) in 𝔼 such that n_{X}.V (λ_{T_{Y} (X)}) = V (λ_{X}); indeed we get first: n_{X}.V T (λ_{Y} X)) = w_{V T (X)}.V T (λ̄_{X}).V T (λ_{Y} X) = w_{V T (X)}.V T (m_{X}).V T (λ_{X}) = V (μ_{X}).V T (λ_{X}) = 1_{V T (X)}. Now, from V (λ_{T_{Y} (X)}) = V T (λ_{Y} X).V (λ_{X}), we get: n_{X}.V (λ_{T_{Y} (X)}) = n_{X}.V T (λ_{Y} X).V (λ_{X}) = V (λ_{X}). The map μ_{Y} X is the factorization of W̃ (n_{X}).λ̄_{T_{Y} X} through the ¶_{𝔼}-cartesian map λ̄_{X}. In spite of appearances, the map μ_{X} is involved in the process, since it is involved in the definition of the map m_{X}. The monad (T_{Y} , λ_{Y} , μ_{Y}) is now precisely described by the following diagram in 𝔼:

[figure: X; (f,1); f; Y × X; p_{Y}; Y × (Y × X); p_{Y}; Y ×p_{X}; Y; s; (1,s); (s_{0},s)]

2.2. The T_{Y} -algebras. We shall determine now the nature of the T_{Y} -algebra structures on a split epimorphism (f, s) : X ⇆ Y and show how, in some specific context, they are controlling the decompositions of the form X ≃ Y × T (see also [7]). Actually the monad (T_{Y} , λ_{Y} , μ_{Y}) is generated by an adjunction: let us denote by Y \𝔼 the usual coslice category and by Υ_{Y} : Y \𝔼 → P t_{Y} 𝔼 the functor defined, for any map t : Y → T by Υ_{Y} (t) = (p_{Y} , (1, t)) : Y × T ⇆ Y.

2.3. Proposition. The functor Υ_{Y} has a left adjoint Ψ_{Y} : P t_{Y} 𝔼 → Y \𝔼 defined by Ψ_{Y} (f, s) = s. It is left exact and produces the monad (T_{Y} , λ_{Y} , μ_{Y}).

Proof. Straightforward checking.

So, there is a canonical left exact comparison functor Γ_{Y}:

\xymatrix{Y \𝔼 \ar[rr]^{Γ_{Y}} \ar[dr]_{Υ_{Y}} &  & Alg^{T_{Y}} \ar[dl]^{U^{T_{Y}}} \\  & P t_{Y} 𝔼}

The upper level of the canonical sequence yielded by the adjunction in Y \𝔼:

\xymatrix{(Ψ_{Y} .Υ_{Y})^{2}(t) \ar[r] \ar[r] & Ψ_{Y} .Υ_{Y} (t) \ar[r] & t}

is, in 𝔼:

\xymatrix{Y × Y × T \ar[r]^{p_{1}^{Y} ×T} \ar[r]_{p_{0}^{Y} ×T} & Y × T \ar[r]^{p_{T}} & T}

It is clear that, when 𝔼 is pointed, it is a coequalizer diagram, and that consequently the factorization Γ_{Y} is fully faithful. The same property holds, when 𝔼 is regular [1] and the object Y has global support. Let us now characterize the T_{Y} -algebras. For that let us recall the following: given any pair (R, S) of equivalence relations on an object X consider the following pullback:

[figure: R□S; S × S; (d_{0}×d_{0},d_{1}×d_{1}); R × R; (d_{0},d_{1})×(d_{0},d_{1}); X^{4}]

which determines the double parallelistic relation associated to (R, S) (see [6] and also [10]):

\xymatrix{R□S \ar[d]_{p_{0}^{R}} \ar[d]^{p_{1}^{R}} \ar[r]_{p_{0}^{S}} \ar[r]^{p_{1}^{S}} & S \ar[d]_{d_{0}} \ar[d]^{d_{1}} \ar[l] \\ R \ar[r]_{d_{0}} \ar[r]^{d_{1}} \ar[u] & X \ar[u] \ar[l]}

2.4. Definition. We say that the pair (R, S) is strictly centralizing when one of the downward and rightward commutative square is a pullback.

In this circumstance, all the other commutative squares are pullbacks since the one in question determines a discrete fibration between groupoids. In the set theoretical context this means that for any triple xRySz, there is a unique t such that xStRz. In any category 𝔼, and for any pair (Y, T) of objects the pair (R[p_{Y}], R[p_{T}]) is a strictly centralizing pair on the object Y ×T. A strictly centralizing pair (R, S) is necessarily such that: R ∩S = Δ_{X}.

2.5. Proposition. A T_{Y} -algebra structure ψ : Y × X → X on (f, s) is equivalent to the data of an equivalence relation S on X such that the pair (R[f], S) is strictly centralizing and the equivalence relation S satisfies: f(S) = ∇_{Y}.

Proof. The map ψ satisfies: f.ψ = p_{Y}, ψ.(f, 1) = 1_{X} and ψ.p_{0} × X = ψ.Y × ψ. Notice that the map s is not involved. Actually the previous equations show that the map ψ completes the following diagram into a vertical discrete fibration between groupoids:

[figure: Y × Y × X; Y ×Y ×f; Y ×ψ; p_{0}×X; p_{1}×X; Y × X; Y ×f; ψ; p_{X}; X; f; (f,1); Y × Y × Y; p_{0}; p_{1}; p_{2}; Y × Y; p_{0}; p_{1}; Y; s_{0}]

This implies that the right hand side dotted square is a pullback. Since the lower groupoid is an equivalence relation, so is the upper one which produces an equivalence relation S on X, which is such that: f(S) = ∇_{Y} since f and Y × f are (split) epimorphisms. If we complete the previous diagram by the kernel equivalence relations:

\xymatrix{R[ϕ] \ar[d]_{p_{0}} \ar[d]^{p_{1}} \ar[r]_{R(p_{0})} \ar[r]^{R(p_{1})} & R[f] \ar[d]_{d_{0}} \ar[d]^{d_{1}} \ar[l] \\ S \ar[r]_{d_{0}} \ar[r]^{d_{1}} \ar[u] \ar@{->>}[d]_{ϕ} & X \ar[u] \ar[l] \ar@{->>}[d]_{f} \\ Y × Y \ar[r]_{p_{0}} \ar[r]^{p_{1}} & Y \ar[l]}

the upper part of the diagram is the parallelistic double relation associated with the pair (S, R[f]). The lower squares being pullbacks, so are the upper ones, and the pair (S, R[f]) is strictly centralizing. Conversely, suppose there is an equivalence relation S satisfying the previous conditions. Let us consider the following diagram:

\xymatrix{R[f]□S \ar[d]_{p_{0}^{R}} \ar[d]^{p_{1}^{R}} \ar[r]_{p_{0}^{S}} \ar[r]^{p_{1}^{S}} & S \ar[d]_{d_{0}} \ar[d]^{d_{1}} \ar[l] \ar@{->>}[r]^{q} & T \ar@{.>}[d] \ar@{.>}[d] \\ R[f] \ar[r]_{d_{0}} \ar[r]^{d_{1}} \ar[u] & X \ar[u] \ar[l] \ar[r]^{f} & Y \ar@{.>}[l]^{s} \ar@{.>}[u]}

Since we have, on the left hand side, discrete fibrations between equivalence relations and since f is a split epimorphism, we have the quotient T of the upper horizontal equivalence relation, which produces the dotted right hand side pullbacks. Since R[f] ∩ S = Δ_{X}, the vertical right hand side groupoid T is actually an equivalence relation on Y. Now we have T = f(S) = ∇_{Y} as equivalence relations, and consequently T ≃ Y × Y as objects. Accordingly we have S ≃ Y × X, whence the splitting ψ : Y × X ≃ S ⟶^{d_{0}} X (and the T_{Y} -algebra) we were looking for.

2.6. Corollary. When 𝔼 is pointed, any comparison functor Γ_{Y} is an equivalence of categories; in other words, we have Alg^{T_{Y}} ≃ Y \𝔼. When 𝔼 is exact [1] (or even efficiently regular), it is the case provided that the object Y has global support.

Proof. We know that, in both situations, the comparison Γ_{Y} is fully faithful. We are going to show that it is essentially surjective. Given any T_{Y} -algebra ψ let us consider the following diagram where the left hand side part is a vertical discrete fibration between equivalence relations:

[figure: Y × X; Y ×f; ψ; p_{X}; X; f; (f,1); q; T; Y × Y; p_{0}; p_{1}; Y; s_{0}; τ_{Y}; 1]

In both cases the upper equivalence relation is effective and we can complete the diagram with its quotient map q which produces the right hand side pullback. Accordingly we get an isomorphism γ in P t_{Y} 𝔼:

\xymatrix{X \ar[r]^{γ} \ar[dr]_{f} & Y × T \ar[d]_{p_{Y}} \\  & Y \ar[ul]_{s} \ar[u]_{(1,γ.s)}}

In the pointed case, the splitting of the terminal map τ_{Y} imposes (actually by construction) T ≃ Kerf.

2.7. Special T_{Y} -algebras. But it does not impose that we have γ.s = 0. We already noticed that the section s of f is not directly involved in the equations defining a T_{Y} - algebra.

2.8. Definition. A T_{Y} -algebra ψ is said to be special when, in addition, it satisfies: ψ.Y × s = s.p_{0}.

Accordingly a T_{Y} -algebra ψ is special, when the section s of f determines a section of the discrete fibration described in the proof of Proposition 2.5; it consequently determines a section of the terminal map τ_{T} : T → 1 in the subsequent corollary. Whence the straightforward following:

2.9. Proposition. Suppose the category 𝔼 is pointed. Then a split epimorphism (f, s) : X ⇆ Y is isomorphic to the split epimorphism (p_{Y} , ι_{Y}) : Y × Kerf ⇆ Y if and only if it is endowed with the structure of a special T_{Y} -algebra.

3. Mal’cev context

Mal’cev categories were introduced in [9] and [10] as those categories ℂ in which any reflexive relation is an equivalence relation. They are characterized by the fact that any fibre P t_{Y} ℂ of the fibration ¶_{ℂ} is unital, a context in which there is an intrinsic notion of commutation [3]. On the other hand, in this same context, there is also an intrinsic notion of commutation of equivalence relations, see [16] and [6]; thanks to Definition 2.4, we are allowed to say now that two equivalence relations R and S on an object X are strictly centralizing if and only if we have: R ∩ S = Δ_{X}.

3.1. Mal’cev monads. Recall also that a monad (T, λ, μ) on any category 𝔼 is said to be a Mal’cev monad, when the pair (λT, T λ) is jointly strongly epic [5]. The main property of a Mal’cev monad is that any T -algebra is characterized by the only unit axiom. First we get the following:

3.2. Proposition. Let ℂ be a Mal’cev category. Then the monads (T, λ, μ) on P tℂ and (T_{Y} , λ_{Y} , μ_{Y}) on P t_{Y} ℂ for any Y are Mal’cev monads.

Proof. 1) The first statement comes from the fact that, in a Mal’cev category, any kernel equivalence relation of a split epimorphism:

[figure: R[f]; d_{1}; d_{0}; X; f; s_{0}; s_{1}; Y; s]

is such that the pair (s_{0}, s_{1}) is jointly strongly epic.

2) Suppose, now, we have a subobject W ↣ Y × Y × X containing s_{0} × X and Y × (f, 1). This can be understood as a relation (y, y′)W x such that (y, y)W x for all (y, x) and (y, f(x))W x for all (y, x). Let us recall that, in a Mal’cev category, any relation is difunctional. Consider, for any triple (y, y′, x), the following diagram:

[figure: (y, y); x; (y, y′); s(y′)]

It shows that ((y, y′), x) is in W, and that w is an isomorphism.

Accordingly, any splitting ψ : Y ×X → X of the map (f, 1) such that f.ψ = p_{Y} determines a T_{Y} -algebra and produces the conditions of Proposition 2.5 which can be reformulated:

3.3. Proposition. Let ℂ be a Mal’cev category. A T_{Y} -algebra structure ψ : Y × X → X on (f, s) is equivalent to the data of an equivalence relation S on X such that we have: R[f] ∩ S = Δ_{X} and f(S) = ∇_{Y}.

3.4. Internal groupoids in the Mal’cev context. Since (T, λ, μ) becomes a Mal’cev monad, a reflexive graph (d_{0}, d_{1}) : X_{1} ⇉ X_{0} is a groupoid if and only if there is a map d_{2}:

[figure: R[d_{0}]; d_{2}; d_{1}; d_{0}; X_{1}; d_{1}; d_{0}; X_{0}]

such that d_{0}.d_{2} = d_{1}.d_{0} and d_{2}.s_{1} = 1_{X_{1}}. On the other hand it is well known [10] that, in this context, any internal category is a groupoid. Now, some aspects of the monad T_{Y} on P t_{Y} ℂ will allow us to shed a new light on another well known property of Mal’cev categories concerning what is called multiplicative reflexive graph in [10]. From any reflexive graph (d_{0}, d_{1}) : X_{1} ⇉ X_{0} in ℂ, we get, by the monad T_{X_{0}}, two subobjects in P t_{X_{0}}ℂ:

[figure: X_{1}; (d_{0},1); d_{0}; X_{0} × X_{1}; p_{X_{0}}; X_{1}; (d_{1},1); d_{1}; X_{0}; s_{0}; (1,s_{0}); s_{0}]

We can assert now the following:

3.5. Proposition. Let ℂ be a Mal’cev category. The reflexive graph in question is a groupoid if and only if these two subobjects commute in P t_{X_{0}}ℂ.

Proof. These two subobjects commute in the fibre P t_{X_{0}}ℂ when they have a cooperator ϕ : X_{1} ×_{X_{0}} X_{1} → X_{0} × X_{1} (i.e. a map satisfying: ϕ.s_{0} = (d_{1}, 1) and ϕ.s_{1} = (d_{0}, 1)):

[figure: X_{1} ×_{X_{0}} X_{1}; ϕ; d_{2}; d_{0}; X_{1}; (d_{0},1); d_{0}; s_{1}; X_{0} × X_{1}; p_{X_{0}}; X_{1}; (d_{1},1); d_{1}; s_{0}; X_{0}; s_{0}; (1,s_{0}); s_{0}]

where the whole quadrangle is a pullback. The map ϕ is consequently a pair (d_{0}.d_{2}, d_{1}), where d_{1} : X_{1} ×_{X_{0}} X_{1} → X_{1} is such that: d_{1}.s_{0} = 1_{X_{1}} and d_{1}.s_{1} = 1_{X_{1}}. This map d_{1} with these two identities makes multiplicative in the sense of [10] the reflexive graph in question. And, according to Theorem 2.2 in [10], in a Mal’cev category, any reflexivemultiplicative graph is a groupoid.

It is well known also that, in this context of Mal’cev categories, an internal reflexive graph (d_{0}, d_{1}) : X_{1} ⇉ X_{0} in ℂ is actually an internal groupoid if and only if the equivalence relations R[d_{0}] and R[d_{1}] commute in ℂ, i.e. if and only if we have [R[d_{0}], R[d_{1}]] = 0, see [10] and [6]. So, in the Mal’cev context, a reflexive graph is a groupoid if and only if either the equivalence relations R[d_{0}] and R[d_{1}] commute in ℂ or the subobjects (d_{0}, 1) and (d_{0}, 1) commute in the fibre P t_{X_{0}}ℂ. Further properties of internal groupoids in a Mal’cev categories are developed in [12].

3.6. When Mal’cev categories are protomodular. The unit of the monad T_{Y} on the fibre P t_{Y} 𝔼 is actually the kernel of a split epimorphism in this pointed category:

[figure: X; (f,1); f; Y × X; p_{Y}; Y ×f; Y × Y; Y ×s; p_{Y}; Y; s; s_{0}]

In the Mal’cev context this split sequence is actually a split exact sequence:

3.7. Proposition. Let ℂ be any Mal’cev category. Then the previous split sequence in P t_{Y} ℂ is a strongly split exact sequence in P t_{Y} ℂ, or equivalently it produces a strongly split epimorphism P t_{Y} ℂ (see Definition 1.8).

Proof. Let w : W ↣ Y × X be a subobject containing (f, 1) and Y × s. Since ℂ is Mal’cev, the relation W on Y × X is difunctional. Consider, for any pair (y, x) ∈ Y × X, the following diagram:

[figure: f(x); x; y; s.f(x)]

It shows that (y, x) is in W, and that w is an isomorphism.

We have now the following characterization:

3.8. Lemma. Let ℂ be a Mal’cev category and (f, s) : X ⇆ Y any split epimorphism. A map ψ : Y × X → X is a special T_{Y} -algebra if and only if we have ψ.(f, 1) = 1_{X} and ψ.Y × s = s.p_{0}. Accordingly there is at most one special T_{Y} -algebra structure on it.

Proof. These conditions are necessary. Conversely it remains to show that ψ.f = p_{Y} which is a consequence of the fact that, thanks to the asserted conditions, the compositions of these two maps by the strongly epic pair ((f, 1), Y × s) are the same.

3.9. Proposition. Let ℂ be a Mal’cev category. The two subobjects (f, 1) and Y × s commute in P t_{Y} ℂ if and only if the split epimorphism (f, s) is endowed with the structure of a special T_{Y} -algebra. When the category ℂ is pointed, the only split epimorphisms inducing this commutation are the trivial ones (p_{Y} , ι_{Y}) : Y × T ⇆ T.

Proof. Suppose the two subobjects (f, 1) and Y × s commute in P t_{Y} ℂ, and denote by (α, β) : X × Y → Y × X the associated cooperator:

[figure: X × Y; f×Y; p_{X}; Y × X; Y × Y; p_{0}; Y ×s; s×Y; X; f; (f,1); (1,f); Y; s_{0}; s]

such that (α, β).(1, f) = (f, 1) and (α, β).s × Y = Y × s. This means that β.(1, f) = 1_{X} and β.s×Y = s.p_{1}. If we denote by tw_{Y,X} : Y ×X → X ×Y the twisting isomorphism, we get (β.tw_{Y,X}).(f, 1) = 1_{X} and (β.tw_{Y,X}).Y × s = s.p_{0}. According to the previous lemma, the map ψ = β.tw_{Y,X} is then a special T_{Y} -algebra. Conversely if ψ is a special T_{Y} -algebra structure on (f, s), then β = ψ.tw_{X,Y} is the cooperator which makes commute the pair ((f, 1), Y × s). The last assertion is a straightforward consequence of Proposition 2.9.

In [13], was made the remarkable observation that, in the category Gp of groups, any change of base functor with respect to the fibration of points has a right adjoint. The same property holds in the category R-Lie of Lie R-algebras, where R is a commutative ring, see [14]. Such a kind of category was called locally algebraically cartesian closed in [8]. The specific cohesion generated by this local algebraic cartesian closedness can be measured by the following:

3.10. Theorem. Let ℂ be a pointed Mal’cev category. When, in addition, it is locally algebraically cartesian closed, then it is protomodular.

Proof. Let (f, s) : X ⇆ Y be a split epimorphism in ℂ. In the pointed Mal’cev fibre P t_{Y} ℂ we get the previous strongly split exact sequence. Now the change of base functor α_{Y}^{∗} along the initial map, having a right adjoint, preserves the jointly strongly epic pairs, see Proposition 1.4. Accordingly the image by α_{Y}^{∗} of this strongly split exact sequence, namely the following one:

[figure: Kerf; k; X; f; Y; s]

is still a strongly split exact sequence, which is the characterization of a pointed protomodular category [3].

Now, what does happen in the non-pointed case? The answer is the following:

3.11. Theorem. Let ℂ be a regular Mal’cev category. When, in addition, it is locally algebraically cartesian closed, then the full subcategory ℂ_{♯} of ℂ whose objects are those which have global support is protomodular.

Proof. Let y : Y′ → Y be any map in ℂ; we have to show that the change of base functor h^{∗} : P t_{Y} ℂ → P t_{Y′}ℂ is conservative. Since h^{∗} is left exact, it is enough to check it on monomorphisms. So let u : (f̄, s̄) ↣ (f, s) be a monomorphism in P t_{Y} ℂ. We get a morphism of strong split exact sequences in this fibre:

[figure: X̄; (f̄,1); u; Y × X̄; Y ×u; Y ×f̄; Y × Y; Y ×s̄; X; (f,1); Y × X; Y ×f; Y × Y; Y ×s]

Suppose that h^{∗}(u) is an isomorphism. Since ℂ is lacc, the change of base functor h^{∗} preserves the strong split exact sequences. Accordingly the monomorphism h^{∗}(Y × u) is an isomorphism as well in the fibre P t_{Y′}ℂ; that means that the monomorphism Y′ × u : Y′×X̄ ↣ Y′×X is an isomorphism in ℂ. Now, when Y′ has global support, the following square has vertical regular epimorphisms:

[figure: Y′ × X̄; Y′×u; p_{X′}; Y′ × X; p_{X}; X̄; u; X]

Consequently, if Y′ × u is an isomorphism, so is u.

The two previous results are even more striking, if we recall that, according to Theorem 4.3 in [8], any locally algebraically cartesian closed protomodular category is necessarily strongly protomodular.

References

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[11] S. Eilenberg and J.C. Moore, Adjoint functors and triples, Illinois J. Math, 9, 1965, 381-398.

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Université du Littoral, Laboratoire de Mathématiques Pures et Appliquées, Bat. H. Poincaré, 50 Rue F. Buisson, BP 699, 62228 Calais Cedex, France

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