Some remarks on connectors and groupoids in Goursat categories

TL;DR

This paper demonstrates that connectors are stable under quotients in Goursat categories, leading to a new characterization of Goursat categories via properties of internal groupoids, thus advancing the understanding of categorical structures.

Contribution

It proves the stability of connectors under quotients in Goursat categories and characterizes Goursat categories through internal groupoid properties.

Findings

Connectors are stable under quotients in Goursat categories.

The category of connectors inherits Goursat properties.

Goursat categories can be characterized by internal groupoid properties.

Abstract

We prove that connectors are stable under quotients in any (regular) Goursat category. As a consequence, the category $\mathsf{Conn}(\mathbb{C})$ of connectors in $\mathbb{C}$ is a Goursat category whenever $\mathbb C$ is. This implies that Goursat categories can be characterised in terms of a simple property of internal groupoids.

Equations

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Full text

The third author acknowledges partial financial assistance by Centro de Matemática da Universidade de Coimbra—UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.

Over the last twenty years the property of $n$-permutability of congruences in a variety of universal algebras has been investigated from a categorical perspective (see [7, 13, 19], for instance, and references therein). When $\mathbb{C}$ is a regular category, the $2$-permutability property, usually called the Mal’tsev property, is a concept giving rise to a beautiful theory, whose main features are collected in [1]. Many important results still hold when a regular category $\mathbb{C}$ satisfies the strictly weaker property of $3$-permutability, namely the Goursat property. A nice feature of a (regular) Goursat category $\mathbb{C}$ is that the lattice of equivalence relations on any object in $\mathbb{C}$ is a modular lattice [7], a property that plays a crucial role in commutator theory [10, 17].

The aim of this paper is twofold: first of all we establish some basic properties of Goursat categories in terms of connectors [4], as it was done in [4] for the case of Mal’tsev categories. These results have turned out to be useful to develop a monoidal approach to internal structures [12]. We then give a new characterisation of Goursat categories in terms of properties of (internal) groupoids, on the model of what was done in [11] in the case of Mal’tsev categories.

In the first section, we recall the main properties of Goursat categories that will be needed throughout the paper. In Section $2$ we prove that for any Goursat category $\mathbb{C}$, the category $\mathrm{Equiv}(\mathbb{C})$ of equivalence relations in $\mathbb{C}$ is also a Goursat category (Proposition 2, see also [2]). We use this result to give some properties of Goursat categories in terms of connectors in Section $3$. More precisely, we show that, when $\mathbb{C}$ is a Goursat category, then connectors are stable under quotients in $\mathbb{C}$ (Proposition 3), and this implies that the category $\mathrm{Conn}(\mathbb{C})$ of connectors in $\mathbb{C}$ is again a Goursat category (Theorem 3).

We conclude the paper by giving a new characterisation of Goursat categories in terms of properties of groupoids and internal categories (Theorem 3). It turns out that a regular category $\mathbb{C}$ is a Goursat category if and only if the category $\mathrm{Grpd}(\mathbb{C})$ of groupoids (equivalently, the category $\mathrm{Cat}(\mathbb{C})$ of internal categories) in $\mathbb{C}$ is closed under quotients in the category $\mathrm{RG}(\mathbb{C})$ of reflexive graphs in $\mathbb{C}$.

1. Preliminaries

In this section we recall some basic definitions and properties of (regular) Goursat categories, needed throughout the article. We shall always assume that the category $\mathbb{C}$ in which we are working is a regular category: this means that $\mathbb{C}$ is finitely complete, regular epimorphisms are stable under pullbacks, and kernel pairs have coequalisers. Equivalently, any arrow $f:A\longrightarrow B$ has a unique factorisation $f=i\circ r$ (up to isomorphism), where $r$ is a regular epimorphism and $i$ is a monomorphism and this factorisation is pullback stable; the subobject corresponding to $i$ is called the image of $f$.

A relation $R$ from $X$ to $Y$ is a subobject $\langle r_{1},r_{2}\rangle:R\rightarrowtail X\times Y$. The opposite relation of $R$, denoted $R^{o}$, is the relation from $Y$ to $X$ given by the subobject $\langle r_{2},r_{1}\rangle:R\rightarrowtail Y\times X$. A relation $R$ from $X$ to $X$ is called a relation on $X$. We shall identify a morphism $f:X\longrightarrow Y$ with the relation $\langle 1_{X},f\rangle:X\rightarrowtail X\times Y$ and write $f^{o}$ for its opposite relation. Given another relation $\langle s_{1},s_{2}\rangle:S\rightarrowtail Y\times Z$ from $Y$ to $Z$, one can define the composite relation $SR$ of $R$ and $S$ as the image of the arrow $(r_{1}\circ p_{1},s_{2}\circ p_{2}):R\times_{Y}S\longrightarrow X\times Z$, where $(R\times_{Y}S,p_{1},p_{2})$ is the pullback of $r_{2}:R\longrightarrow Y$ along $s_{1}:S\longrightarrow Y$. With the above notations, any relation $\langle r_{1},r_{2}\rangle:R\rightarrowtail X\times Y$ can be seen as the relational composite $r_{2}r_{1}^{o}$.

The following properties are well known and easy to prove. We collect them in the following lemma:

Lemma \thethm.

Let $f:X\longrightarrow Y$ be an arrow in a regular category $\mathbb{C}$, and let $i\circ r$ be its (regular epimorphism, monomorphism) factorisation. Then:

1. (1)

$f^{o}f$* is the kernel pair of $f$, thus $1_{X}\leqslant f^{o}f$; moreover, $1_{X}=f^{o}f$ if and only if $f$ is a monomorphism;* 2. (2)

$ff^{o}$* is $(i,i)$, thus $ff^{o}\leqslant 1_{Y}$; moreover, $ff^{o}=1_{Y}$ if and only if $f$ is a regular epimorphism;* 3. (3)

$ff^{o}f=f$* and $f^{o}ff^{o}=f^{o}$.*

{defi}

A relation $(R,r_{1},r_{2})$ on an object $X$ is said to be :

• •

reflexive when there is an arrow $r:X\longrightarrow R$ such that $r_{1}\circ r=1_{X}=r_{2}\circ r$;

• •

symmetric when there is an arrow $\sigma:R\longrightarrow R$ such that $r_{2}=r_{1}\circ\sigma$ and $r_{1}=r_{2}\circ\sigma$;

• •

transitive when, by considering the following pullback

[TABLE]

there is an arrow $t:R\times_{X}R\longrightarrow R$ such that $r_{1}\circ t=r_{1}\circ p_{1}$ and $r_{2}\circ t=r_{2}\circ p_{2}$.

• •

an equivalence relation if $R$ is reflexive, symmetric and transitive.

In particular, a kernel pair $\langle f_{1},f_{2}\rangle:\mathrm{Eq}(f)\rightarrowtail X\times X$ of a morphism $f:X\longrightarrow Y$ is an equivalence relation. The equivalence relations that occur as kernel pairs of some morphism in $\mathbb{C}$ are called effective. Let $\mathrm{Equiv}(\mathbb{C})$ be the category whose objects are equivalence relations in $\mathbb{C}$ and arrows from $\langle r_{1},r_{2}\rangle:R\rightarrowtail X\times X$ to $\langle s_{1},s_{2}\rangle:S\rightarrowtail Y\times Y$ are pairs $(f,g)$ of arrows in $\mathbb{C}$ making the following diagram commute

[TABLE]

When $\mathbb{C}$ is a regular category, $(R,r_{1},r_{2})$ is an equivalence relation on $X$ and $f:X\twoheadrightarrow Y$ a regular epimorphism, we define the regular image of $R$ along $f$ to be the relation $f(R)$ on $Y$ induced by the (regular epimorphism, monomorphism) factorisation $\langle s_{1},s_{2}\rangle\circ\psi$ of the composite $(f\times f)\circ\langle r_{1},r_{2}\rangle$:

[TABLE]

Note that the regular image $f(R)$ can be obtained as the relational composite $f(R)=fRf^{o}=fr_{2}r_{1}^{o}f^{o}$. When $R$ is an equivalence relation, $f(R)$ is also reflexive and symmetric. In a general regular category $f(R)$ is not necessarily an equivalence relation. This is the case in a Goursat category (Theorem 1).

Definition \thethm [8, 7].

A regular category $\mathbb{C}$ is called a Goursat category when the equivalence relations in $\mathbb{C}$ are $3$-permutable, i.e. $RSR=SRS$ for any pair of equivalence relations $R$ and $S$ on the same object.

The following characterisation will be useful in the sequel:

Theorem \thethm [7].

A regular category $\mathbb{C}$ is a Goursat category if and only if for any regular epimorphism $f:X\twoheadrightarrow Y$ and any equivalence relation $R$ on $X$, the regular image $f(R)=fRf^{o}$ of $R$ along $f$ is an equivalence relation.

There are many important algebraic examples of Goursat categories. Indeed, by a classical theorem in [18], a variety of universal algebras is a Goursat category precisely when its theory has two ternary operations $r$ and $s$ such that the identities $r(x,y,y)=x$, $r(x,x,y)=s(x,y,y)$ and $s(x,x,y)=y$ hold. Accordingly, the categories of groups, abelian groups, modules over some fixed ring, crossed modules, quasi-groups, rings, associative algebras, Heyting algebras and implication algebras are all Goursat categories.

Any regular Mal’tsev category is a Goursat category, thus, in particular, so is any semi-abelian category.

Many interesting properties of Goursat categories can be found in the literature (see [7, 15, 16] and references therein). In particular, the following characterisations will be useful for the development of this work:

Theorem \thethm [15].

Let $\mathbb{C}$ be a regular category. The following conditions are equivalent:

1. (i)

$\mathbb{C}$* is a Goursat category;* 2. (ii)

any commutative diagram where $\alpha$ and $\beta$ are regular epimorphisms and $f$ and $g$ are split epimorphisms in $\mathbb{C}$

[TABLE]

(which is necessarily a pushout) is a Goursat pushout: the morphism $\lambda:\mathrm{Eq}(f)\longrightarrow\mathrm{Eq}(g)$ induced by the universal property of kernel pair $\mathrm{Eq}(g)$ of $g$ is a regular epimorphism.

We recall part of Theorem 1.3 in [16]:

Theorem \thethm [16].

Let $\mathbb{C}$ be a regular category. The following conditions are equivalent:

1. (i)

$\mathbb{C}$* is a Goursat category;* 2. (ii)

for any commutative cube

[TABLE]

where the left square is a pullback of split epimorphisms, the right square is a commutative square of split epimorphisms and the horizontal arrows $\alpha$, $\beta$, $\gamma$ and $\delta$ are regular epimorphisms (commuting also with the splittings), then the right square is a pullback.

2. Equivalence relations in Goursat categories

In this section we prove that $\mathrm{Equiv}(\mathbb{C})$ is a Goursat category for any Goursat category $\mathbb{C}$.

The category $\mathrm{Equiv}(\mathbb{C})$ is finitely complete whenever $\mathbb{C}$ is: the terminal object in $\mathrm{Equiv}(\mathbb{C})$ is the discrete equivalence relation

[TABLE]

on the terminal object $1$ of $\mathbb{C}$, and pullbacks are computed “levelwise”. In particular, the kernel pair of a morphism $(f,g)$ in $\mathrm{Equiv}(\mathbb{C})$ is given by the kernel pairs $\mathrm{Eq}(f)$ of $f$ and $\mathrm{Eq}(g)$ of $g$ in $\mathbb{C}$

[TABLE]

Consequently, a morphism $(f,g)$ is a monomorphism in $\mathrm{Equiv}(\mathbb{C})$ if and only if $f$ and $g$ are monomorphisms in $\mathbb{C}$. When $\mathbb{C}$ is a Goursat category, a similar property holds with respect to regular epimorphisms:

Lemma \thethm.

Let $R$ and $S$ be two equivalence relations in a Goursat category $\mathbb{C}$ and $(f,g):R\rightarrow S$ a morphism

[TABLE]

in $\mathrm{Equiv}(\mathbb{C})$. Then $(f,g)$ is a regular epimorphism in $\mathrm{Equiv}(\mathbb{C})$ if and only if $f$ and $g$ are regular epimorphisms in $\mathbb{C}$.

Proof.

When $f$ and $g$ are regular epimorphisms in $\mathbb{C}$, it is not difficult to check that $(f,g)$ is necessarily the coequaliser of its kernel pair in $\mathrm{Equiv}(\mathbb{C})$ given in (1) (one uses the fact that $g=coeq(g_{1},g_{2})$ and $f=coeq(f_{1},f_{2})$ in $\mathbb{C}$).

Conversely, let $(f,g)$ be a morphism in $\mathrm{Equiv}(\mathbb{C})$ as in (2) that is a regular epimorphism in $\mathrm{Equiv}(\mathbb{C})$. Consider the kernel pairs of $f$ and $g$, the (regular epimorphism, monomorphism) factorisation $f=i\circ q$ of $f$, and the regular image $(q(R),t_{1},t_{2})$ of $(R,r_{1},r_{2})$ along $q$. We obtain the following commutative diagram

[TABLE]

where $(q(R),t_{1},t_{2})\in\mathrm{Equiv}(\mathbb{C})$ (by Theorem 1) and $(i,j)$ is the morphism in $\mathrm{Equiv}(\mathbb{C})$ such that $(i,j)\circ(q,\alpha)=(f,g)$. Note that $j$ is induced from the fact that $(i\times i)\circ\langle t_{1},t_{2}\rangle\circ\alpha$ is the (regular epimorphism, monomorphism) factorisation of $\langle s_{1},s_{2}\rangle\circ g$, thus it is a monomorphism

[TABLE]

From the fact that $(f,g)$ is the coequaliser of its kernel pair in $\mathrm{Equiv}(\mathbb{C})$ it easily follows that $(i,j)$ is an isomorphism in $\mathrm{Equiv}(\mathbb{C})$. This implies that $f$ and $g$ are regular epimorphisms in $\mathbb{C}$. ∎

Proposition \thethm.

$\mathrm{Equiv}(\mathbb{C})$* is a Goursat category whenever $\mathbb{C}$ is.*

Proof.

As mentioned above, the category $\mathrm{Equiv}(\mathbb{C})$ is finitely complete because $\mathbb{C}$ is so. Lemma 2 implies that regular epimorphisms in $\mathrm{Equiv}(\mathbb{C})$ are stable under pullbacks since regular epimorphisms are stable in $\mathbb{C}$, and regular epimorphisms in $\mathrm{Equiv}(\mathbb{C})$ are “levelwise” regular epimorphisms. The existence of the (regular epimorphism, monomorphism) factorisation of a morphism $(f,g)$ as in (2) in the category $\mathrm{Equiv}(\mathbb{C})$ follows from the construction of diagram (3): the (regular epimorphism, monomorphism) factorisation $f=i\circ q$ of $f$ in $\mathbb{C}$ gives rise to the (regular epimorphism, monomorphism) factorisation $g=j\circ\alpha$ of $g$ in $\mathbb{C}$. Thus $(q,\alpha)\circ(i,j)$ is the (regular epimorphism, monomorphism) factorisation of $(f,g)$ in $\mathrm{Equiv}(\mathbb{C})$. To see that $\mathrm{Equiv}(\mathbb{C})$ has the Goursat property one uses Theorem 1: to check that the regular image of an equivalence relation in the category $\mathrm{Equiv}(\mathbb{C})$ is again an equivalence in $\mathrm{Equiv}(\mathbb{C})$ one mainly uses the same (“levelwise”) property in the category $\mathbb{C}$. ∎

3. Connectors and groupoids in Goursat categories

In this section we prove that connectors are stable under quotients in any Goursat category $\mathbb{C}$. We then define the category $\mathrm{Conn}(\mathbb{C})$ of connectors in $\mathbb{C}$ whose objects are pairs of equivalence relations equipped with a connector, and prove that $\mathrm{Conn}(\mathbb{C})$ is a Goursat category whenever the base category $\mathbb{C}$ is. We conclude by giving a new characterisation of Goursat categories in terms of properties of groupoids and internal categories.

{defi}

Let $(R,r_{1},r_{2})$ and $(S,s_{1},s_{2})$ be two equivalence relations on an object $X$ and $R\times_{X}S$ the pullback of $r_{2}$ along $s_{1}$. A connector [4] between $R$ and $S$ is an arrow $p:R\times_{X}S\longrightarrow X$ in $\mathbb{C}$ such that

1. (1)

$xSp(x,y,z)Rz$; 2. (2)

$p(x,x,y)=y$; 3. (3)

$p(x,y,y)=x$; 4. (4)

$p(x,y,p(z,u,v))=p(p(x,y,z),u,v)$,

when each term is defined.

Remark \thethm.

Given two regular epimorphisms $d\colon X\twoheadrightarrow Y$ and $c\colon X\twoheadrightarrow Z$, a connector on the effective equivalence relations $\mathrm{Eq}(d)$ and $\mathrm{Eq}(c)$ is the same thing as an internal pregroupoid in the sense of Kock [20, 21] (see also the introduction of [5], for instance, for a comparison between these two related notions and some additional references). In the context of Mal’tsev or Goursat categories connectors are useful to develop a centrality theory of non-effective equivalence relations.

Example \thethm.

If $\nabla_{X}$ is the largest equivalence relation on an object $X$, then an associative Mal’tsev operation

[TABLE]

is precisely a connector between $\nabla_{X}$ and $\nabla_{X}$.

Connectors provide a way to distinguish groupoids amongst reflexive graphs:

Proposition \thethm [9].

Given a reflexive graph

[TABLE]

in a finitely complete category $\mathcal{C}$ (i.e. $d\circ e=1_{X_{0}}=c\circ e$) then the connectors between $\mathrm{Eq}(c)$ and $\mathrm{Eq}(d)$ are in bijections with the groupoid structures on this reflexive graph.

It is well known that Goursat categories satisfy the so-called Shifting Property [17, 6]. In this context connectors are unique when they exist (Theorem 2.13 and Proposition 5.1 in [6]): accordingly, for a given pair of equivalence relations on the same object the fact of having a connector becomes a property.

{defi}

Let $R$ and $S$ be two equivalence relations on an object X. A double equivalence relation on $R$ and $S$ is given by an object $C\in\mathbb{C}$ equipped with two equivalence relations $(\pi_{1},\pi_{2}):C\rightrightarrows R$ and $(p_{1},p_{2}):C\rightrightarrows S$ such that the following diagram

[TABLE]

commutes (in the “obvious” way).

A double equivalence relation $C$ on $R$ and $S$ is called a centralizing relation [9] when the square

[TABLE]

is a pullback. Under this assumption it follows that any of the commutative squares in the definition of a centralizing relation is a pullback.

The following lemma gives the relationship between connectors and centralizing relations.

Lemma \thethm [4].

If $\mathbb{C}$ is a category with finite limits and $R$ and $S$ are two equivalence relations on the same object $X$, then the following conditions are equivalent:

1. (i)

there exists a connector between $R$ and $S$; 2. (ii)

there exists a centralizing relation on $R$ and $S$.

When $\mathbb{C}$ is a Mal’tsev category, $R$ and $S$ are equivalence relations on an object $X$ with a connector and $i:I\rightarrowtail X$ is a monomorphism, then the inverse images $i^{-1}(R)$ and $i^{-1}(S)$ also have a connector [4]. We establish a similar property for Goursat categories, with respect to regular epimorphisms:

Proposition \thethm.

Let $\mathbb{C}$ be a Goursat category, $R$ and $S$ two equivalence relations on an object $X$, and let $f:X\twoheadrightarrow Y$ be a regular epimorphism. If there exists a connector between $R$ and $S$, then there exists a connector between the regular images $f(R)$ and $f(S)$.

Proof.

Suppose that there exists a connector between $R$ and $S$. This implies that there exists a centralizing relation $(C,(\pi_{1},\pi_{2}),(p_{1},p_{2}))$ on $R$ and $S$. Consider the regular image $(f(R),a,b)$ and $(f(S),c,d)$ of $R$ and $S$ along $f$. We obtain the following diagram

[TABLE]

where $(f_{R}(C),\beta_{1},\beta_{2})$ is the regular image of the equivalence relation $(C,\pi_{1},\pi_{2})$ along the regular epimorphism $f_{R}$. The fact that the square

[TABLE]

commutes, $\alpha$ is a strong epimorphism and $\langle c,d\rangle$ is a monomorphism, implies the existence of an arrow $\alpha_{1}:f_{R}(C)\longrightarrow f(S)$ making the above diagram commute. Similarly, from the commutativity of the third diagram

[TABLE]

we obtain an arrow $\alpha_{2}:f_{R}(C)\longrightarrow f(S)$.

The relations $(f_{R}(C),\beta_{1},\beta_{2})$, $(f(R),a,b)$ and $(f(S),c,d)$ are all equivalence relations by Theorem 1. It is then easy to check that the relation $(f_{R}(C),\alpha_{1},\alpha_{2})$ is an equivalence relation on $f(S)$. In fact, the morphism $\langle\alpha_{1},\alpha_{2}\rangle\colon f_{R}(C)\to f(S)\times f(S)$ is a monomorphism since $\langle c\times c,d\times d\rangle\circ\langle\alpha_{1},\alpha_{2}\rangle=\langle a,b\rangle\times\langle a,b\rangle\circ\langle\beta_{1},\beta_{2}\rangle$. So, $\langle\alpha_{1},\alpha_{2}\rangle$ is the regular image of $\langle p_{1},p_{2}\rangle$ along $f_{S}$, thus it is an equivalence relation on $f(S)$ by Theorem 1.

By assumption all the left squares of (4) are pullbacks, so it follows that all the right squares of (4) are pullbacks as well by Theorem 1 (ii). Then $(f_{R}(C),(\alpha_{1},\alpha_{2}),(\beta_{1},\beta_{2}))$ is a centralizing relation on $f(R)$ and $f(S)$. By Lemma 3 there is a connector between $f(R)$ and $f(S)$. ∎

We are now going to show that the category whose objects are pairs of equivalence relations equipped with a connector is a Goursat category whenever the base category is a Goursat category. For this, let us first fix some notation: if $\mathbb{C}$ is a finitely complete category, we write 2-$\mathrm{Eq}(\mathbb{C})$ for the category whose objects $(R,S,X)$ are pairs of equivalence relations $R$ and $S$ on the same object $X$

[TABLE]

and arrows are triples $(f_{R},f_{S},f)$ making the following diagram commute:

[TABLE]

We write $\mathrm{Conn}(\mathbb{C})$ for the category whose objects $(R,S,X,p)$ are pairs of equivalence relations $R$ and $S$ on an object $X$ with a given connector $p:R\times_{X}S\rightarrow X$; arrows in $\mathrm{Conn}(\mathbb{C})$ are arrows in 2-$\mathrm{Eq}(\mathbb{C})$ respecting the connectors. This means that, given a diagram (5) where both $(R,S,X)$ and $(\bar{R},\bar{S},\bar{X})$ are in $\mathrm{Conn}(\mathbb{C})$, with $p:R\times_{X}S\rightarrow X$ and $\bar{p}:\bar{R}\times_{Y}\bar{S}\rightarrow Y$ the corresponding connectors, then the diagram

[TABLE]

commutes, where $\bar{f}$ is the natural map induced by the universal property of the pullback $\bar{R}\times_{\bar{X}}\bar{S}$.

We say that a subcategory $\mathbb{P}$ is closed under (regular) quotients in a category $\mathbb{Q}$ if, for any regular epimorphism $f:A\twoheadrightarrow B$ in $\mathbb{Q}$ such that $A\in\mathbb{P}$, then $B\in\mathbb{P}$.

Proposition \thethm.

If $\mathbb{C}$ is a Goursat category, then $\mathrm{Conn}(\mathbb{C})$ is a full subcategory of 2-$\mathrm{Eq}(\mathbb{C})$, that is closed in 2-$\mathrm{Eq}(\mathbb{C})$ under quotients.

Proof.

The fullness of the forgetful functor $\mathrm{Conn}(\mathbb{C})\rightarrow$ 2-$\mathrm{Eq}(\mathbb{C})$ follows from Corollary 5.2 in [6], by taking into account the fact that any Goursat category satisfies the Shifting Property.

Let us then consider a regular epimorphism in 2-$\mathrm{Eq}(\mathbb{C})$

[TABLE]

(this means that $f$, $f_{R}$ and $f_{S}$ are regular epimorphisms in $\mathbb{C}$) such that its domain $(R,S,X)$ belongs to $\mathrm{Conn}(\mathbb{C})$. The equalities $f(R)=\bar{R}$ and $f(S)=\bar{S}$, together with Proposition 3, imply that there exists a connector between $\bar{R}$ and $\bar{S}$. ∎

Lemma \thethm.

Let $\mathbb{D}$ be a finitely complete category, and $\mathbb{C}$ a full subcategory of $\mathbb{D}$ closed in $\mathbb{D}$ under finite limits and quotients. Then:

1. (1)

$\mathbb{C}$* is regular whenever $\mathbb{D}$ is regular.* 2. (2)

$\mathbb{D}$* is a Goursat category whenever $\mathbb{C}$ is a Goursat category.*

Proof.

The (regular epimorphism, monomorphism) factorisation in $\mathbb{D}$ of an arrow in $\mathbb{C}$ is also its factorisation in $\mathbb{C}$, since $\mathbb{C}$ is closed in $\mathbb{D}$ under quotients. Since finite limits in $\mathbb{C}$ are calculated as in $\mathbb{D}$, it follows that regular epimorphisms are stable under pullbacks. Now the second statement easily follows from the fact that the composition of relations is computed in the same way in $\mathbb{C}$ and in $\mathbb{D}$. ∎

Theorem \thethm.

If $\mathbb{C}$ is a Goursat category then $\mathrm{Conn}(\mathbb{C})$ is a Goursat category.

Proof.

Using similar arguments as those given in the proof of Proposition 2 with respect to $\mathrm{Equiv}(\mathbb{C})$, one may deduce that 2-$\mathrm{Eq}(\mathbb{C})$ is a Goursat category. The result then follows from Proposition 3 and Lemma 3. ∎

We finally prove that internal categories and groupoids can be used to characterise Goursat categories. Recall that an internal category in a category $\mathbb{C}$ with pullbacks is a reflexive graph with a multiplication $m\colon X_{1}\times_{X_{0}}X_{1}\rightarrow X_{1}$

[TABLE]

(where $X_{1}\times_{X_{0}}X_{1}$ is the pullback of $d$ and $c$) such that:

• •

$d\circ m=d\circ p_{2}$,   $c\circ m=c\circ p_{1}$,   $m\circ\langle e\circ d,1_{X_{1}}\rangle=1_{X_{1}}=m\circ\langle 1_{X_{1}},e\circ c\rangle$;

• •

$m\circ(1\times m)=m\circ(m\times 1)$.

An internal category

[TABLE]

is a groupoid when there is an additional morphism $i\colon X_{1}\longrightarrow X_{1}$ satisfying the axioms:

• •

$d\circ i=c$,   $c\circ i=d$;

• •

$m\circ\langle i,1_{X_{1}}\rangle=e\circ c$   and   $m\circ\langle 1_{X_{1}},i\rangle=e\circ d$.

We write $\mathrm{Cat}(\mathbb{C})$ for the category of internal categories in $\mathbb{C}$ (and internal functors as morphisms), $\mathrm{Grpd}(\mathbb{C})$ for the category of groupoids in $\mathbb{C}$, and $\mathrm{RG}(\mathbb{C})$ for the category of reflexive graphs in $\mathbb{C}$ (with obvious morphisms).

An equivalence relation is a special kind of groupoid, where its domain and codomain morphisms are jointly monomorphic; also any reflexive and transitive relation is in particular an internal category. If $\mathbb{C}$ is a Goursat category, then any reflexive and transitive relation is an equivalence relation or, equivalently, any internal category is a groupoid (Theorem 1 in [22]). Then Theorem  1, which could equivalently be stated through the property that $\mathrm{Equiv}(\mathbb{C})$ (or the category of reflexive and transitive relations in $\mathbb{C}$) is closed in the category of reflexive relations in $\mathbb{C}$ under quotients, has an extended counterpart given below. This characterisation leads to the observation that the structural aspects of Goursat categories mainly concern groupoids (rather than equivalence relations).

Theorem \thethm.

Let $\mathbb{C}$ be a regular category. Then the following conditions are equivalent:

1. (i)

$\mathbb{C}$* is a Goursat category;* 2. (ii)

$\mathrm{Grpd}(\mathbb{C})$* is closed in $\mathrm{RG}(\mathbb{C})$ under quotients;* 3. (iii)

$\mathrm{Cat}(\mathbb{C})$* is closed in $\mathrm{RG}(\mathbb{C})$ under quotients.*

Proof.

**(i) $\Rightarrow$ (ii): **

Let

[TABLE]

be a regular epimorphism $(f,g)$ in $\mathrm{RG}(\mathbb{C})$ (which means that $f$ and $g$ are regular epimorphisms in $\mathbb{C}$), with

[TABLE]

a groupoid in $\mathbb{C}$. By Proposition 3, there exists a connector between $\mathrm{Eq}(d)$ and $\mathrm{Eq}(c)$. Let $\mathrm{Eq}(d)$, $\mathrm{Eq}(c)$, $\mathrm{Eq}(d^{\prime})$ and $\mathrm{Eq}(c^{\prime})$ be the kernel pairs of the arrows $d$, $c$, $d^{\prime}$ and $c^{\prime}$, respectively. Let $\lambda:\mathrm{Eq}(d)\rightarrow\mathrm{Eq}(d^{\prime})$ and $\beta:\mathrm{Eq}(c)\rightarrow\mathrm{Eq}(c^{\prime})$ be the arrows induced by the universal property of kernel pairs $\mathrm{Eq}(d^{\prime})$ and $\mathrm{Eq}(c^{\prime})$, respectively. By Theorem 1, $\lambda$ and $\beta$ are regular epimorphisms, so that $g(\mathrm{Eq}(d))=\mathrm{Eq}(d^{\prime})$ and $g(\mathrm{Eq}(c))=\mathrm{Eq}(c^{\prime})$. By Proposition 3 there is then a connector between $\mathrm{Eq}(d^{\prime})$ and $\mathrm{Eq}(c^{\prime})$, thus

[TABLE]

is a groupoid (Proposition 3).

**(ii) $\Rightarrow$ (i): **

This implication follows from Theorem 1 and the fact that equivalence relations are in particular groupoids (whose domain and codomain morphisms are jointly monomorphic).

**(i) $\Rightarrow$ (iii): **

This implication follows from (i) $\Rightarrow$ (ii) and the fact that $\mathrm{Grpd}({\mathbb{C}})\cong\mathrm{Cat}(\mathbb{C})$ in a Goursat context, as recalled above.

**(iii) $\Rightarrow$ (i): **

Let $(R,r_{1},r_{2})$ be an equivalence relation on $X$, $f:X\twoheadrightarrow Y$ a regular epimorphism and $(f(R),t_{1},t_{2})$ the regular image of $R$ along $f$

[TABLE]

$(f(R),t_{1},t_{2})$ is reflexive and symmetric being the image of the equivalence relation $R$ along a regular epimorphism $f$. By assumption, $(f(R),t_{1},t_{2})$ is an internal category, thus it is an equivalence relation. It follows that $\mathbb{C}$ is a Goursat category (by Theorem 1). ∎

Remark \thethm.

Observe that Theorem 3 also implies that $\mathrm{Grpd}(\mathbb{C})$ and $\mathrm{Cat}(\mathbb{C})$ are Goursat categories whenever $\mathbb{C}$ is, again thanks to Lemma 3, the category $\mathrm{RG}(\mathbb{C})$ obviously being a Goursat category. This simplifies and slightly extends Proposition $4.3$ in [14], where the existence of coequalizers in $\mathbb{C}$ was assumed.

Remark \thethm.

A result analogous to Theorem 3 holds in the Mal’tsev context: a category $\mathbb{C}$ is a Mal’tsev category if and only if $\mathrm{Grpd}(\mathbb{C})$ (or, equivalently, $\mathrm{Cat}(\mathbb{C})$) is closed in $\mathrm{RG}(\mathbb{C})$ under subobjects [3]. Together with the comments made before Proposition 3 we observe the existence of a sort of duality between Mal’tsev categories and Goursat categories. Similar results hold for Mal’tsev categories with respect to monomorphisms and for Goursat categories with respect to regular epimorphisms.