Some remarks on connectors and groupoids in Goursat categories
Marino Gran Diana Rodelo Idriss Tchoffo Nguefeu

Some remarks on connectors and groupoids in Goursat categories]
Some remarks on connectors and groupoids
 in Goursat categories

Dedicated to Jiří  Adámek on the occasion of his seventieth birthday

Marino Gran]Marino Grana
aInstitut de Recherche en Mathématique et Physique, Université Catholique de Louvain,
Belgium

{marino.gran,idriss.tchoffo}@uclouvain.be

Idriss Tchoffo Nguefeu]Idriss Tchoffo Nguefeua





Diana Rodelo]Diana Rodelob
bDepartamento de Matemática, Universidade
do Algarve, 3001-501 Coimbra, Portugal



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.
drodelo@ualg.pt



[2000]
08C05, 
08B05, 
08A30, 
08B10, 
18C05, 
18B99, 
18E10 




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






[
    [
    
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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 <cit.>, for instance, and references therein). When 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 <cit.>. Many important results still hold when a regular category ℂ satisfies the strictly weaker property of 3-permutability, namely the Goursat property. A nice feature of a (regular) Goursat category ℂ is that the lattice of equivalence relations on any object in ℂ is a modular lattice <cit.>, a property that plays a crucial role in commutator theory <cit.>.


The aim of this paper is twofold: first of all we establish some basic properties of Goursat categories in terms of connectors <cit.>, as it was done in <cit.> for the case of Mal'tsev categories. These results have turned out to be useful to develop a monoidal approach to internal structures <cit.>. We then give a new characterisation of Goursat categories in terms of properties of (internal) groupoids, on the model of what was done in <cit.> 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 ℂ, the category (ℂ) of equivalence
relations in ℂ is also a Goursat category (Proposition
<ref>, see also <cit.>). We use this result to give some properties of Goursat categories in terms of connectors in Section 3. More precisely, we show that, when ℂ is a Goursat category, then connectors are stable under quotients in ℂ (Proposition <ref>), and this implies that the category () of connectors in  is again a Goursat category (Theorem <ref>).

We conclude the paper by giving a new characterisation of Goursat categories in terms of properties of groupoids and internal categories (Theorem <ref>). It turns out that a regular category ℂ is a Goursat category if and only if the category (ℂ) of groupoids (equivalently, the category (ℂ) of internal categories) in ℂ is closed under quotients in the category (ℂ) of reflexive graphs in .






§ 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  ℂ in which we are working is a regular category: this means that  is finitely complete, regular epimorphisms are stable under pullbacks, and kernel pairs have coequalisers. Equivalently, any arrow f: A⟶ B has a unique factorisation f=i∘ 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 ⟨ r_1,r_2 ⟩ : R ↣ X × Y. The opposite relation of R, denoted R^o, is the relation from Y to X given by the subobject ⟨ r_2,r_1 ⟩ : R ↣ Y × X. A relation R from X to X is called a relation on X. We shall identify a morphism f: X ⟶ Y with the relation ⟨ 1_X,f ⟩: X ↣ X × Y and write f^o for its opposite relation. Given another relation ⟨ s_1, s_2 ⟩ : S ↣ Y × Z from Y to Z, one can define the composite relation SR  of R and S as the image of the arrow (r_1 ∘ p_1, s_2 ∘ p_2): R ×_Y S ⟶ X × Z, where (R ×_Y S, p_1, p_2) is the pullback of r_2: R⟶ Y along s_1: S ⟶ Y. With the above notations, any relation ⟨ r_1, r_2 ⟩: R ↣ X × Y can be seen as the relational composite r_2r_1^o.

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



Let f: X ⟶ Y be an arrow in a regular category  ℂ,
and let  i ∘ r be its (regular epimorphism, monomorphism) factorisation. Then:

  
  * f^of is the kernel pair of f, thus 1_X ⩽ f^of; moreover, 1_X = f^of if and only if f is a monomorphism;
  
  * ff^o is (i,i), thus ff^o ⩽ 1_Y; moreover, ff^o = 1_Y if and only if f is a  regular epimorphism;
  
  * ff^of = f and f^off^o = f^o.




  A relation (R, r_1,r_2) on an object X  is said to be :

  
  * reflexive when there is an arrow r: X ⟶ R such that r_1 ∘ r = 1_X = r_2 ∘ r;
  
  * symmetric when there is an arrow  σ :  R ⟶ R such that r_2 = r_1 ∘σ and r_1 = r_2 ∘σ;
  
  * transitive when, by considering the following pullback
  
    R ×_X R [r]^-p_2@[dr]|(0.25)⌟[d]_p_1    R [d]^r_1
     R [r]_r_2    X,

there is an arrow t : R ×_X R ⟶ R such that r_1 ∘ t = r_1 ∘ p_1 and r_2 ∘ t = r_2 ∘ p_2.
 
  * an equivalence relation if R is  reflexive, symmetric and transitive.

In particular, a kernel pair ⟨ f_1,f_2 ⟩: (f)↣ X × X of a morphism f: X ⟶ Y is an equivalence relation.
The equivalence relations that occur as kernel pairs of some morphism in ℂ are called effective.
Let (ℂ) be the category whose objects are equivalence relations in  ℂ and arrows from ⟨ r_1,r_2 ⟩ : R ↣ X × X to ⟨ s_1,s_2 ⟩ : S ↣ Y × Y are pairs (f,g) of arrows in ℂ  making the following diagram commute


    R  [r]^g@<.5ex>[d]^r_2@<-.5ex>[d]_r_1    S @<.5ex>[d]^s_2@<-.5ex>[d]_s_1
     X  [r]_f     Y.



When ℂ is a regular category, (R,r_1,r_2) is an equivalence relation on X and f: X ↠ 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 ⟨ s_1, s_2 ⟩∘ψ of the composite (f× f)∘⟨ r_1,r_2⟩:

    R @.>>[r]^ψ@ >->[d]_⟨ r_1,r_2⟩    f(R)  @ >.>[d]^⟨ s_1,s_2⟩
    
    X × X @>>[r]_f × f    Y × Y.

Note that the regular image f(R) can be obtained as the relational composite f(R)=fRf^o=fr_2r_1^of^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 <ref>).

[<cit.>]
A regular category ℂ is called a Goursat category when the equivalence relations in 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:

[<cit.>]   A regular category ℂ is a Goursat category if and only if for any regular epimorphism f: X ↠ 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 <cit.>, 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 <cit.> and references therein).  In particular, the following characterisations will be useful for the development of this work:

[<cit.>]
Let ℂ be a regular category. The following conditions are equivalent:

 
  (i) ℂ is a Goursat category;
 
  (ii) any commutative diagram where α and β are regular epimorphisms and f and g are split epimorphisms in ℂ
  
    X @->>[r]^α@<3pt>[d]^f    U @<3pt>[d]^g
     Y @->>[r]_β@<3pt>[u]^s    W @<3pt>[u]^t

(which is necessarily a pushout) is a Goursat pushout: the morphism λ : (f) ⟶(g) induced by the universal property of kernel pair (g) of g is a regular epimorphism.


We recall part of Theorem 1.3 in <cit.>:

[<cit.>]
Let ℂ be a regular category. The following conditions are equivalent:

  
  (i) ℂ is a Goursat category;
  
  (ii) for any commutative cube
  
    X ×_Y Z @->>[rr]^δ@<3pt>[rd] @<3pt>[dd]       A @<3pt>[rd] @<3pt>@–>[dd]    
        Z @<3pt>[ul]  @->>[rr]^(0.3)γ@<3pt>[dd]        V @<3pt>[ul]  @<3pt>[dd]
    
         X @–>>[rr]^(0.7)α@<3pt>[rd] @<3pt>[uu]       U @<3pt>@–>[rd] @<3pt>@–>[uu]    
        Y @->>[rr]_β@<3pt>[ul] @<3pt>[uu]         W @<3pt>@–>[ul] @<3pt>[uu]

where the left square is a pullback of split epimorphisms, the right square is a commutative square of split epimorphisms and the horizontal arrows α, β, γ and δ are regular epimorphisms (commuting also with the splittings), then the right square is a pullback.







§ EQUIVALENCE RELATIONS IN GOURSAT CATEGORIES

In this section we prove that (ℂ) is a Goursat category for any Goursat category ℂ.

The category () is finitely complete whenever  is: the terminal object in () is the discrete equivalence relation

    1 @<.5ex>[r]^@<-.5ex>[r]_    1

on the terminal object 1 of , and pullbacks are computed “levelwise”. In particular, the kernel pair of a morphism (f,g) in () is given by the kernel pairs (f) of f and (g) of g in 

    (g) @<.5ex>[r]^g_1@<-.5ex>[r]_g_2@<.5ex>[d]^r̅_̅2̅@<-.5ex>[d]_r̅_̅1̅    R  [r]^g@<.5ex>[d]^r_2@<-.5ex>[d]_r_1    S @<.5ex>[d]^s_2@<-.5ex>[d]_s_1
    (f) @<.5ex>[r]^f_1@<-.5ex>[r]_f_2    X  [r]_f     Y.

 Consequently, a morphism (f,g) is a monomorphism in () if and only if f and g are monomorphisms in . When  is a Goursat category, a similar property holds with respect to regular epimorphisms:


Let R and S be two equivalence relations in a Goursat category ℂ and (f,g) : R → S a morphism

    R  [r]^g@<.5ex>[d]^r_2@<-.5ex>[d]_r_1    S @<.5ex>[d]^s_2@<-.5ex>[d]_s_1
     X  [r]_f     Y

in (ℂ). Then (f,g) is a regular epimorphism in (ℂ) if and only if f and g are regular epimorphisms in ℂ.


When f and g are regular epimorphisms in ℂ, it is not difficult to check that (f,g) is necessarily the coequaliser of its kernel pair in (ℂ) given in (<ref>) (one uses the fact that g = coeq(g_1,g_2) and f = coeq(f_1,f_2) in ℂ).

Conversely, let (f,g) be a morphism in (ℂ) as in (<ref>) that is a regular epimorphism in (ℂ). Consider the kernel pairs of f and g, the (regular epimorphism, monomorphism) factorisation f=i ∘ 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

    (g) @<.5ex>[r]^g_1@<-.5ex>[r]_g_2@<.5ex>[dd]^r̅_̅2̅@<-.5ex>[dd]_r̅_̅1̅     R @->>[rd]_α[rr]^g@<.5ex>[dd]^r_2@<-.5ex>[dd]_r_1       S @<.5ex>[dd]^s_2@<-.5ex>[dd]_s_1
           q(R)  @<.5ex>[dd]^(0.3)t_2@<-.5ex>[dd]_(0.3)t_1@.>@<3pt>[ru]_(0.6)j   
    (f)@<.5ex>[r]^f_1@<-.5ex>[r]_f_2     X @->>[rd]_q [rr]^(0.8)f       Y,  
           Z @ >->[ru]_(0.6)i

where (q(R), t_1, t_2) ∈(ℂ) (by Theorem <ref>) and (i, j) is the morphism in (ℂ) such that (i, j) ∘ (q, α) = (f,g). Note that j is induced from the fact that (i× i) ∘⟨ t_1, t_2 ⟩∘α is the (regular epimorphism, monomorphism) factorisation of ⟨ s_1, s_2 ⟩∘ g, thus it is a monomorphism

    @C=30pt@R=25pt
        R @>>[r]^-α[d]_-g     q(R) @ >->[d]^-(i× i)∘⟨ t_1,t_2⟩@ >.>[ld]_-j 
    
        S @ >->[r]_-⟨ s_1,s_2⟩    Y× Y.

From the fact that (f,g) is the coequaliser of its kernel pair in (ℂ) it easily follows that (i,j) is an isomorphism in (ℂ).
This implies that f and g are regular epimorphisms in ℂ.



    (ℂ) is a Goursat category whenever  is.

 As mentioned above, the category (ℂ) is finitely complete because ℂ is so. Lemma <ref> implies that regular epimorphisms in () are stable under pullbacks since regular epimorphisms are stable in ℂ, and regular epimorphisms in (ℂ) are “levelwise” regular epimorphisms. The existence of the (regular epimorphism, monomorphism) factorisation of a morphism (f,g) as in (<ref>) in the category (ℂ) follows from the construction of diagram (<ref>): the (regular epimorphism, monomorphism) factorisation f=i∘ q of f in  gives rise to the (regular epimorphism, monomorphism) factorisation g=j∘α of g in . Thus (q,α)∘ (i,j) is the (regular epimorphism, monomorphism) factorisation of (f,g) in ().
To see that (ℂ) has the Goursat property one uses Theorem <ref>: to check that the regular image of an equivalence relation in the category (ℂ) is again an equivalence in (ℂ) one mainly uses the same (“levelwise”) property in the category ℂ.





§ CONNECTORS AND GROUPOIDS IN GOURSAT CATEGORIES

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



Let (R,r_1,r_2) and (S, s_1,s_2) be two equivalence relations on an object X and R×_X S the pullback of r_2 along s_1. A connector <cit.> between R and S  is an arrow p: R ×_X S ⟶  X in ℂ such that


  * x S p(x,y,z)R z;

  * p(x,x,y)= y;

  * p(x,y,y)= x;

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

when each term is defined.


Given two regular epimorphisms d  X ↠ Y and c  X ↠ Z, a connector on the effective equivalence relations (d) and (c) is the same thing as an internal pregroupoid in the sense of Kock <cit.> (see also the introduction of <cit.>, 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.



If ∇_X is the largest equivalence relation on an object X, then an associative Mal'tsev operation 
    p: X× X × X ⟶ X
 is precisely a connector between ∇_X and ∇_X.


Connectors provide a way to distinguish groupoids amongst reflexive graphs:
[<cit.>] Given a reflexive graph 
    X_1 @<6pt>[r]^d@<-6pt>[r]_c    X_0 [l]|-e

in a finitely complete category 𝒞
  (i.e. d ∘ e = 1_X_0 = c∘ e)  then the connectors between (c) and (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 <cit.>. In this context connectors are unique when they exist (Theorem 2.13 and Proposition 5.1 in <cit.>):
accordingly, for a given pair of equivalence relations on the same object the fact of having a connector becomes a property.


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 ∈ℂ equipped with two equivalence relations (π_1,π_2): C ⇉ R and (p_1,p_2): C ⇉ S such that the following diagram

    C  @<.5ex>[r]^π_1@<-.5ex>[r]_π_2@<.5ex>[d]^p_2@<-.5ex>[d]_p_1    R @<.5ex>[d]^r_2@<-.5ex>[d]_r_1
     S  @<.5ex>[r]^s_1@<-.5ex>[r]_s_2    X

commutes (in the “obvious” way).


A double equivalence relation C on R and S is called a centralizing relation <cit.> when the square

    C @[dr]|(.25)⌟[r]^π_1[d]_p_1    R [d]^r_1
     S  [r]_s_1    X

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.
[<cit.>] 
If 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:

  
  (i) there exists a connector between R and S;
  
  (ii) there exists a centralizing relation on R and S.



When  is a Mal'tsev category, R and S are equivalence relations on an object X with a connector  and i:I↣ X is a monomorphism, then the inverse images i^-1(R) and i^-1(S) also have a connector <cit.>.  We establish a similar property for Goursat categories, with respect to regular epimorphisms:


Let ℂ be a Goursat category, R and S two equivalence relations on an object X, and let f: X ↠ 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).


 Suppose that there exists a connector between R and S. This implies that there exists a centralizing relation (C,(π_1,π_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

    C @->>[rr]^α@<.5ex>[rd]^π_2@<-.5ex>[rd]_π_1@<.5ex>[dd]^p_2@<-.5ex>[dd]_p_1       f_R(C) @<.5ex>@–>[dd]^(0.7)α_2@<-.5ex>@–>[dd]_(0.7)α_1@<.5ex>[rd]^β_2@<-.5ex>[rd]_β_1   
        R @<.5ex>[dd]^(0.3)r_2@<-.5ex>[dd]_(0.3)r_1@->>[rr]^(0.3)f_R       f(R) @<.5ex>[dd]^b@<-.5ex>[dd]_a
    
         S @–>>[rr]^(0.7)f_S@<.5ex>[rd]^s_2@<-.5ex>[rd]_s_1       f(S) @<.5ex>@–>[rd]^d@<-.5ex>@–>[rd]_c   
        X @->>[rr]_f        Y,

where (f_R(C), β_1,β_2) is the regular image of the equivalence relation (C,π_1,π_2) along the regular epimorphism f_R. The fact that the square

    C @->>[r]^α[d]_f_Sp_1    f_R(C) [d]^⟨ aβ_1,aβ_2⟩@.>[ld]_-α_1
    f(S) @ >->[r]_-⟨ c,d⟩    Y × Y

 commutes, α is a strong epimorphism and ⟨ c,d⟩ is a
 monomorphism, implies the existence of an arrow α_1:
 f_R(C)⟶ f(S) making the above diagram commute. Similarly, from
 the commutativity of the third diagram

    C @->>[r]^α[d]_f_Sp_2    f_R(C) [d]^⟨ bβ_1,bβ_2⟩@.>[ld]_-α_2
    f(S)  @ >->[r]_-⟨ c,d⟩    Y × Y

we obtain an arrow α_2: f_R(C)⟶ f(S).

The relations (f_R(C), β_1,β_2), (f(R), a, b) and (f(S),c,d) are all equivalence relations by Theorem <ref>. It is then easy to check that the relation (f_R(C),α_1,α_2) is an equivalence relation on f(S). In fact, the morphism ⟨α_1,α_2⟩ f_R(C)→ f(S)× f(S) is a monomorphism since ⟨ c× c, d× d⟩∘⟨α_1,α_2 ⟩= ⟨ a,b⟩×⟨ a,b⟩∘⟨β_1, β_2⟩. So, ⟨α_1,α_2⟩ is the regular image of ⟨ p_1,p_2⟩ along f_S, thus it is an equivalence relation on f(S) by Theorem <ref>.

By assumption all the left squares of (<ref>) are pullbacks, so it follows that all the right squares of (<ref>) are pullbacks as well by Theorem <ref> (ii). Then (f_R(C), (α_1,α_2),(β_1,β_2)) is a centralizing relation on f(R) and f(S). By Lemma <ref> 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 ℂ is a finitely complete category, we write 2-(ℂ) for the category whose objects (R,S,X) are pairs of equivalence relations R and S on the same object X

    R @<.5ex>[r]^r_1@<-.5ex>[r]_r_2    X     S @<.5ex>[l]^s_2@<-.5ex>[l]_s_1

and arrows are triples (f_R,f_S,f) making the following diagram commute:

    R @<.5ex>[r]^r_1@<-.5ex>[r]_r_2[d]_f_R    X  [d]_f     S @<.5ex>[l]^s_2@<-.5ex>[l]_s_1[d]^f_S
    R̅@<.5ex>[r]^r̅_̅1̅@<-.5ex>[r]_r̅_̅2̅   X̅   S̅. @<.5ex>[l]^s̅_̅2̅@<-.5ex>[l]_s̅_̅1̅

We write (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 ×_X S → X; arrows in (ℂ) are arrows in 2-(ℂ) respecting the connectors. This means that, given a diagram (<ref>) where both (R,S,X) and (R̅,S̅,X̅) are in (ℂ), with p: R ×_X S → X and p̅: R̅×_Y S̅→ Y the corresponding connectors,  then the diagram

    R ×_X S [r]^f̅[d]_p   R̅×_X̅S̅[d]^p̅
     X  [r]_f   X̅

commutes, where f̅ is the natural map induced by the universal property of the pullback R̅×_X̅S̅.

We say that a subcategory ℙ is closed under (regular) quotients in a category ℚ if, for any regular epimorphism f: A ↠ B in ℚ such that A ∈ℙ, then B ∈ℙ.



If ℂ is a Goursat category, then (ℂ) is a full subcategory of  2-(ℂ), that is closed in 2-(ℂ) under quotients.


The fullness of the forgetful functor (ℂ) → 2-(ℂ) follows from Corollary 5.2 in <cit.>, by taking into account the fact that any Goursat category satisfies the Shifting Property.

Let us then consider a regular epimorphism in 2-(ℂ)

    R @<.5ex>[r]^r_1@<-.5ex>[r]_r_2@>>[d]_f_R    X  @>>[d]_f     S @<.5ex>[l]^s_2@<-.5ex>[l]_s_1@>>[d]^f_S
    R̅@<.5ex>[r]^r̅_̅1̅@<-.5ex>[r]_r̅_̅2̅   X̅   S̅@<.5ex>[l]^s̅_̅2̅@<-.5ex>[l]_s̅_̅1̅

 (this means that f, f_R and f_S are regular epimorphisms in ℂ) such that its domain (R,S,X) belongs to (ℂ). The equalities f(R)= R̅ and f(S)= S̅, together with Proposition <ref>, imply that there exists a connector between R̅ and  S̅.



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


  * ℂ is regular whenever 𝔻 is regular.

  * 𝔻 is a Goursat category whenever ℂ is a Goursat category.


 The (regular epimorphism, monomorphism) factorisation in  𝔻 of an arrow in  ℂ is also its factorisation in  ℂ, since ℂ is closed in 𝔻 under quotients. Since finite limits in ℂ are calculated as in 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 ℂ and in 𝔻. 



 If ℂ is a Goursat category then (ℂ) is a Goursat category.
 

  Using similar arguments as those given in the proof of Proposition <ref> with respect to (), one may deduce that 2-( ℂ) is a Goursat category. The result then follows from Proposition <ref> and Lemma <ref>.


We finally prove that internal categories and groupoids can be used to characterise Goursat categories. Recall that an internal category in a category ℂ with pullbacks is a reflexive graph with a multiplication m  X_1 ×_X_0X_1 → X_1

    X_1 ×_X_0X_1   @<6pt>[r]^-p_1@<-6pt>[r]_-p_2[r]|-m    X_1 @<6pt>[r]^d@<-6pt>[r]_c    X_0, [l]|-e

(where X_1 ×_X_0X_1 is the pullback of d and c) such that:


  * d∘ m=d ∘ p_2,   c ∘ m=c∘ p_1,   m ∘⟨ e∘ d, 1_X_1⟩ = 1_X_1= m ∘⟨ 1_X_1,e∘ c⟩;

  * m∘ (1× m) = m∘ (m × 1).

An internal category

    X_1 ×_X_0X_1   @<6pt>[r]^-p_1@<-6pt>[r]_-p_2[r]|-m    X_1  @<6pt>[r]^d@<-6pt>[r]_c    X_0, [l]|-e

is a groupoid when there is an additional morphism i   X_1 ⟶ X_1
satisfying the axioms:


  * d∘ i = c,   c∘ i=d;

  * m∘⟨ i, 1_X_1⟩= e ∘ c   and   m∘⟨ 1_X_1 , i⟩= e∘ d.


We write (ℂ) for the category of internal categories in  ℂ (and internal functors as morphisms),  (ℂ) for the category of groupoids in ℂ,  and (ℂ) for the category of reflexive graphs in 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  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 <cit.>). Then Theorem  <ref>, which could equivalently be stated through the property that () (or the category of reflexive and transitive relations in ) is closed in the category of reflexive relations in  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).



Let ℂ be a regular category. Then the following conditions are equivalent:

  
  (i) ℂ is a Goursat category;
  
  (ii) (ℂ) is closed in (ℂ) under quotients;
  
  (iii) (ℂ) is closed in (ℂ) under quotients.




  
  (i) ⇒ (ii) Let

    X_1@>>[r]^g@<6pt>[d]^c@<-6pt>[d]_d   X'_1@<6pt>[d]^c'@<-6pt>[d]_d'
     X_0 [u]|-e@>>[r]_f    X'_0 [u]|-e'

be a regular epimorphism (f,g) in (ℂ) (which means that f and g are regular epimorphisms in ), with

    X_1 @<6pt>[r]^d@<-6pt>[r]_c    X_0 [l]|-e

a groupoid in ℂ. By Proposition <ref>, there exists a connector between (d) and (c). Let (d), (c), (d') and (c') be the kernel pairs of the arrows d, c, d' and c', respectively. Let  λ : (d)→(d') and β : (c)→(c') be the arrows induced by the universal property of kernel pairs (d') and (c'), respectively. By Theorem <ref>, λ and β are regular epimorphisms, so that g((d)) = (d') and g((c)) = (c'). By Proposition <ref> there is then a connector between (d') and (c'), thus

    X'_1 @<6pt>[r]^d'@<-6pt>[r]_c'    X'_0 [l]|-e'

is a groupoid (Proposition <ref>).


  (ii) ⇒ (i) This implication follows from Theorem <ref> and the fact that equivalence relations are in particular groupoids (whose domain and codomain morphisms are jointly monomorphic).


  (i) ⇒ (iii) This implication follows from (i) ⇒ (ii) and the fact that (ℂ) ≅(ℂ) in a Goursat context, as recalled above.
















  (iii) ⇒ (i) Let (R, r_1, r_2) be an equivalence relation on X, f: X ↠ Y a regular epimorphism and (f(R), t_1, t_2) the regular image of R along f

    R @->>[r]^g@<.5ex>[d]^r_2@<-.5ex>[d]_r_1    f(R) @<.5ex>[d]^t_2@<-.5ex>[d]_t_1
     X  @->>[r]^f    Y.


(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 ℂ is a Goursat category (by Theorem <ref>).






Observe that Theorem <ref> also implies that (ℂ) and (ℂ) are Goursat categories whenever ℂ is, again thanks to Lemma <ref>, the category (ℂ) obviously being a Goursat category. This simplifies and slightly extends Proposition 4.3 in <cit.>, where the existence of coequalizers in ℂ was assumed.


A result analogous to Theorem <ref> holds in the Mal'tsev context: a category ℂ is a Mal'tsev category if and only if (ℂ) (or, equivalently, ()) is closed in (ℂ) under subobjects <cit.>. Together with the comments made before Proposition <ref> 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.


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