Page 136 - Textos de Matemática Vol. 46

126 M. GRAN AND D. RODELO
property of regular Gumm categories (Proposition 4.1). Given a commutative diagram
Z⇥V U ////X //U OO OO
12
✏✏✏✏ // // ✏✏✏✏ // ✏✏ ZYV
in a regular Gumm category such that the whole rectangle is a pullback and the left square is composed by vertical split epimorphisms and horizontal regular epimorphisms, then both squares and are pullbacks. This property is known to hold in any regular Mal’tsev category, and has been used, for example, in the categorical theory of central extensions [10, 7].
In the present article we also show that this property can be used to give a new proof of a remarkable property of exact Goursat categories, namely the fact that central extensions and normal extensions relative to any (admissible) Birkho↵ subcategory X of C coincide [15]. Let us recall that a full reflective subcategory X of an exact category C is called a Birkho↵ subcategory when X is closed in C under subobjects and regular quotients. In particular, a Birkho↵ subcategory of a variety of universal algebras is just a subvariety. A Birkho↵ subcategory X is admissible, from the point of view of Categorical Galois The- ory, when the reflector I : X ! C preserves pullbacks of regular epimorphisms in X along any morphism in C. The notions of central extension and of normal extension are defined relatively to the choice of the admissible Birkho↵ sub- category X of C, as recalled in Section 5. It is precisely the useful property of pullbacks in regular Gumm categories stated above which allows one to find a simple proof of the coincidence of these two notions in the exact Goursat context (Theorem 5.2 and Corollary 5.3).
In [15] G. Janelidze and G.M. Kelly proved that every Birkho↵ subcategory X of an exact category C with modular lattice of equivalence relations (on any object in C) is always admissible. It was later shown by V. Rossi in [20] that the same admissibility property still holds in the more general context of Gumm categories which are almost exact, a notion introduced by G. Janelidze and M. Sobral in [16]. We conclude the article by relating our observations on Gumm categories with these results concerning the admissibility of Galois structures.
2. Preliminaries
In the present paper the term regular category [1] will be used for a finitely complete category such that any kernel pair has a coequaliser and, moreover, regular epimorphisms are stable under pullbacks. Any morphism f : A ! B in a regular category C has a factorisation f = m·p, with p a regular epimorphism
1
1
2