Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Abstract

Given a regular Gumm category C such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory X in C gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal'cev category C, we first characterize the coverings of the Galois structurē 1 induced by the subcategory C Ab of the abelian objects in C. Then we consider C as a subcategory of the category Eq(C) of the equivalence relations in C, and we characterize the coverings of the corresponding Galois structure¯ 2 . By composing the Galois structures¯ 1 and¯ 2 we obtain the Galois structure¯ induced by C Ab as a subcategory of Eq(C). We give the characterization of the¯ -coverings in terms of the coverings of¯ 1 and¯ 2 .

Some remarks on pullbacks in Gumm categories

“…Kelly proved that every Birkhoff 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.…”

“…where w : U / / / / V is a regular epimorphism of X (see Proposition 3.3 of [15]). In [20] V. Rossi proved that any Birkhoff subcategory of an almost exact Gumm category is admissible, extending a result due to G. Janelidze and G.M. Kelly [15].…”

“…Kelly [15]. The proof of the more general Proposition 4.2 above is actually similar to the one given in [20]. In order to deduce the admissibility result from Proposition 4.2 it suffices to decompose the square (5.7) above as…”

Relative commutator theory in semi-abelian categories

“…However, looking at the diagram in the introduction, this is not entirely satisfactory, because: · Central extensions were defined in [32] in the context of exact categories A, relative to a choice of admissible Birkhoff subcategory; and it was shown that if A is Mal'tsev (every reflexive relation internal in A is an equivalence relation) then any Birkhoff subcategory is admissible. More recently, V. Rossi proved in [42] the admissibility of Birkhoff subcategories in a context which includes every regular Mal'tsev category that is "almost exact" in the sense that every regular epimorphism is an effective descent morphism. · The Huq commutator can be considered in a context, as general as that of finitely cocomplete unital categories; in particular, in any finitely cocomplete pointed Mal'tsev category [9].…”

A New Characterisation of Goursat Categories

“…Many important results still hold when a regular category C satisfies the strictly weaker 3-permutability property, also known as the Goursat property: in this case, C is called a Goursat category. Any Goursat category C is such that the semi-lattice of equivalence relations on any object in C is a modular lattice [4], a fact which is no longer true for the weaker 4-permutability property, and that has some important consequences in categorical Galois theory [10,16]. Other aspects of the theory of Goursat categories which have been recently considered are: the validity of the denormalised 3-by-3 Lemma [14], the regularity of the category of internal groupoids [7], an explicit description of the closure operator associated with any Birkhoff subcategory of an exact Goursat category [2].…”