Galois theory and double central extensions

Abstract

We define a Galois structure between central extensions and extensions in a Maltsev variety. By using the theory of commutators we introduce double central extensions. We then prove that the covering morphisms relative to this Galois structure are precisely the double central extensions.

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

“…The pair (Ext(V), Centr(V)) also satisfies the conditions of Theorem 3.5, and gives rise to an admissible Galois structure. The coverings relative to this Galois structure are precisely the double central extensions described in [13] and [19].…”

Higher commutator conditions for extensions in Mal'tsev categories

“…These conditions are a generalization of the ones given by G. Janelidze in [21] characterizing double central extensions in Grp, and later extended to Mal'tsev varieties by the second author and V. Rossi in [20], and to exact Mal'tsev categories by T. Everaert and T. Van der Linden [17]. In fact, the proofs given below are suitably adapted from the ones appearing in these two papers.…”

“…Using again the equivalence between B-precrossed modules and internal reflexive graphs over B in Lie, we find that (21) where k ∈ Ker(f ) and k ′ ∈ Ker(g). Since the commutator in equation (22) can be treated exactly as the commutator in (20), we find that the double extension (21) where both sides are ideals of L 1 .…”

“…As a consequence, we find that [Ker(1 B ⋉ f ), Ker(π B )] is the ideal of B ⋉ L generated by terms of the form [(0, k), (0, l)] = (0, [k, l]) for k ∈ Ker(f ) and l ∈ L, while [Ker(1 B ⋉ f ), Ker ((1 B , ∂))] by terms of the form [(0, k), (−∂(l), l)] = (0, ∂(l) k + [k, l]). It follows that the commutator described in (20) can be seen as an ideal of L, more precisely the subspace generated by terms of the form [k, l] or ∂(l) k, for k ∈ Ker(f ) and l ∈ L. By analogy with the case of precrossed modules of groups [11], we call this subobject the Peiffer commutator Ker(f ), L of Ker(f ) and L.…”

Galois Groups, Abstract Commutators, and Hopf Formula

“…It shows that the Brown-Ellis-Hopf formulae, as well as the classical Hopf formula, can be presented as a connection between homology in semi-abelian and more general categories and categorical Galois theory. Let us also mention recent related work of M. Gran and V. Rossi [12] (see also [28]) on generalized double central extensions, and of T. Everaert [8] on relative commutators for -groups related to their central extensions studied by A. Fröhlich, A. S.-T. Lue, and J. Furtado-Coelho long time ago, and therefore to central extensions in the sense of [21]; various closely related topics of categorical algebra have been developed in a number of papers of D. Bourn, M. Gran, T. Van der Linden, and T. Everaert, many of which are included in the lists of references in [9,11,28].…”