Galois theory and commutators

Abstract

Abstract. We prove that the relative commutator with respect to a subvariety of a variety of Ω-groups introduced by the first author can be described in terms of categorical Galois theory. This extends the known correspondence between the Fröhlich-Lue and the Janelidze-Kelly notions of central extension.As an example outside the context of Ω-groups we study the reflection of the category of loops to the category of groups where we obtain an interpretation of the associator as a relative commutator.

Relative commutator theory in semi-abelian categories

“…To prove (7), first of all recall that the square It was shown in [22] that two normal subobjects of an Ω-group commute in the sense of [15] if and only if they commute in the sense of our Definition 3.1. Since both notions of relative commutator satisfy the same universal property (see Theorem 3.9 (8)), we find: [15]: precrossed modules vs. crossed modules, where the relative commutator is the Peiffer commutator, for instance.…”

“…Since both notions of relative commutator satisfy the same universal property (see Theorem 3.9 (8)), we find: [15]: precrossed modules vs. crossed modules, where the relative commutator is the Peiffer commutator, for instance. An example which is not a consequence of this theorem-loops vs. groups, where the relative commutator is an associator-was considered in the article [22]. Another example which falls outside the scope of [15] is the case of compact Hausdorff topological groups vs. profinite groups.…”

Higher central extensions and cohomology

“…This interpretation of cohomology is part of a bigger programme which intends to understand homological algebra in a non-abelian environment from the viewpoint of (categorical) Galois theory. Related results include, for instance, higher Hopf formulae for homology in semi-abelian categories [40], higher-dimensional torsion theories [39], a theory of satellites for homology without projectives [47], and higherdimensional commutator theory based on a notion of higher centrality [43,44].…”

Resolutions, higher extensions and the relative Malʼtsev axiom

“…where H 2 pB, gpq is the second homology of B relative to the category of groups (i.e., with coefficients in the reflector gp : Loop Ñ Gp) and the brackets on the right hand side are associators [17]. The article [15] gives calculations of the homology objects for groups vs. abelian groups, rings vs. zero rings, precrossed modules vs. crossed modules, Lie algebras vs. modules, groups vs. groups of a certain nilpotency or solvability class, etc., in all dimensions.…”

“…The article [15] gives calculations of the homology objects for groups vs. abelian groups, rings vs. zero rings, precrossed modules vs. crossed modules, Lie algebras vs. modules, groups vs. groups of a certain nilpotency or solvability class, etc., in all dimensions. This approach to homology was extended to cover other examples [14,13] and several theoretical perspectives were explored: slightly different approaches [11,9,25], links with relative commutator theory [10,12,17,18], first steps towards an interpretation of cohomology [21,37], the characterisation of higher central extensions [16], and satellites [19,20].…”