Galois theory and commutators

TL;DR

This paper explores the relationship between Galois theory and commutators in algebraic structures, extending known concepts and providing new interpretations of associators in loops.

Contribution

It introduces a categorical description of relative commutators in Omega-groups and extends the correspondence between different notions of central extension.

Findings

Categorical description of relative commutators in Omega-groups

Extension of Froehlich-Lue and Janelidze-Kelly central extension correspondence

Interpretation of associator as a relative commutator in loops

Abstract

We prove that the relative commutator with respect to a subvariety of a variety of Omega-groups introduced by the first author can be described in terms of categorical Galois theory. This extends the known correspondence between the Froehlich-Lue and the Janelidze-Kelly notions of central extension. As an example outside the context of Omega-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.