Relative Commutator Theory in Semi-Abelian Categories

TL;DR

This paper develops a relative commutator theory within semi-abelian categories, generalizing existing concepts and characterizing B-central extensions through categorical Galois theory.

Contribution

It introduces a new notion of commutator relative to Birkhoff subcategories, unifying several existing commutator concepts in a categorical framework.

Findings

Characterizes B-central extensions categorically.

Recovers Huq's commutator in abelian cases.

Aligns with the relative commutator in omega-groups.

Abstract

Basing ourselves on the concept of double central extension from categorical Galois theory, we study a notion of commutator which is defined relative to a Birkhoff subcategory B of a semi-abelian category A. This commutator characterises Janelidze and Kelly's B-central extensions; when the subcategory B is determined by the abelian objects in A, it coincides with Huq's commutator; and when the category A is a variety of omega-groups, it coincides with the relative commutator introduced by the first author.