Relative Commutator Theory in Semi-Abelian Categories

Everaert, Tomas [Vrije Universiteit Brussel] Van der Linden, Tim [UCL]

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 Beta of a semi-abelian category Alpha. This commutator characterises Janelidze and Kelly's Beta-central extensions ; when the subcategory Beta is determined by the abelian objects in Alpha, it coincides with Huq's commutator ; and when the category Alpha is a variety of Oméga-groups, it coincides with the relative commutator introduced by the first author.