  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.
