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