Central Extensions in Mal’tsev Varieties

We show that every algebraically–central extension in a Mal’tsev variety — that is, every surjective homomorphism f : A−→ B whose kernel–congruence is contained in the centre of A, as defined using the theory of commutators — is also a central extension in the sense of categorical Galois theory; this was previously known only for varieties of Ω-groups, while its converse is easily seen to hold for any congruence–modular variety.