We show that every algebraically-central extension in a Mal'tsev variety - that is, every surjective homomorphism $f : A \longrightarrow B$ 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 $\Omega$-groups, while its converse is easily seen to hold for any congruence-modular variety.
2000 MSC: 08B05, 08C05, 18G50.
Theory and Applications of Categories, Vol. 7, 2000, No. 10, pp 219-226.