We prove that all semi-abelian categories with the the Smith is Huq property satisfy the Commutator Condition(CC): higher central extensions may be characterised in terms of binary (Huq or Smith) commutators. In fact, even Higgins commutators suffice. As a consequence, in the presence of enough projectives we obtain explicit Hopf formulae for homology with coefficients in the abelianisation functor, and an interpretation of cohomology with coefficients in an abelian object in terms of equivalence classes of higher central extensions. We also give a counterexample against (CC) in the semi-abelian category of (commutative) loops.
Keywords: Higgins, Huq, Smith commutator; higher central extension; semi-abelian, exact Mal'tsev category; Hopf formula; (co)homology
2010 MSC: 18G50, 18G60, 18G15, 20J, 55N
Theory and Applications of Categories, Vol. 27, 2012, No. 9, pp 189-209.