Tim Van der Linden: Higher
central extensions via binary commutators
We prove that all semi-abelian categories with the Smith is Huq
property satisfy the Commutator Condition: higher central extensions
may be characterised in terms of binary (Huq or Smith) commutators. In
fact, even binary Higgins commutators suffice. As a consequence, in
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 the Commutator Condition in the semi-abelian
category of loops.