On the second cohomology group in semi-abelian categories

Abstract

We develop some new aspects of cohomology in the context of semi-abelian categories: we establish a Hochschild–Serre 5-term exact sequence extending the classical one for groups and Lie algebras; we prove that an object is perfect if and only if it admits a universal central extension; we show how the second Barr–Beck cohomology group classifies isomorphism classes of central extensions; we prove a universal coefficient theorem to explain the relationship with homology.