  We establish a Galois-theoretic interpretation of cohomology in semi-abelian
categories: cohomology with trivial coefficients classifies central extensions,
also in arbitrarily high degrees. This allows us to obtain a duality, in a
certain sense, between "internal" homology and "external" cohomology in
semi-abelian categories. These results depend on a geometric viewpoint of the
concept of a higher central extension, as well as the algebraic one in terms of
commutators.
