  We use Janelidze's Categorical Galois Theory to extend Brown and Ellis's
higher Hopf formulae for homology of groups to arbitrary semi-abelian monadic
categories. Given such a category A and a chosen Birkhoff subcategory B of A,
thus we describe the Barr-Beck derived functors of the reflector of A onto B in
terms of centralization of higher extensions. In case A is the category Gp of
all groups and B is the category Ab of all abelian groups, this yields a new
proof for Brown and Ellis's formulae. We also give explicit formulae in the
cases of groups vs. k-nilpotent groups, groups vs. k-solvable groups and
precrossed modules vs. crossed modules.
