Higher Hopf formulae for homology via Galois Theory

Everaert, Tomas; Gran, Marino; Van der Linden, Tim

Higher Hopf formulae for homology via Galois Theory

Authors

Everaert, Tomas

Gran, Marino

Van der Linden, Tim

Description

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.
35 pages; major changes in section 5, minor changes elsewhere