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Higher Hopf formulae for homology via Galois Theory

Tomas Everaert·Marino Gran·Tim Van der Linden
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Abstract

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.

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Tomas Everaert, Marino Gran, Tim Van der Linden. 2007-05-21. Higher Hopf formulae for homology via Galois Theory. https://doi.org/10.1016/j.aim.2007.11.001

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