Higher Hopf formulae for homology via Galois Theory

Details Higher Hopf formulae for homology via Galois Theory Higher Hopf formulae for homology via Galois Theory

2007-01-29

arxiv.org/abs/math/0701815v2

Advances in Mathematics 217 (2008) 2231-2267

Mathematics

Algebraic Topology

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

Scientific paper

10.1016/j.aim.2007.11.001

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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