Higher Hopf formulae for homology via Galois Theory

Everaert, Tomas;Gran, Marino;Van der Linden, Tim
(2008) Advances in mathematics — Vol. 217, n° 5, p. 2231-2267 (2008)

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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 Alpha and a chosen Birkhoff subcategory Beta of Alpha, thus we describe the Barr–Beck derived functors of the reflector of Alpha onto Beta in terms of centralization of higher extensions. In case Alpha is the category Gp of all groups and Beta 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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Everaert, T., Gran, M., & Van der Linden, T. (2008). Higher Hopf formulae for homology via Galois Theory. Advances in mathematics, 217(5), 2231-2267. https://doi.org/10.1016/j.aim.2007.11.001 (Original work published 2008)