"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." . "Higher Hopf formulae for homology via Galois Theory"@en . "217" . . . "5" . . "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."@en . "Higher Hopf formulae for homology via Galois Theory"@en . "14410" . . "RIV/00216224:14410/08:00027953!RIV10-MSM-14410___" . . "RIV/00216224:14410/08:00027953" . . "37"^^ . "Higher Hopf formulae for homology via Galois Theory" . "Gran, Marino" . . "Van der Linden, Tim" . "Higher Hopf formulae for homology via Galois Theory" . . "P(LC505)" . . "Everaert, Thomas" . "Advances in Mathematics" . "370102" . "1"^^ . . "US - Spojen\u00E9 st\u00E1ty americk\u00E9" . "3"^^ . "0001-8708" . . "000254098200013" . "Van der Linden, Tim" . "Semi-abelian category; Hopf formula; Homology; Galois Theory"@en . . . "[210A45BB6123]" . . .