Higher Hopf formulae for homology via Galois Theory

Tomas Everaert, Marino Gran, Tim Van Der Linden

    Research output: Contribution to journalArticlepeer-review

    43 Citations (Scopus)


    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.
    Original languageEnglish
    Pages (from-to)2231-2267
    Number of pages37
    JournalAdvances in Mathematics
    Issue number5
    Publication statusPublished - 2008


    • semi-abelian category
    • Hopf formula
    • Galois theory
    • homology


    Dive into the research topics of 'Higher Hopf formulae for homology via Galois Theory'. Together they form a unique fingerprint.

    Cite this