Higher Hopf formulae for homology via Galois Theory

Warning

This publication doesn't include Institute of Computer Science. It includes Faculty of Education. Official publication website can be found on muni.cz.

Authors	EVERAERT Thomas GRAN Marino VAN DER LINDEN Tim
Year of publication	2008
Type	Article in Periodical
Magazine / Source	Advances in Mathematics
MU Faculty or unit	Faculty of Education
Citation
web	http://dx.doi.org/10.1016/j.aim.2007.11.001
Field	General mathematics
Keywords	Semi-abelian category; Hopf formula; Homology; Galois Theory
Description	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.
Related projects:	Eduard Čech Center for Algebra and Geometry