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Group cohomology with coefficients in a crossed module

Part of: Categories with structure Connections with homological algebra and category theory Homological algebra

Published online by Cambridge University Press:  17 June 2010

Behrang Noohi
Behrang Noohi
Affiliation:
King's College London, Strand, London WC2R 2LS, UK (behrang.noohi@kcl.ac.uk)
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Abstract

We compare three different ways of defining group cohomology with coefficients in a crossed module: (1) explicit approach via cocycles; (2) geometric approach via gerbes; (3) group theoretic approach via butterflies. We discuss the case where the crossed module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed modules and also prove the ‘long’ exact cohomology sequence associated to a short exact sequence of crossed modules and weak morphisms.

Keywords

equivariant crossed moduleequivariant categorical groupequivariant 2-groupgroup cohomologyprincipal 2-bundlebutterfly

MSC classification

Secondary: 18G50: Nonabelian homological algebra 18G60: Other (co)homology theories 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 20J06: Cohomology of groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Issue 2 , April 2011 , pp. 359 - 404
Copyright
Copyright © Cambridge University Press 2010

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