Group cohomology with coefficients in a crossed-module

2009/02/01 by Behrang Noohi, Noohi, Behrang
Mathematics · #Category Theory (math.CT) #FOS: Mathematics #K-Theory and Homology (math.KT) #math.CT #math.KT

paper · pdf · doi:10.48550/arxiv.0902.0161

A discussion on braided 2-crossed-modules added to Section 5.3. References added

arxiv created 2010/01/26 · arxiv updated 2010/02/08

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.

Related

• On the nonabelian cohomology with coefficients in a crossed module
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• (Co)homology of crossed modules with coefficients in a π₁-module
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