GROUP COHOMOLOGY WITH COEFFICIENTS IN A CROSSED MODULE

Behrang Noohi

doi:10.1017/S1474748010000186

GROUP COHOMOLOGY WITH COEFFICIENTS IN A CROSSED MODULE

Behrang Noohi

Mathematics

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

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.

Original language	English
Pages (from-to)	359 - 404
Number of pages	46
Journal	Journal Of The Institute Of Mathematics Of Jussieu
Volume	10
Issue number	2
DOIs	https://doi.org/10.1017/S1474748010000186
Publication status	Published - Apr 2011

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title = "GROUP COHOMOLOGY WITH COEFFICIENTS IN A CROSSED MODULE",

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.",

author = "Behrang Noohi",

year = "2011",

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language = "English",

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journal = "Journal Of The Institute Of Mathematics Of Jussieu",

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AB - 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.

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DO - 10.1017/S1474748010000186

M3 - Article

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SP - 359

EP - 404

JO - Journal Of The Institute Of Mathematics Of Jussieu

JF - Journal Of The Institute Of Mathematics Of Jussieu

IS - 2

ER -

GROUP COHOMOLOGY WITH COEFFICIENTS IN A CROSSED MODULE

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