Group cohomology with coefficients in a crossed-module

TL;DR

This paper compares three methods of defining group cohomology with coefficients in a crossed-module, exploring their relationships and properties, including functoriality and exact sequences, especially in braided and symmetric cases.

Contribution

It introduces a unified comparison of cocycle, gerbe, and butterfly approaches to group cohomology with crossed-module coefficients, including functoriality and exact sequence results.

Findings

Established functoriality of cohomologies under weak morphisms

Proved long exact sequences in cohomology for crossed-modules

Analyzed cases with braided and symmetric braidings

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.