On the nonabelian cohomology with coefficients in a crossed module

TL;DR

This paper develops a theory for nonabelian group cohomology with coefficients in crossed modules, extending classical extension theories and classifying important algebraic objects.

Contribution

It introduces a Schreier-like obstruction theory for nonabelian cohomology with crossed modules, advancing the understanding of classified objects.

Findings

Established a framework for nonabelian cohomology with crossed modules

Developed a Schreier obstruction theory for these cohomologies

Connected cohomology classes to classification of algebraic structures

Abstract

This paper is concerned with the nonabelian cohomology of groups with coefficients in crossed modules. These objects were introduced by Dedecker and studied by Breen, Borovoi, Noohi and many others. In this paper we study several important objects that are classified by these cohomologies and for them we develop a theory similar to the Schreier obstruction theory of group extensions.