Abelian crossed modules and strict Picard categories

TL;DR

This paper establishes an equivalence between the category of abelian crossed modules and strict Picard categories, utilizing symmetric cohomology to address group extension problems.

Contribution

It introduces morphisms in abelian crossed modules and proves their categorical equivalence to strict Picard categories, applying symmetric cohomology to group extensions.

Findings

Category of abelian crossed modules is equivalent to strict Picard categories

Symmetric cohomology groups are used to analyze group extension problems

The paper provides a framework for understanding morphisms in these categories

Abstract

In this paper, we state the notion of morphisms in the category of abelian crossed modules and prove that this category is equivalent to the category of strict Picard categories and regular symmetric monoidal functors. The theory of obstructions for symmetric monoidal functors and symmetric cohomology groups are applied to show a treatment of the group extension problem of the type of an abelian crossed module.