Group-groupoid actions and liftings of crossed modules

TL;DR

This paper introduces the concept of lifting crossed modules via group morphisms, explores their properties, and establishes categorical equivalences with covers and group-groupoid actions, advancing the theoretical understanding of these structures.

Contribution

It defines the notion of lifting for crossed modules, provides criteria for their existence, and shows the categorical equivalence with covers and group-groupoid actions.

Findings

Lifting of crossed modules forms a category

Categorical equivalence with cover categories

Characterization of liftings via criteria

Abstract

The aim of this paper is to define the notion of lifting of a crossed module via a group morphism and give some properties of this type of the lifting. Further we obtain a criterion for a crossed module to have a lifting of crossed module. We also prove that the liftings of a certain crossed module constitute a category; and that this category is equivalent to the category of covers of that crossed module and hence to the category of group-groupoid actions of the corresponding groupoid to that crossed module.