Coverings and liftings of generalized crossed modules

TL;DR

This paper extends the theory of crossed modules by introducing generalized crossed modules via self-actions, and explores coverings and liftings, establishing an equivalence between their categories.

Contribution

It provides a formal definition of generalized cat$^1$-groups and constructs a functor to generalized crossed modules, also analyzing coverings and liftings within this framework.

Findings

Established a functor from generalized cat$^1$-groups to generalized crossed modules

Defined and analyzed coverings and liftings of generalized crossed modules

Proved an equivalence between the categories of coverings and liftings

Abstract

In the theory of crossed modules, considering arbitrary self-actions instead of conjugation allows for the extension of the concept of crossed modules and thus the notion of generalized crossed module emerges. In this paper we give a precise definition for generalized cat$^1$-groups and obtain a functor from the category of generalized cat$^1$-groups to generalized crossed modules. Further, we introduce the notions of coverings and liftings for generalized crossed modules and investigate properties of these structures. Main objective of this study is to obtain an equivalence between the category of coverings and the category of liftings of a given generalized crossed module $(A,B,\alpha)$.