# Further remarks on liftings of crossed modules

## TL;DR

This paper introduces the concept of pullback liftings of crossed modules, explores their properties in group-groupoid actions, and establishes criteria for homotopy lifting, advancing the understanding of crossed module morphisms.

## Contribution

It defines pullback liftings of crossed modules, interprets them as pullback actions, and provides a homotopy lifting property criterion for crossed module morphisms.

## Key findings

- Defined pullback lifting of crossed modules
- Established homotopy lifting property for morphisms
- Analyzed properties of derivations in lifting crossed modules

## Abstract

In this paper we define the notion of pullback lifting of a lifting crossed module over a crossed module morphism and interpret this notion in the category of group-groupoid actions as pullback action. Moreover, we give a criterion for the lifting of homotopic crossed module morphisms to be homotopic, which will be called homotopy lifting property for crossed module morphisms. Finally, we investigate some properties of derivations of lifting crossed modules according to base crossed module derivations.