# Categories internal to crossed modules

## TL;DR

This paper characterizes internal categories within the category of crossed modules and establishes an equivalence with crossed squares, providing new insights and examples in this mathematical framework.

## Contribution

It introduces a natural equivalence between crossed squares and internal categories in crossed modules, expanding the understanding of their structural relationships.

## Key findings

- Established equivalence between crossed squares and internal categories in crossed modules
- Provided examples of crossed squares using this equivalence
- Enhanced the theoretical framework of crossed modules and their internal categories

## Abstract

In this study, internal categories in the category of the crossed modules are characterized and it has been shown that there is a natural equivalence between the category of the crossed modules over crossed modules, i.e. crossed squares, and the category of the internal categories within the category of crossed modules. Finally, we obtain examples of crossed squares using this equivalence.