Derived crossed modules

TL;DR

This paper explores the homotopy theory of crossed modules within categories of groups with operations, establishing new characterizations and constructions through derivations and categorical equivalences.

Contribution

It introduces a categorical interpretation of homotopy in crossed modules and develops methods to generate new modules via derivations.

Findings

Categorical interpretation of homotopy in crossed modules

Characterization of derivations in categories of groups with operations

Construction of new crossed modules using regular derivations

Abstract

In this study, we interpret the notion of homotopy of morphisms in the category of crossed modules in a category $\mathsf{C}$ of groups with operations using the categorical equivalence between crossed modules and internal categories in $\mathsf{C}$. Further, we characterize the derivations of crossed modules in a category $\mathsf{C}$ of groups with operations and obtain new crossed modules using regular derivations of old one.