The Category of Crossed Modules of Crossed Modules and Its Associated Double Groupoids

TL;DR

This paper explores the structure of Whitehead sequences in the category of crossed modules, examining conditions under which they form internal groupoids, and investigates the 'Smith is Huq' condition in this context.

Contribution

It provides a detailed analysis of Whitehead sequences and their relation to crossed squares and internal groupoids within the category of crossed modules.

Findings

Whitehead sequences correspond to crossed squares

Conditions identified for Whitehead sequences to form internal groupoids

Analysis of the 'Smith is Huq' condition in crossed modules

Abstract

In this work we study the notion of Whitehead sequence in the category of crossed modules and actions of crossed modules. As expected, Whitehead sequences in that context are the same as crossed squares. We investigate under which conditions a Whitehead sequence of crossed modules gives rise to an internal groupoid in the category of crossed modules. In other words, we explicitly investigate the so called "Smith is Huq" condition in the category of crossed modules.