Crossed-modules and Whitehead sequences

TL;DR

This paper introduces Whitehead sequences in a categorical setting, generalizing crossed-modules from groups to broader contexts, and establishes conditions for their equivalence with internal groupoids.

Contribution

It generalizes the concept of crossed-modules to arbitrary categories with actions and provides criteria for their equivalence with internal groupoids.

Findings

Whitehead sequences generalize crossed-modules.

Conditions for categorical equivalence with internal groupoids are established.

Framework applies to various algebraic and categorical structures.

Abstract

We introduce the notion of Whitehead sequence which is defined for a base category together with a system of abstract actions over it. In the classical case of groups and group actions the Whitehead sequences are precisely the crossed-modules of groups. For a general setting we give sufficient conditions for the existence of a categorical equivalence between internal groupoids and Whitehead sequences.