Three Crossed Modules

TL;DR

This paper introduces 3-crossed modules, extending previous concepts, and establishes their equivalence with certain simplicial groups, connecting them to models of algebraic homotopy 4-types.

Contribution

It defines 3-crossed modules, relates them to simplicial groups with Moore complex length 3, and links them to existing algebraic models of homotopy 4-types.

Findings

3-crossed modules generalize 1- and 2-crossed modules

Category of 3-crossed modules is equivalent to certain simplicial groups

Connections made with cat$^{3}$-groups and 3-hyper-complexes

Abstract

We introduce the notion of 3-crossed module, which extends the notions of 1-crossed module (Whitehead) and 2-crossed module (Conduch\'e). We show that the category of 3-crossed modules is equivalent to the category of simplicial groups having a Moore complex of length 3. We make explicit the relationship with the cat$^{3}$-groups (Loday) and the 3-hyper-complexes (Cegarra-Carrasco), which also model algebraically homotopy 4-types.