Three Crossed Modules

Understand

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.