We introduce the notion of a 3-crossed module, which extends the notions of a 1-crossed module (Whitehead) and a 2-crossed module (Conduché). 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 cat3-groups (Loday) and the 3-hypercomplexes (Cegarra-Carrasco), which also model algebraically homotopy 4-types.
Homology, Homotopy and Applications, Vol. 11 (2009), No. 2, pp.161-187.
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