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 cat 3 -groups (Loday) and the 3-hypercomplexes (Cegarra-Carrasco), which also model algebraically homotopy 4-types.
We construct an actor of a precat 1 -algebra and then by using the natural equivalence between the categories of precat 1 -algebras and that of precrossed modules, we construct the split extension classifier of the corresponding precrossed module, which gives rise to the representability of actions in the category of precrossed modules of commutative algebras under certain conditions.
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