Homology, Homotopy and Applications
Volume 11 (2009)
Pages: 161 – 187
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.
crossed module; 2-crossed module; simplicial group; Moore complex
2010 Mathematics Subject Classification
18D35, 18G30, 18G50, 18G55