Homology, Homotopy and Applications

Volume 11 (2009)

Number 2

Three-crossed modules

Pages: 161 – 187

DOI: https://dx.doi.org/10.4310/HHA.2009.v11.n2.a8


Z. Arvasi (Department of Mathematics and Computer Sciences, Osmangazi University, Eskişehir, Turkey)

T. S. Kuzpinari (Department of Mathematics, Aksaray University, Aksaray, Turkey)

E. Ö. Uslu (Department of Mathematics, Afyon Kocatepe University, Afyonkarahisar, Turkey)


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

Published 1 January 2009