Relations between simplicial groups, 3-crossed modules and 2-quadratic modules (Q2453851)

In this paper, the notion of two-qadratic module is defined and related to 3-crossed modules and simplicial groups. The notion of crossed module as an algebraic model was defined by \textit{J. H. C. Whitehead} [Bull. Am. Math. Soc. 55, 453--496 (1949; Zbl 0040.38801)] (see also [\textit{H. J. Baues} and \textit{D. Conduché}, J. Algebra 133, No. 1, 1--34 (1990; Zbl 0704.55008)]) during his study on the second homotopy groups of topological spaces. The category of crossed modules and the category of simplicial groups with Moore complexes of length 1 are equivalent; and similarly 2-crossed modules and simplicial groups with Moore complexes of length 2 are equivalent. In [\textit{Z. Arvasi} et al., Homology Homotopy Appl. 11, No. 2, 161--187 (2009; Zbl 1307.18008)] the equivalence is also proved for 3-types. On the other hand, \textit{H. J. Baues} [Combinatorial homotopy and 4-dimensional complexes. Berlin etc.: Walter de Gruyter (1991; Zbl 0716.55001)] defined quadratic models which is related to 2-crossed modules. In this paper 2-quadratic module is defined and related to 3-crossed modules and simplicial groups.