We introduce a pairing structure within the Moore complex NG of a simplicial group G and use it to investigate generators for N G n ∩ D n where D n is the subgroup generated by degenerate elements. This is applied to the study of algebraic models for homotopy types.
Generalising a result of Brown and Loday, we give for n = 3 and 4, a decomposition of the group, d n NG n , of boundaries of a simplicial group G as a product of commutator subgroups. Partial results are given for higher dimensions. Applications to 2-crossed modules and quadratic modules are discussed.
Using free simplicial groups, it is shown how to construct a free or totally free 2-crossed module on suitable construction data. 2-crossed complexes are introduced and similar freeness results for these are discussed.
In this paper we give a construction of free 2-crossed modules. By the use of a `step-by-step' method based on the work of Andr e, we will give a description of crossed algebraic models for the steps in the construction of a free simplicial resolution of an algebra. This involves the introduction of the notion of a free 2-crossed module of algebras.