The homotopy category of parity quasi-complexes is introduced. The homotopy structure is compatible with the non-abelian homology of parity quasi-complexes.
Parity contracting homotopies are defined, determining the parity free resolutions in a canonical way, enabling the nonabelian bar construction.
In this way, the even/odd grouping of the simplicial maps in the cocycle conditions of nonabelian cohomology is explained.
Homology, Homotopy and Applications, Vol. 6(2004), No. 1, pp. 49-58