Abstract
The homotopy category of parity quasi-complexes is introduced. The homotopy structure is compatible with the nonabelian 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.
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