Non-abelian cohomology via parity quasi-complexes

Abstract

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.