On Parity Complexes and Non-abelian Cohomology

To characterize categorical constraints associativity, commutativity and monoidality in the context of quasimonoidal categories, from a cohomological point of view, we define the notion of a parity (quasi)complex. Applied to groups gives non-abelian cohomology. The categorification functor from groups to monoidal categories provides the correspondence between the respective parity (quasi)complexes and allows to interpret 1-cochains as functors, 2-cocycles monoidal structures, 3-cocycles associators. The cohomology spaces H, H, H, H correspond as usual to quasi-extensions, extensions, split extensions and invariants, as in the abelian case. A larger class of commutativity constraints for monoidal categories is identified. It is naturally associated with coboundary Hopf algebras.