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 H3, H2, H1, H0 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. 
Lucian M. Ionescu (Kansas State University)