  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.
