On Parity Complexes and Non-abelian Cohomology

Details On Parity Complexes and Non-abelian Cohomology On Parity Complexes and Non-abelian Cohomology

1998-08-15

arxiv.org/abs/math/9808068v1

Mathematics

Category Theory

AMS-LaTex, 31 pages

Scientific paper

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.

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