Cohomology of monoidal categories and non-Abelian group cohomology.

摘要

To characterize the categorical constraints—associativity, commutativity and monoidality—in the wider context of quasimonoidal categories, from a multiplicative cohomological point of view, we define the notion of a parity quasi-complex.; We introduce the standard parity complex of a group G with non abelian coefficients and define the corresponding cohomology spaces in all dimensions. In dimensions less then two it coincides with the non-abelian cohomology spaces as agreed by various authors.; The cohomology space H3 corresponds to quasi-extensions and H2, H 1, H0 corresponds as usual to group extensions, split extensions and invariants, as in the abelian case.; We introduce two new categorification procedures, which are functors from the category of groups to the category of monoidal categories. Fiber categorification associates to a group a monoidal groupoid and bundle categorification associates to a group extension a fibration of monoidal groupoids. The correspondences are functorial and compatible with discrete categorification, a procedure used in [CY2] in connection with fusion rings.; The categorification is compatible with the parity quasi-complexes of the corresponding groups and monoidal categories. It allows to interpret (1) cochains as functors, (2) cocycles—monoidal structures, (3) cocycles—associators, representing a correspondence between non-abellan group cohomology and the cohomology of a functor.; A class of commutativity constraints in monoidal categories is identified. They are equivalences of the associators of the monoidal category and its opposite. It extends the class of braidings. It is proved that the category of representations of an almost cocomutative Hopf algebra is a commutative monoidal category iff the latter is a coboundary Hopf algebra.; As a model for the categorical setup, we investigate the Hochschild cohomological construction in the non-associative case. The notion of a N-coherent algebra is introduced and their cohomology is defined using the construction from [Ka].; The non-associativity condition is naturally interpreted as a curvature. The square of the Hochschild quasi-differential is proved to have the properties related to the curvature from the context of differential geometry. A possible geometric viewpoint is mentioned: torsion algebras. The non-associative multiplication is interpreted as a formal connection and the elements of the algebra are thought of as formal vector fields. We determine conditions for the existence of an algebra of “functions” and representing the formal vector fields as derivations.