Twisting of monoidal structures

TL;DR

This paper explores the deformation of monoidal structures through twisting, revealing their cohomological nature and constructing maps to Hochschild cohomology, with examples including bimodules and free tensor categories.

Contribution

It introduces a cohomological framework for understanding twistings of monoidal structures and provides explicit examples involving bimodules and tensor categories.

Findings

Twisting sets have a non-abelian cohomological structure.

Maps from twistings to Hochschild cohomology are constructed.

Examples include bimodules, modules, comodules, and free tensor categories.

Abstract

This article is devoted to the investigation of the deformation (twisting) of monoidal structures, such as the associativity constraint of the monoidal category and the monoidal structure of monoidal functor. The sets of twistings have a (non-abelian) c ohomological nature. Using this fact the maps from the sets of twistings to some cohomology groups (Hochschild cohomology of K-theory) are constructed. The examples of monoidal categories of bimodules over some algebra, modules and comodules over bialgeb ra are examined. We specially concentrate on the case of free tensor category.