Tensor-Hochschild complex

TL;DR

This paper introduces a complex that governs deformations of monoidal dg-categories, generalizing known complexes like Davydov-Yetter and Gerstenhaber-Schack, with applications to $A$-modules and bialgebras.

Contribution

It constructs a new complex controlling deformations of monoidal dg-categories, unifying and extending existing deformation complexes in special cases.

Findings

Reduces to Davydov-Yetter complex for semisimple categories.

Computes operadic $E_2$-cohomology for $A$-modules.

Recovers Gerstenhaber-Schack complex for representations of bialgebras.

Abstract

Let $(\mathcal{C}, \otimes)$ be a monoidal dg-category. We construct a complex controlling the deformation of the monoidal structure on $\mathcal{C}$ together with the deformation of the underlying dg-category itself. We show that in the case of a semisimple category $\mathcal{C}$ it reduces to the Davydov-Yetter complex. Furthermore, we study this complex in several special cases, in particular, in the case of the category of $A$-modules over a commutative algebra $A$ we obtain a complex computing operadic $E_2$-cohomology of $A$. And in the case of the category of representations of an associative bialgebra we recover the Gerstenhaber-Schack complex. In the latter case our construction can be considered as a generalization of the Gerstenhaber-Schack complex to quasi-bialgebras.