Tensor-Hochschild complex

TL;DR

The paper introduces the tensor-Hochschild complex $TC^\bullet(\mathcal{C},\otimes)$ to simultaneously deform a monoidal dg-category and its underlying dg-category. It proves that $TC^\bullet$ extends Hochschild theory by incorporating higher coherence data via admissible paths in associahedra and recovers known deformation theories in key special cases, such as $E_2$-cohomology for a single-object endomorphism algebra and Gerstenhaber-Schack cohomology for bialgebras. In semisimple settings, the Davydov-Yetter complex suffices, while in general $TC^\bullet$ contains richer information and admits spectral sequences linking to Hochschild and DY cohomologies. The work also analyzes concrete instances including smooth schemes and quiver-derived categories, yielding degeneracy results, rigidity phenomena, and pathways toward higher operadic actions, with implications for infinity-monoidal structures.

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.

Theorems & Definitions (12)

• Proposition 1.1
• Lemma 1.2
• Proposition 1.3
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• Theorem 2.4
• Corollary 2.5
• Corollary 2.6
• Proposition 2.7