Sheaves of categories with local actions of Hochschild cochains

TL;DR

This paper develops a categorified theory of sheaves of categories associated with Hochschild cochains, extending the notion of D-modules to a higher categorical setting and establishing its properties and applications to algebraic stacks.

Contribution

It introduces the functor $ShvCat^{ ext H}$, studies its properties, and proves $ ext H$-affineness for algebraic stacks, advancing the understanding of sheaves of categories in derived algebraic geometry.

Findings

$ShvCat^{ ext H}$ is functorial and satisfies descent.

Algebraic stacks are $ ext H$-affine under mild conditions.

Categories with $ ext H(Y)$-action admit singular support in $Sing(Y)$.

Abstract

The notion of Hochschild cochains induces an assignment from $Aff$, affine DG schemes, to monoidal DG categories. We show that this assignment extends, under some appropriate finiteness conditions, to a functor $\mathbb H: Aff \to AlgBimod(DGCat)$, where the latter denotes the category of monoidal DG categories and bimodules. Now, any functor $\mathbb A: Aff \to AlgBimod(DGCat)$ gives rise, by taking modules, to a theory of sheaves of categories $ShvCat^{\mathbb A}$. In this paper, we study $ShvCat^{\mathbb H}$. Vaguely speaking, this theory categorifies the theory of D-modules, in the same way as Gaitsgory's original $ShvCat$ categorifies the theory of quasi-coherent sheaves. We develop the functoriality of $ShvCat^{\mathbb H}$, its descent properties and, most importantly, the notion of $\mathbb H$-affineness. We then prove the $\mathbb H$-affineness of algebraic stacks: for $Y$ a stack satisfying some mild conditions, the $\infty$-category $ShvCat^{\mathbb H}(Y)$ is equivalent to the $\infty$-category of modules for $\mathbb H(Y)$, the monoidal DG category defined in arXiv:1709.07867. As an application, consider a quasi-smooth stack $Y$ and a DG category $C$ with an action of $\mathbb H(Y)$. Then $C$ admits a theory of singular support in $Sing(Y)$, where $Sing(Y)$ is the space of singularities of $Y$.