Sheaves of categories with local actions of Hochschild cochains

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 $Y$.