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 $ShvCa{t}^{\mathbb{A}}$. In this paper, we study $ShvCa{t}^{\mathbb{H}}$. 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 $ShvCa{t}^{\mathbb{H}}$, its descent properties and, most importantly, the notion of $\mathbb{H}$-affineness. We then prove the $\mathbb H$-affinenessofalgebraicstacks:for$Y$ a stack satisfying some mild conditions, the $\mathrm{\infty }$-category $ShvCa{t}^{\mathbb{H}}(Y)$ is equivalent to the $\mathrm{\infty }$-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 $Y$.