The notion of Hochschild cochains induces an assignment from
$\mathsf{Aff}$
, affine DG schemes, to monoidal DG categories. We show that this assignment extends, under appropriate finiteness conditions, to a functor
$\mathbb{H}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$
, where the latter denotes the category of monoidal DG categories and bimodules. Any functor
$\mathbb{A}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$
gives rise, by taking modules, to a theory of sheaves of categories
$\mathsf{ShvCat}^{\mathbb{A}}$
. In this paper, we study
$\mathsf{ShvCat}^{\mathbb{H}}$
. Loosely speaking, this theory categorifies the theory of
$\mathfrak{D}$
-modules, in the same way as Gaitsgory’s original
$\mathsf{ShvCat}$
categorifies the theory of quasi-coherent sheaves. We develop the functoriality of
$\mathsf{ShvCat}^{\mathbb{H}}$
, its descent properties and the notion of
$\mathbb{H}$
-affineness. We then prove the
$\mathbb{H}$
-affineness of algebraic stacks: for
${\mathcal{Y}}$
a stack satisfying some mild conditions, the
$\infty$
-category
$\mathsf{ShvCat}^{\mathbb{H}}({\mathcal{Y}})$
is equivalent to the
$\infty$
-category of modules for
$\mathbb{H}({\mathcal{Y}})$
, the monoidal DG category of higher differential operators. The main consequence, for
${\mathcal{Y}}$
quasi-smooth, is the following: if
${\mathcal{C}}$
is a DG category acted on by
$\mathbb{H}({\mathcal{Y}})$
, then
${\mathcal{C}}$
admits a theory of singular support in
$\operatorname{Sing}({\mathcal{Y}})$
, where
$\operatorname{Sing}({\mathcal{Y}})$
is the space of singularities of
${\mathcal{Y}}$
. As an application to the geometric Langlands programme, we indicate how derived Satake yields an action of
$\mathbb{H}(\operatorname{LS}_{{\check{G}}})$
on
$\mathfrak{D}(\operatorname{Bun}_{G})$
, thereby equipping objects of
$\mathfrak{D}(\operatorname{Bun}_{G})$
with singular support in
$\operatorname{Sing}(\operatorname{LS}_{{\check{G}}})$
.
On the Notion of Conductor in the Local
Geometric Langlands Correspondence
