2.1.1

First, let us fix some terminology. Algebra objects in a symmetric monoidal $\infty$ -category are always unital in this paper. In particular, monoidal DG categories are unital. Given $A$ an algebra, denote by $A^{\operatorname{rev}}$ the algebra obtained by reversing the order of the multiplication. For a left $A$ -module $M$ and a right $A$ -module $N$ , we denote by $\mathsf{pr}:N\otimes M\rightarrow N\otimes _{A}M$ the tautological functor.

Our conventions regarding bimodules are as follows: an $(A,B)$ -bimodule $M$ is acted on on the left by $A$ and on the right by $B$ . Hence, endowing $C\in \mathsf{DGCat}$ with the structure of an $(A,B)$ -bimodule amounts to endowing it with the structure of a left $A\otimes B^{\operatorname{rev}}$ -module.

2.1.2

Let $M$ be an $(A,B)$ -bimodule. We say that $M$ is left dualizable (as an $(A,B)$ -bimodule) if there exists a $(B,A)$ -bimodule $M^{L}$ (called the left dual of $M$ ) realizing an adjunction

Similarly, $M$ is right dualizable if there exists $M^{R}\in (B,A)\operatorname{ -}\mathsf{bimod}$ (the right dual of $M$ ) realizing an adjunction

We say that an $(A,B)$ -bimodule $M$ is ambidextrous if both $M^{L}$ and $M^{R}$ exist and are equivalent as $(B,A)$ -bimodules.

2.1.4

Let $M$ be an $(A,B)$ -bimodule which is dualizable as a DG category. Then we can contemplate three $(B,A)$ -bimodules: $M^{L},M^{R}$ (if they exist) as well as $M^{\ast }$ , the dual of $\mathsf{oblv}_{A,B}(M)$ equipped with the dual actions.

In particular, a monoidal DG category $A$ is called proper if it is dualizable as a plain DG category. In this case, we denote by $S_{A}:=A^{\ast }$ its dual, equipped with the tautological $(A,A)$ -bimodule structure.

2.1.5

Recall the notion of rigid monoidal DG category; see [Reference GaitsgoryGai15b, Appendix D]. Any rigid $A$ is automatically proper. Furthermore, its dual $S_{A}:=A^{\ast }$ comes equipped with the canonical object $1_{A}^{\text{fake}}:=(u^{R})^{\vee }(k)$ , where $u^{R}$ is the (continuous) right adjoint to the unit functor $u:\operatorname{Vect}\rightarrow A$ . The left $A$ -linear functor

$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{A}:A\longrightarrow S_{A},\quad a{\rightsquigarrow}a\star 1_{A}^{\text{fake}}\end{eqnarray}$$

is an equivalence; in particular, any rigid monoidal category is self-dual. We say that $A$ is very rigid if the canonical equivalence $\unicode[STIX]{x1D70E}_{A}:A\rightarrow S_{A}$ admits a lift to an equivalence of $(A,A)$ -bimodules.Footnote ⁴

2.2.1

We assume familiarity with the notion of $(\infty ,2)$ -category and with the notion of (lax) $(\infty ,2)$ -functor between $(\infty ,2)$ -categories; see, for example, [Reference Gaitsgory and RozenblyumGR17, Appendix A]. For an $(\infty ,2)$ -category $\mathbf{C}$ , we denote by $\mathbf{C}^{1-\operatorname{op}}$ the $(\infty ,2)$ -category obtained from $\mathbf{C}$ by flipping the $1$ -arrows. Similarly, we denote by $\mathbf{C}^{2-\operatorname{op}}$ the $(\infty ,2)$ -category obtained by flipping the directions of the $2$ -arrows.

2.2.2 Correspondences

Let ${\mathcal{C}}$ be an $\infty$ -category equipped with fibre products. We refer to [Reference Gaitsgory and RozenblyumGR17, ch. V.1] for the construction of the $\infty$ -category of correspondences associated to ${\mathcal{C}}$ . In particular, for $\text{vert}$ and $\text{horiz}$ two subsets of the space morphisms of ${\mathcal{C}}$ satisfying some natural requirements, one considers the $\infty$ -category $\mathsf{Corr}({\mathcal{C}})_{\text{vert};\text{horiz}}$ , defined in the usual way: objects of $\mathsf{Corr}({\mathcal{C}})_{\text{vert};\text{horiz}}$ coincide with the objects of ${\mathcal{C}}$ , while $1$ -morphisms in $\mathsf{Corr}({\mathcal{C}})_{\text{vert};\text{horiz}}$ are given by correspondences

$$\begin{eqnarray}[c\leftarrow h\rightarrow d]\end{eqnarray}$$

with left leg in $\text{vert}$ and right leg in $\text{horiz}$ .

To enhance $\mathsf{Corr}({\mathcal{C}})_{\text{vert};\text{horiz}}$ to an $(\infty ,2)$ -category, we must further choose a subset $\text{adm}\subset \text{vert}\cap \text{horiz}$ of admissible arrows, closed under composition. Then, following [Reference Gaitsgory and RozenblyumGR17, ch. V.1], we define the $(\infty ,2)$ -category

$$\begin{eqnarray}\mathsf{Corr}({\mathcal{C}})_{\text{vert};\text{horiz}}^{\text{adm}}.\end{eqnarray}$$

This is one of the most important $(\infty ,2)$ -categories of the present paper.

To fix notation, recall that a $2$ -arrow

$$\begin{eqnarray}[c\leftarrow h\rightarrow d]\;\Longrightarrow \;[c\leftarrow h^{\prime }\rightarrow d]\end{eqnarray}$$

in $\mathsf{Corr}({\mathcal{C}})_{\text{vert};\text{horiz}}^{\text{adm}}$ is by definition an admissible arrow $h\rightarrow h^{\prime }$ compatible with the maps to $c\times d$ .

As explained in [Reference Gaitsgory and RozenblyumGR17, ch. V.3], $\mathsf{Corr}({\mathcal{C}})_{\text{vert};\text{horiz}}^{\text{adm}}$ is symmetric monoidal with tensor product induced by the cartesian symmetric monoidal product on ${\mathcal{C}}$ .

2.2.3 Algebras and bimodules

The other important $(\infty ,2)$ -category of this paper is $\mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat})$ , the $(\infty ,2)$ -category of monoidal DG categories, bimodules, and natural transformations. We refer to [Reference HaugsengHau17] for a rigorous construction. More generally, that paper gives a construction of $\mathsf{ALG}^{\text{bimod}}({\mathcal{S}})$ for any (nice enough) symmetric monoidal $(\infty ,2)$ -category ${\mathcal{S}}$ .

We denote by $\mathsf{Alg}^{\text{bimod}}({\mathcal{S}})$ the $(\infty ,1)$ -category underlying $\mathsf{ALG}^{\text{bimod}}({\mathcal{S}})$ : that is, the former is obtained from the latter by discarding non-invertible $2$ -morphisms.

2.2.4

There is an obvious functor

(2.1) $$\begin{eqnarray}\unicode[STIX]{x1D704}_{\mathsf{Alg}\rightarrow \mathsf{Bimod}}:\mathsf{Alg}(\mathsf{DGCat})^{\operatorname{op}}\longrightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})\end{eqnarray}$$

that is the identity on objects and that sends a monoidal functor $A\rightarrow B$ to the $(B,A)$ -bimodule  $B$ .

The tautological functor

$$\begin{eqnarray}\mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})^{\operatorname{op}}\xrightarrow[{}]{\operatorname{ -}\mathbf{mod}}\mathsf{Cat}_{\infty }\end{eqnarray}$$

upgrades to a (strict) $(\infty ,2)$ -functor

$$\begin{eqnarray}\mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat})^{1-\operatorname{op}}\xrightarrow[{}]{\operatorname{ -}\mathbf{mod}}\mathsf{Cat}_{\infty },\end{eqnarray}$$

where now $\mathsf{Cat}_{\infty }$ is considered as an $(\infty ,2)$ -category.

2.2.5

Let ${\mathcal{C}}$ denote an $(\infty ,1)$ -category admitting fibre products and equipped with the cartesian symmetric monoidal structure. Let $F:{\mathcal{C}}^{\operatorname{op}}\longrightarrow \mathsf{DGCat}$ be a lax-monoidal functor. (The example we have in mind is ${\mathcal{C}}=\mathsf{PreStk}$ and $F=\operatorname{QCoh}$ .)

These data give rise to a lax $(\infty ,2)$ -functor

$$\begin{eqnarray}\widetilde{F}:\big(\mathsf{Corr}({\mathcal{C}})_{\operatorname{all};\operatorname{all}}^{\operatorname{all}}\big)^{2-\operatorname{op}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat}),\end{eqnarray}$$

described informally as follows:

1. – an object $c\in {\mathcal{C}}$ gets sent to $F(c)$ , with its natural monoidal structure;

2. – a correspondence $[c\leftarrow h\rightarrow d]$ gets sent to the $(F(c),F(d))$ -bimodule $F(h)$ ;

3. – a map between correspondences, given by an arrow $h^{\prime }\rightarrow h$ over $c\times d$ , gets sent to the associated $(F(c),F(d))$ -linear arrow $F(h)\rightarrow F(h^{\prime })$ ;

4. – for two correspondences $[c\leftarrow h\rightarrow d]$ and $[d\leftarrow k\rightarrow e]$ , the lax composition is encoded by the natural $(F(c),F(e))$ -linear arrow

   $$\begin{eqnarray}F(h)\underset{F(d)}{\otimes }F(k)\longrightarrow F(h\times _{d}k).\end{eqnarray}$$

2.2.6

Here is another example of the interaction between lax-monoidal functors and lax $(\infty ,2)$ -functors. Let $F:{\mathcal{C}}\rightarrow {\mathcal{D}}$ be a lax-monoidal functor between ‘well-behaved’ monoidal $(\infty ,1)$ -categories. Then $F$ induces a lax $(\infty ,2)$ -functor

$$\begin{eqnarray}\widetilde{F}:\mathsf{ALG}^{\text{bimod}}({\mathcal{C}})\longrightarrow \mathsf{ALG}^{\text{bimod}}({\mathcal{D}}).\end{eqnarray}$$

To define it, it suffices to recall that, since $F$ is lax monoidal, it preserves algebra and bimodule objects. The fact that $\widetilde{F}$ is a lax $(\infty ,2)$ -functor comes from the natural map (not necessarily an isomorphism)

$$\begin{eqnarray}F(c^{\prime })\otimes _{F(c)}F(c^{\prime \prime })\longrightarrow F(c^{\prime }\otimes _{c}c^{\prime \prime }).\end{eqnarray}$$

2.2.7

Recall the $\infty$ -category $\operatorname{Mod}(\mathsf{DGCat})$ whose objects are pairs $(A,M)$ with $A$ a monoidal DG category and $M$ an $A$ -module. Morphisms $(A,M)\rightarrow (B,N)$ consist of pairs $(\unicode[STIX]{x1D719},f)$ where $\unicode[STIX]{x1D719}:A\rightarrow B$ is a monoidal functor and $f:M\rightarrow N$ an $A$ -linear functor.

There is a lax $(\infty ,2)$ -functor

(2.2) $$\begin{eqnarray}\mathsf{LOOP}_{\operatorname{Mod}}:\operatorname{Mod}(\mathsf{DGCat})^{\operatorname{op}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat}),\end{eqnarray}$$

described informally as follows:

1. – an object $(A,M)\in \operatorname{Mod}(\mathsf{DGCat})$ goes to the monoidal DG category $\mathsf{End}_{A}(M):=\mathsf{Fun}_{A}(M,M)$ ;

2. – a morphism $(A,M)\xrightarrow[{}]{(\unicode[STIX]{x1D719},f)}(B,N)$ gets sent to the $(\mathsf{End}_{B}(N),\mathsf{End}_{A}(M))$ -bimodule $\mathsf{Fun}_{A}(M,N)$ ;

3. – a composition $(A,M)\xrightarrow[{}]{(\unicode[STIX]{x1D719},f)}(B,N)\xrightarrow[{}]{(\unicode[STIX]{x1D713},g)}(C,P)$ goes over to the $(\mathsf{End}_{C}(P),\mathsf{End}_{A}(M))$ -bimodule

   $$\begin{eqnarray}\mathsf{Fun}_{B}(N,P)\underset{\mathsf{End}_{B}(N)}{\otimes }\mathsf{Fun}_{A}(M,N);\end{eqnarray}$$

4. – the lax structure comes from the tautological morphism (not invertible, in general)

   (2.3) $$\begin{eqnarray}\mathsf{Fun}_{B}(N,P)\underset{\mathsf{End}_{B}(N)}{\otimes }\mathsf{Fun}_{A}(M,N)\longrightarrow \mathsf{Fun}_{A}(M,P)\end{eqnarray}$$
   induced by composition.

2.2.8

For later use, we record here the following tautological observation. Let ${\mathcal{I}}$ be an $(\infty ,1)$ -category and $\mathbb{A}:{\mathcal{I}}\rightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat})$ be a lax $(\infty ,2)$ -functor. Assume given the following data:

1. – for each $i\in {\mathcal{I}}$ , a monoidal subcategory $\mathbb{A}^{\prime }(i){\hookrightarrow}\mathbb{A}(i)$ ;

2. – for each $i\rightarrow j$ , a full subcategory $\mathbb{A}_{i\rightarrow j}^{\prime }{\hookrightarrow}\mathbb{A}_{i\rightarrow j}$ preserved by the $(\mathbb{A}^{\prime }(i),\mathbb{A}^{\prime }(j))$ -action.

Assume also that, for each string $i\rightarrow j\rightarrow k$ , the functor

$$\begin{eqnarray}\mathbb{A}_{i\rightarrow j}^{\prime }\otimes \mathbb{A}_{j\rightarrow k}^{\prime }{\hookrightarrow}\mathbb{A}_{i\rightarrow j}\otimes \mathbb{A}_{j\rightarrow k}\xrightarrow[{}]{\;\mathsf{pr}\;}\mathbb{A}_{i\rightarrow j}\otimes _{\mathbb{A}(j)}\mathbb{A}_{j\rightarrow k}\xrightarrow[{}]{\;\unicode[STIX]{x1D702}_{i\rightarrow j\rightarrow k}\,}\mathbb{A}_{i\rightarrow k}\end{eqnarray}$$

lands in $\mathbb{A}_{i\rightarrow k}^{\prime }\subseteq \mathbb{A}_{i\rightarrow k}$ . Then the assignment

$$\begin{eqnarray}i{\rightsquigarrow}\mathbb{A}^{\prime }(i),\quad (i\rightarrow j){\rightsquigarrow}\mathbb{A}_{i\rightarrow j}^{\prime }\end{eqnarray}$$

naturally upgrades to a lax $(\infty ,2)$ -functor $\mathbb{A}^{\prime }:{\mathcal{I}}\rightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat})$ .

3.1.1

Let $\mathsf{Stk}$ denote the $\infty$ -category of perfect quasi-compact algebraic stacks of finite type and with affine diagonal; see, for example, [Reference Ben-Zvi, Francis and NadlerBFN10]. Inside $\mathsf{Stk}$ , we single out the subcategory $\mathsf{Stk}_{\text{lfp}}^{{<}\infty }$ consisting of those stacks that are bounded and with perfect cotangent complex (both properties can be checked on an atlas).

3.1.2

For ${\mathcal{C}}$ an $\infty$ -category, denote by $\mathsf{Arr}({\mathcal{C}}):={\mathcal{C}}^{\unicode[STIX]{x1D6E5}^{1}}$ the $\infty$ -category whose objects are arrows in ${\mathcal{C}}$ and whose $1$ -morphisms are commutative squares. We will be interested in the $\infty$ -category $\mathsf{Arr}(\mathsf{Stk}_{\text{lfp}}^{{<}\infty })$ and in the functor

(3.1) $$\begin{eqnarray}\operatorname{IndCoh}_{0}:\mathsf{Arr}(\mathsf{Stk}_{\text{lfp}}^{{<}\infty })^{\operatorname{op}}\longrightarrow \mathsf{DGCat}\end{eqnarray}$$

defined by

$$\begin{eqnarray}[{\mathcal{Y}}\rightarrow {\mathcal{Z}}]{\rightsquigarrow}\operatorname{IndCoh}_{0}({\mathcal{Z}}_{{\mathcal{Y}}}^{\wedge }).\end{eqnarray}$$

Recall from [Reference Arinkin and GaitsgoryAG18] or [Reference BeraldoBer17b] that $\operatorname{IndCoh}_{0}({\mathcal{Z}}_{{\mathcal{Y}}}^{\wedge })$ is defined by the pullback square

(3.2)

In particular, when writing $\operatorname{IndCoh}_{0}({\mathcal{Z}}_{{\mathcal{Y}}}^{\wedge })$ we are committing a potentially dangerous abuse of notation: it would be better to write $\operatorname{IndCoh}_{0}({\mathcal{Y}}\rightarrow {\mathcal{Z}}_{{\mathcal{Y}}}^{\wedge })$ , as the latter category depends on the formal moduli problem ${\mathcal{Y}}\rightarrow {\mathcal{Z}}_{{\mathcal{Y}}}^{\wedge }$ and in particular on the derived structure of ${\mathcal{Y}}$ .

3.1.3

For two objects $[{\mathcal{Y}}_{1}\rightarrow {\mathcal{Z}}_{1}]$ and $[{\mathcal{Y}}_{2}\rightarrow {\mathcal{Z}}_{2}]$ in $\mathsf{Arr}(\mathsf{Stk}_{\text{lfp}}^{{<}\infty })$ , a morphism $\unicode[STIX]{x1D709}$ from the former to the latter is given by a commutative square

(3.3)

The structure pullback functor

$$\begin{eqnarray}\unicode[STIX]{x1D709}^{!,0}:\operatorname{IndCoh}_{0}(({\mathcal{Z}}_{2})_{{\mathcal{Y}}_{2}}^{\wedge })\longrightarrow \operatorname{IndCoh}_{0}(({\mathcal{Z}}_{1})_{{\mathcal{Y}}_{1}}^{\wedge })\end{eqnarray}$$

is the obvious one induced by the pullback functor $\unicode[STIX]{x1D709}^{!}:\operatorname{IndCoh}(({\mathcal{Z}}_{2})_{{\mathcal{Y}}_{2}}^{\wedge })\rightarrow \operatorname{IndCoh}(({\mathcal{Z}}_{1})_{{\mathcal{Y}}_{1}}^{\wedge })$ , where we are abusing notation again by confusing $\unicode[STIX]{x1D709}$ with the map $({\mathcal{Z}}_{1})_{{\mathcal{Y}}_{1}}^{\wedge }\rightarrow ({\mathcal{Z}}_{2})_{{\mathcal{Y}}_{2}}^{\wedge }$ . We will do this throughout the paper, and hope it will not be too unpleasant for the reader.

3.1.4

Let us now recall the extension of (3.1) to a functor out of a category of correspondences. Notice that $\mathsf{Arr}(\mathsf{PreStk})$ admits fibre products, computed objectwise; its subcategory $\mathsf{Arr}(\mathsf{Stk}_{\text{lfp}}^{{<}\infty })$ is closed under products, but not under fibre products. Thus, to have a well-defined category of correspondences, we must choose appropriate classes of horizontal and vertical arrows.

We say that a commutative diagram (3.1.3), thought of as a morphism in $\mathsf{Arr}(\mathsf{Stk}_{\text{lfp}}^{{<}\infty })$ , is schematic (or bounded, or proper) if so is the top horizontal map. It is clear that

(3.4) $$\begin{eqnarray}\mathsf{Corr}\big(\mathsf{Arr}(\mathsf{Stk}_{\text{lfp}}^{{<}\infty })\big)_{\text{schem}\& \text{bdd};\operatorname{all}}\end{eqnarray}$$

is well defined.

For the theorem below, we will need to further upgrade (3.4) to an $(\infty ,2)$ -category by allowing as admissible arrows (see § 2.2.2 for the terminology) those $\unicode[STIX]{x1D709}$ that are schematic, bounded and proper. We denote by

$$\begin{eqnarray}\mathsf{Corr}\big(\mathsf{Arr}(\mathsf{Sch}_{\text{lfp}}^{{<}\infty })\big)_{\text{schem}\& \text{bdd};\operatorname{all}}^{\text{schem}\& \text{bdd}\& \text{proper}}\end{eqnarray}$$

the resulting $(\infty ,2)$ -category.

3.1.5

If $\unicode[STIX]{x1D709}$ is bounded and schematic in the above sense, then the pushforward $\unicode[STIX]{x1D709}_{\ast }^{\operatorname{IndCoh}}:\operatorname{IndCoh}(({\mathcal{Z}}_{1})_{{\mathcal{Y}}_{1}}^{\wedge })\rightarrow \operatorname{IndCoh}(({\mathcal{Z}}_{2})_{{\mathcal{Y}}_{2}}^{\wedge })$ is continuous and preserves the $\operatorname{IndCoh}_{0}$ -subcategories, thereby descending to a functor $\unicode[STIX]{x1D709}_{\ast ,0}$ . For the proof, see [Reference BeraldoBer17b].

Theorem 3.1.6. The above pushforward functors upgrade the functor $\operatorname{IndCoh}_{0}^{!}$ of (3.1) to an $(\infty ,2)$ -functor

$$\begin{eqnarray}\operatorname{IndCoh}_{0}:\mathsf{Corr}\big(\mathsf{Arr}(\mathsf{Sch}_{\text{lfp}}^{{<}\infty })\big)_{\text{schem}\& \text{bdd};\operatorname{all}}^{\text{schem}\& \text{bdd}\& \text{proper}}\longrightarrow \mathsf{DGCat},\end{eqnarray}$$

where $\mathsf{DGCat}$ is viewed as an $(\infty ,2)$ -category in the obvious way.

Remark 3.1.7. The existence of the above $(\infty ,2)$ -functor is deduced (essentially formally) by the $(\infty ,2)$ -functor

(3.5)

constructed in [Reference Gaitsgory and RozenblyumGR17, ch. III.3]. For later use, we will also need another fact from the same book: the above $(\infty ,2)$ -category of correspondences possesses a symmetric monoidal structure, and (3.5) is naturally symmetric monoidal; see [Reference Gaitsgory and RozenblyumGR17, ch. V.3]. It follows that the $(\infty ,2)$ -functor on Theorem 3.1.6 is symmetric monoidal, too.

3.1.8 Example

For $f:{\mathcal{Y}}\rightarrow {\mathcal{Z}}$ , the admissible arrow ${\mathcal{Y}}\rightarrow {\mathcal{Z}}_{{\mathcal{Y}}}^{\wedge }$ yields an adjuction

(3.6)

Let us also recall that $\operatorname{IndCoh}_{0}({\mathcal{Z}}_{{\mathcal{Y}}}^{\wedge })$ is self-dual and that these two adjoints $(\text{}^{\prime }f)_{\ast ,0}$ and $(\text{}^{\prime }f)^{!,0}$ are dual to each other.

3.2.2

We begin by observing that, for any ${\mathcal{X}}\in \mathsf{Stk}$ , the DG category

$$\begin{eqnarray}\mathbb{I}^{\wedge ,\text{geom}}({\mathcal{X}}):=\operatorname{IndCoh}({\mathcal{X}}\times _{{\mathcal{X}}_{\operatorname{ dR}}}{\mathcal{X}})\end{eqnarray}$$

possesses a convolution monoidal structure and that, for any correspondence $[{\mathcal{Y}}\leftarrow {\mathcal{W}}\rightarrow {\mathcal{Z}}]$ in $\mathsf{Stk}$ , the DG category

$$\begin{eqnarray}\mathbb{I}_{{\mathcal{Y}}\leftarrow {\mathcal{W}}\rightarrow {\mathcal{Z}}}^{\wedge ,\text{geom}}:=\operatorname{IndCoh}(({\mathcal{X}}\times {\mathcal{Y}})_{{\mathcal{W}}}^{\wedge })\simeq \operatorname{IndCoh}({\mathcal{Y}}\times _{{\mathcal{Y}}_{\operatorname{ dR}}}{\mathcal{W}}_{\operatorname{dR}}\times _{{\mathcal{Z}}_{\operatorname{dR}}}{\mathcal{Z}})\end{eqnarray}$$

admits the structure of an $(\mathbb{I}^{\wedge ,\text{geom}}({\mathcal{Y}}),\mathbb{I}^{\wedge ,\text{geom}}({\mathcal{Z}}))$ -bimodule.

3.2.3

Let us now enhance the assignment

$$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{X}}{\rightsquigarrow}\operatorname{IndCoh}(({\mathcal{X}}\times {\mathcal{X}})_{{\mathcal{X}}}^{\wedge })=:\mathbb{I}^{\wedge ,\text{geom}}({\mathcal{X}}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \text{}[{\mathcal{X}}\leftarrow {\mathcal{W}}\rightarrow {\mathcal{Y}}]{\rightsquigarrow}\operatorname{IndCoh}(({\mathcal{X}}\times {\mathcal{Y}})_{{\mathcal{W}}}^{\wedge })=:\mathbb{I}_{{\mathcal{X}}\leftarrow {\mathcal{W}}\rightarrow {\mathcal{Y}}}^{\wedge ,\text{geom}} & \displaystyle \nonumber\end{eqnarray}$$

to a lax  $(\infty ,2)$ -functor

(3.7) $$\begin{eqnarray}\mathbb{I}^{\wedge ,\text{geom}}:\mathsf{Corr}(\mathsf{Stk})_{\operatorname{ all};\operatorname{all}}^{\text{schem}\& \text{proper}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat}).\end{eqnarray}$$

To construct this, we first appeal to the lax symmetric monoidal structure on (3.5): § 2.2.6 yields a lax $(\infty ,2)$ -functor

All that remains is to precompose with the lax $(\infty ,2)$ -functor

(3.8)

that sends

Observe that the requirement that $f$ be schematic and proper implies that $f_{\operatorname{dR}}$ , and hence $\widetilde{f_{\operatorname{dR}}}$ , is inf-schematic and ind-proper.

3.2.5

Let us now turn to the construction of $\mathbb{H}^{\text{geom}}$ . For ${\mathcal{Y}}\in \mathsf{Stk}_{\text{lfp}}^{{<}\infty }$ , the canonical inclusion

$$\begin{eqnarray}\unicode[STIX]{x1D704}:\operatorname{IndCoh}_{0}({\mathcal{Y}}\times _{{\mathcal{Y}}_{\operatorname{dR}}}{\mathcal{Y}}){\hookrightarrow}\operatorname{IndCoh}({\mathcal{Y}}\times _{{\mathcal{Y}}_{\operatorname{dR}}}{\mathcal{Y}})\end{eqnarray}$$

is monoidal. Moreover, the left action of $\operatorname{IndCoh}_{0}({\mathcal{Y}}\times _{{\mathcal{Y}}_{\operatorname{dR}}}{\mathcal{Y}})$ on $\operatorname{IndCoh}(({\mathcal{Y}}\times {\mathcal{Z}})_{{\mathcal{W}}}^{\wedge })$ preserves the subcategory $\operatorname{IndCoh}_{0}(({\mathcal{Y}}\times {\mathcal{Z}})_{{\mathcal{W}}}^{\wedge })$ . This is an easy diagram chase left to the reader.

Thus, we are in a position to apply the paradigm of § 2.2.8 to obtain a lax $(\infty ,2)$ -functor

(3.9) $$\begin{eqnarray}\mathbb{H}^{\text{geom}}:\mathsf{Corr}\big(\mathsf{Stk}_{\text{lfp}}^{{<}\infty }\big)_{\text{bdd};\operatorname{all}}^{\text{schem}\& \text{bdd}\& \text{proper}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat}),\end{eqnarray}$$

as desired. We repeat here that one of the aims of this paper is to show that such lax $(\infty ,2)$ -functor is actually strict: this is accomplished in Theorem 6.5.3. In the next section, we give an overview of the strategy of the proof of such theorem. This could serve as a guide through the constructions of the remainder of the present paper.

3.3.1

First, we look at the restriction of $\mathbb{H}^{\text{geom}}$ along the functor

$$\begin{eqnarray}\mathsf{Aff}_{\text{lfp}}^{{<}\infty }\rightarrow \mathsf{Corr}\big(\mathsf{Stk}_{\text{lfp}}^{{<}\infty }\big)_{\text{bdd};\operatorname{all}}\end{eqnarray}$$

which is the natural inclusion on objects, and $[S\rightarrow T]{\rightsquigarrow}[S\xleftarrow[{}]{=}S\rightarrow T]$ on $1$ -morphisms.

Using results from the theory of ind-coherent sheaves, we show in Theorem 4.3.4 that such lax $(\infty ,2)$ -functor is strict. By definition, this is simply the functor $\mathbb{H}:\mathsf{Aff}_{\text{lfp}}^{{<}\infty }\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ discussed in § 1.6.3.

3.3.2

Next, we show that the restriction of $\mathbb{H}^{\text{geom}}$ to $\mathsf{Corr}(\mathsf{Aff}_{\text{lfp}}^{{<}\infty })_{\text{bdd};\operatorname{all}}$ is strict (Corollary 5.2.13). We do so in an indirect way, by establishing some important duality properties of $\mathbb{H}$ . Namely, we show that, for each map $U\rightarrow T$ in $\mathsf{Aff}_{\text{lfp}}^{{<}\infty }$ , the bimodule $\mathbb{H}_{U\rightarrow T}$ admits a right dual (which happens to be a left dual as well), denoted by $\mathbb{H}_{T\leftarrow U}$ . Having such right duals allows us to form the bimodules

$$\begin{eqnarray}\mathbb{H}_{S\leftarrow U\rightarrow T}:=\mathbb{H}_{S\leftarrow U}\underset{\mathbb{H}(U)}{\otimes }\mathbb{H}_{U\rightarrow T},\quad \mathbb{H}_{S\rightarrow V\leftarrow T}:=\mathbb{H}_{S\rightarrow V}\underset{\mathbb{H}(V)}{\otimes }\mathbb{H}_{V\leftarrow T}.\end{eqnarray}$$

We also show that $\mathbb{H}_{S\rightarrow V\leftarrow T}\simeq \mathbb{H}_{S\leftarrow S\times _{V}T\rightarrow T}$ naturally, provided that at least one arrow between $S\rightarrow V$ and $T\rightarrow V$ is bounded. This is enough to extend $\mathbb{H}$ to a strict functor

$$\begin{eqnarray}\mathbb{H}^{\mathsf{Corr}}:\mathsf{Corr}(\mathsf{Aff}_{\text{lfp}}^{{<}\infty })_{\text{bdd};\operatorname{all}}\longrightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat}).\end{eqnarray}$$

By inspection, such functor coincides with the restriction of $\mathbb{H}^{\text{geom}}$ to $\mathsf{Corr}(\mathsf{Aff}_{\text{lfp}}^{{<}\infty })_{\text{bdd};\operatorname{all}}$ , whence the latter is also strict.

3.3.4

To study $\mathbb{H}^{\text{geom}}$ on stacks, we introduce the sheaf theory $\mathsf{ShvCat}^{\mathbb{H}}$ , which is the right Kan extension of the functor

$$\begin{eqnarray}(\mathsf{Aff}_{\text{lfp}}^{{<}\infty })^{\operatorname{op}}\rightarrow \mathsf{Cat}_{\infty },\quad S{\rightsquigarrow}\mathbb{H}(S)\operatorname{ -}\mathbf{mod}.\end{eqnarray}$$

Note that Theorem 4.3.4 is essential to make this well defined.

In principle, $\mathsf{ShvCat}^{\mathbb{H}}$ comes equipped only with pullback functors. However, thanks to the existence of the right duals $\mathbb{H}_{T\leftarrow S}$ , there are also $\ast$ -pushforward functors (right adjoints to pullbacks), which turn out to satisfy base-change against pullbacks. Symmetrically, the existence of the left duals provides $!$ -pushforward functors (left adjoints to pullbacks), also satisfying base-change against pullbacks.Footnote ⁵

3.3.5

In Theorem 6.5.1, we prove the $\mathbb{H}$ -affineness theorem, which states that, for any ${\mathcal{Y}}\in \mathsf{Stk}_{\text{lfp}}^{{<}\infty }$ , the $\infty$ -category $\mathsf{ShvCat}^{\mathbb{H}}({\mathcal{Y}})$ is equivalent to $\mathbb{H}^{\text{geom}}({\mathcal{Y}})\operatorname{ -}\mathbf{mod}$ . This theorem, together with the above base-change properties, automatically upgrades the assignment ${\mathcal{Y}}{\rightsquigarrow}\mathbb{H}^{\text{geom}}({\mathcal{Y}})$ to a strict $(\infty ,2)$ -functor out of $\mathsf{Corr}(\mathsf{Stk}_{\text{lfp}}^{{<}\infty })_{\text{bdd};\operatorname{all}}$ . Fortunately, such functor is easily seen to match with $\mathbb{H}^{\text{geom}}$ , thereby proving that the latter is strict, too.

3.3.6

An important technical result, which we use frequently, is the smooth descent property for $\mathsf{ShvCat}^{\mathbb{H}}$ , proven in § 6.1: any object ${\mathcal{C}}\in \mathsf{ShvCat}^{\mathbb{H}}({\mathcal{Y}})$ is determined by its restrictions along smooth maps $S\rightarrow {\mathcal{Y}}$ , with $S$ affine. This is a very convenient simplification. For instance, let $\operatorname{IndCoh}_{/{\mathcal{Y}}}\in \mathsf{ShvCat}^{\mathbb{H}}({\mathcal{Y}})$ be the sheaf corresponding to $\operatorname{IndCoh}({\mathcal{Y}})\in \mathbb{H}^{\text{geom}}({\mathcal{Y}})\operatorname{ -}\mathbf{mod}$ via $\mathbb{H}$ -affineness. In § 6.6 we will show that the restriction of $\operatorname{IndCoh}_{/{\mathcal{Y}}}$ along a smooth map $S\rightarrow {\mathcal{Y}}$ is the $\mathbb{H}(S)$ -module $\operatorname{IndCoh}(S)$ , whereas the restriction along a non-smooth map does not admit such a simple description.

4.1.1

Thus, a lax coefficient system $\mathbb{A}$ consists of:

1. – a monoidal category $\mathbb{A}(S)$ , for each affine scheme $S$ ;

2. – an $(\mathbb{A}(S),\mathbb{A}(T))$ -bimodule $\mathbb{A}_{S\rightarrow T}$ for any map of affine schemes $S\rightarrow T$ ;

3. – an $(\mathbb{A}(S),\mathbb{A}(U))$ -linear functor

   $$\begin{eqnarray}\unicode[STIX]{x1D702}_{S\rightarrow T\rightarrow U}:\mathbb{A}_{S\rightarrow T}\underset{\mathbb{A}(T)}{\otimes }\mathbb{A}_{T\rightarrow U}\longrightarrow \mathbb{A}_{S\rightarrow U}\end{eqnarray}$$
   for any string $S\rightarrow T\rightarrow U$ of affine schemes;

4. – natural compatibilities for higher compositions.

Clearly, such $\mathbb{A}$ is a strict (i.e. non-lax) coefficient system if and only if all functors $\unicode[STIX]{x1D702}_{S\rightarrow T\rightarrow U}$ are equivalences.

4.1.2

One obtains variants of the above definitions by replacing the source $\infty$ -category $\mathsf{Aff}$ with a subcategory $\mathsf{Aff}_{\text{type}}$ , where ‘ $\text{type}$ ’ is a property of affine schemes. For instance, we will often consider $\mathsf{Aff}_{\text{aft}}$ (the full subcategory of affine schemes almost of finite type) or $\mathsf{Aff}_{\text{lfp}}^{{<}\infty }$ (affine schemes that are bounded and locally of finite presentation).

We now give a list of examples of (lax) coefficient systems, in decreasing order of simplicity.

4.1.3 Example 1

Any monoidal DG category ${\mathcal{A}}$ yields a ‘constant’ coefficient system $\text{}\underline{{\mathcal{A}}}$ whose value on $S\rightarrow T$ is ${\mathcal{A}}$ , considered as a bimodule over itself.

4.1.4 Example 2

Slightly less trivial: coefficient systems induced by a functor $\mathsf{Aff}\rightarrow \mathsf{Alg}(\mathsf{DGCat})^{\operatorname{op}}$ via the functor $\unicode[STIX]{x1D704}_{\mathsf{Alg}\rightarrow \mathsf{Bimod}}$ defined in (2.1). These coefficient systems are automatically strict.

For instance, we have the coefficient system $\mathbb{Q}$ which sends

$$\begin{eqnarray}S{\rightsquigarrow}\operatorname{QCoh}(S),\hspace{22.76228pt}[S\rightarrow T]{\rightsquigarrow}\operatorname{QCoh}(S)\in (\operatorname{QCoh}(S),\operatorname{QCoh}(T))\!\operatorname{ -}\!\mathbf{bimod}.\end{eqnarray}$$

Similarly, we have $\mathbb{D}$ , obtained as above using $\mathfrak{D}$ -modules rather than quasi-coherent sheaves. This coefficient system is defined only out of $\mathsf{Aff}_{\text{aft}}\subset \mathsf{Aff}$ .

4.1.5 Example 3

Let us precompose the lax $(\infty ,2)$ -functor

$$\begin{eqnarray}\mathsf{LOOP}_{\operatorname{Mod}}:\operatorname{Mod}(\mathsf{DGCat})^{\operatorname{op}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat})\end{eqnarray}$$

of § 2.2.7 with the functor

$$\begin{eqnarray}\mathsf{Aff}_{\text{aft}}\longrightarrow \operatorname{Mod}(\mathsf{DGCat})^{\operatorname{op}},\quad S{\rightsquigarrow}(\mathfrak{D}(S)\circlearrowright \operatorname{IndCoh}(S))\end{eqnarray}$$

that encodes the action of $\mathfrak{D}$ -modules on ind-coherent sheaves. Since $\operatorname{IndCoh}(S)$ is self-dual as a $\mathfrak{D}(S)$ -module (Corollary 4.2.2), we obtain a lax coefficient system

$$\begin{eqnarray}\mathbb{I}^{\wedge }:\mathsf{Aff}_{\text{aft}}\longrightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})\end{eqnarray}$$

described informally by

$$\begin{eqnarray}\displaystyle & & \displaystyle S{\rightsquigarrow}\operatorname{IndCoh}(S\times _{S_{\operatorname{dR}}}S),\nonumber\\ \displaystyle & & \displaystyle [S\rightarrow T]{\rightsquigarrow}\operatorname{IndCoh}(S\times _{T_{\operatorname{dR}}}T)\in (\operatorname{IndCoh}(S\times _{S_{\operatorname{dR}}}S),\operatorname{IndCoh}(T\times _{T_{\operatorname{dR}}}T))\!\operatorname{ -}\!\mathbf{bimod},\nonumber\\ \displaystyle & & \displaystyle [S\rightarrow T\rightarrow U]{\rightsquigarrow}\operatorname{IndCoh}(S\times _{T_{\operatorname{dR}}}T)\underset{\operatorname{IndCoh}(T\times _{T_{\operatorname{dR}}}T)}{\otimes }\operatorname{IndCoh}(T\times _{U_{\operatorname{dR}}}U)\longrightarrow \operatorname{IndCoh}(S\times _{U_{\operatorname{dR}}}U).\nonumber\end{eqnarray}$$

In other words, $\mathbb{I}^{\wedge }$ is obtained by restricting the very general $\mathbb{I}^{\wedge ,\text{geom}}$ defined in § 3.2.3 to $\mathsf{Aff}_{\text{aft}}$ . We will prove that $\mathbb{I}^{\wedge }$ is strict in Proposition 4.2.5.

4.1.6 Example 4

As a variation on the above example, let $\mathbb{H}$ be the lax coefficient system

$$\begin{eqnarray}\mathbb{H}:\mathsf{Aff}_{\text{lfp}}^{{<}\infty }\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat})\end{eqnarray}$$

defined by

$$\begin{eqnarray}\displaystyle & & \displaystyle S{\rightsquigarrow}\mathbb{H}(S):=\operatorname{IndCoh}_{0}(S\times _{S_{\operatorname{dR}}}S),\nonumber\\ \displaystyle & & \displaystyle [S\rightarrow T]{\rightsquigarrow}\mathbb{H}_{S\rightarrow T}:=\operatorname{IndCoh}_{0}(S\times _{T_{\operatorname{dR}}}T)\in (\mathbb{H}(S),\mathbb{H}(T))\!\operatorname{ -}\!\mathbf{bimod},\nonumber\\ \displaystyle & & \displaystyle [S\rightarrow T\rightarrow U]{\rightsquigarrow}\mathbb{H}_{S\rightarrow T}\underset{\mathbb{H}(T)}{\otimes }\mathbb{H}_{T\rightarrow U}\longrightarrow \mathbb{H}_{S\rightarrow U}.\nonumber\end{eqnarray}$$

Similarly to $\mathbb{I}^{\wedge }$ , this is the restriction of (3.9) to affine schemes. We will show that $\mathbb{H}$ is strict too.

The importance of $\mathbb{H}$ comes from the monoidal equivalence

$$\begin{eqnarray}\mathbb{H}(S)\simeq \operatorname{HC}(S)^{\operatorname{op}}\operatorname{ -}\!\mathsf{mod}.\end{eqnarray}$$

To be precise, we have the following. First, the equivalence $\mathbb{H}(S)\simeq \operatorname{HC}(\operatorname{IndCoh}(S))^{\operatorname{op}}\operatorname{ -}\!\mathsf{mod}$ is obvious. Second, [Reference Arinkin and GaitsgoryAG15, Proposition F.1.5] provides a natural isomorphism $\operatorname{HC}(\operatorname{IndCoh}(S))\simeq \operatorname{HC}(\operatorname{QCoh}(S))=:\operatorname{HC}(S)$ of $E_{2}$ -algebras.

4.1.7 Example 5

One last example arising in a geometric fashion. Let ${\mathcal{Y}}:\mathsf{Aff}\rightarrow \mathsf{Corr}(\mathsf{PreStk})_{\operatorname{all};\operatorname{all}}^{\operatorname{all}}$ be an arbitrary lax $(\infty ,2)$ -functor, described informally by the assignments

$$\begin{eqnarray}S{\rightsquigarrow}{\mathcal{Y}}_{S},\quad [S\rightarrow T]{\rightsquigarrow}{\mathcal{Y}}_{S}\leftarrow {\mathcal{Y}}_{S\rightarrow T}\rightarrow {\mathcal{Y}}_{T}.\end{eqnarray}$$

The lax structure amounts to the data of maps

(4.1) $$\begin{eqnarray}{\mathcal{Y}}_{S\rightarrow T}\underset{{\mathcal{Y}}_{T}}{\times }{\mathcal{Y}}_{T\rightarrow U}\longrightarrow {\mathcal{Y}}_{S\rightarrow U}\end{eqnarray}$$

over ${\mathcal{Y}}_{S}\times {\mathcal{Y}}_{U}$ , for any string $S\rightarrow T\rightarrow U$ . Recalling now the paradigm of § 2.2.5, we obtain a lax $(\infty ,2)$ -functor

$$\begin{eqnarray}\mathsf{Corr}(\mathsf{PreStk})_{\operatorname{all};\operatorname{all}}^{\operatorname{all}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat})\end{eqnarray}$$

defined by sending

$$\begin{eqnarray}{\mathcal{Y}}_{S}{\rightsquigarrow}\operatorname{QCoh}({\mathcal{Y}}_{S}),\quad [{\mathcal{Y}}_{S}\leftarrow {\mathcal{Y}}_{S\rightarrow T}\rightarrow {\mathcal{Y}}_{T}]{\rightsquigarrow}\operatorname{QCoh}({\mathcal{Y}}_{S\rightarrow T}).\end{eqnarray}$$

The combination of this with ${\mathcal{Y}}$ yields a lax coefficient system, which is strict if the maps (4.1) are isomorphisms and the prestacks ${\mathcal{Y}}_{S\rightarrow T}$ are nice enough.Footnote ⁶

4.1.8 Sub-example: singular support

The theory of singular support provides an important example of the above construction: the assignment

$$\begin{eqnarray}[S\rightarrow T]{\rightsquigarrow}\operatorname{Sing}(S)/\mathbb{G}_{m}\leftarrow S\times _{T}\operatorname{Sing}(T)/\mathbb{G}_{m}\rightarrow \operatorname{Sing}(T)/\mathbb{G}_{m},\end{eqnarray}$$

where $\operatorname{Sing}(U):=\operatorname{Spec}(\operatorname{Sym}_{H^{0}(U,{\mathcal{O}}_{U})}H^{1}(U,\mathbb{T}_{U}))$ is equipped with the obvious weight- $2$ dilation action.

We obtain a coefficient system $\mathbb{S}^{\prime }:\mathsf{Aff}_{\text{q-smooth}}\longrightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ defined on quasi-smooth affine schemes. By construction, if ${\mathcal{C}}$ is a module category over $\mathbb{S}^{\prime }(U)$ , then objects of ${\mathcal{C}}$ are equipped with a notion of support in $\operatorname{Sing}(U)$ ; see [Reference Arinkin and GaitsgoryAG15] for more details.

4.2.4

The lax-coefficient system $\mathbb{I}^{\wedge }$ is the restriction of the lax $(\infty ,2)$ -functor $\operatorname{IndCoh}^{\wedge ,\text{geom}}$ to $\mathsf{Aff}_{\text{aft}}$ . Consider now the intermediate lax $(\infty ,2)$ -functor $\mathsf{Sch}_{\text{aft}}\rightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat})$ , also denoted by $\mathbb{I}^{\wedge }$ by an abuse of notation. Our present aim is to prove the following result.

Proposition 4.2.5. The lax $(\infty ,2)$ -functor

$$\begin{eqnarray}\mathbb{I}^{\wedge }:\mathsf{Sch}_{\text{aft}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat})\end{eqnarray}$$

is strict.

The proof of the above proposition will be given after some preparation.

4.2.6

For $Y\in \mathsf{Sch}_{\text{aft}}$ , Corollary 4.2.3 shows that $\operatorname{IndCoh}(Y)$ admits the structure of an $(\operatorname{IndCoh}(Y\times _{Y_{\operatorname{dR}}}Y),\mathfrak{D}(Y))$ -bimodule, as well as the structure of a $(\mathfrak{D}(Y),\operatorname{IndCoh}(Y\times _{Y_{\operatorname{dR}}}Y))$ -bimodule. Now, one verifies directly that the latter bimodule is left dual to the former, that is, there is an adjunction

(4.3)

Lemma 4.2.7. These two adjoint functors form a pair of mutually inverse equivalences. In particular, we also have an adjunction in the other direction:

(4.4)

Lemma 4.2.8. For $Y\in \mathsf{Sch}_{\text{aft}}$ , the functor

$$\begin{eqnarray}\operatorname{IndCoh}(Y)\underset{\mathfrak{D}(Y)}{\otimes }-:\mathfrak{D}(Y)\operatorname{ -}\mathbf{mod}\longrightarrow \mathsf{DGCat}\end{eqnarray}$$

is conservative.

Proof. Let $f:{\mathcal{M}}\rightarrow {\mathcal{N}}$ be a $\mathfrak{D}(Y)$ -linear functor with the property that

$$\begin{eqnarray}\text{id}\otimes f:\operatorname{IndCoh}(Y)\underset{\mathfrak{D}(Y)}{\otimes }{\mathcal{M}}\longrightarrow \operatorname{IndCoh}(Y)\underset{\mathfrak{D}(Y)}{\otimes }{\mathcal{N}}\end{eqnarray}$$

is an equivalence. We need to show that $f$ itself is an equivalence.

Denote by $\widehat{Y}_{\bullet }$ the Čech nerve of $q:Y\rightarrow Y_{\operatorname{dR}}$ . Recall that the natural arrow

$$\begin{eqnarray}\mathfrak{D}(Y):=\operatorname{IndCoh}(Y_{\operatorname{dR}})\longrightarrow \operatorname{IndCoh}(|\widehat{Y}_{\bullet }|)\simeq \lim _{[n]\in \unicode[STIX]{x1D71F}}\operatorname{IndCoh}(\widehat{Y}_{n})\end{eqnarray}$$

is an equivalence and that each of the structure functors composing the above cosimplicial category admits a left adjoint (indeed, each structure map $\widehat{Y}_{m}\rightarrow \widehat{Y}_{n}$ is a nil-isomorphism between inf-schemes). Consequently, the tautological functor

$$\begin{eqnarray}{\mathcal{C}}\longrightarrow \lim _{[n]\in \unicode[STIX]{x1D71F}}\Big(\operatorname{IndCoh}(\widehat{Y}_{n})\underset{\mathfrak{D}(Y)}{\otimes }{\mathcal{C}}\Big)\end{eqnarray}$$

is an equivalence for any ${\mathcal{C}}\in \mathfrak{D}(S)\operatorname{ -}\mathbf{mod}$ . Under these identifications, our functor $f:{\mathcal{M}}\rightarrow {\mathcal{N}}$ is the limit of the equivalences

$$\begin{eqnarray}\text{id}\otimes f:\operatorname{IndCoh}(\widehat{Y}_{n})\underset{\mathfrak{D}(Y)}{\otimes }{\mathcal{M}}\longrightarrow \operatorname{IndCoh}(\widehat{Y}_{n})\underset{\mathfrak{D}(Y)}{\otimes }{\mathcal{N}},\end{eqnarray}$$

whence it is itself an equivalence. ◻

4.2.9

We are now ready for the proof of the proposition left open above.

Proof of Proposition 4.2.5.

Thanks to (4.2), it suffices to prove that, for any $Y\in \mathsf{Sch}_{\text{aft}}$ , the obvious functor $q_{\ast }^{\operatorname{IndCoh}}\circ \unicode[STIX]{x1D6E5}^{!}:\operatorname{IndCoh}(Y)\otimes \operatorname{IndCoh}(Y)\rightarrow \mathfrak{D}(Y)$ induces an equivalence

(4.5) $$\begin{eqnarray}\operatorname{IndCoh}(Y)\underset{\operatorname{IndCoh}(Y\times _{Y_{\operatorname{dR}}}Y)}{\otimes }\operatorname{IndCoh}(Y)\xrightarrow[{}]{\;\;\simeq \;\;}\mathfrak{D}(Y).\end{eqnarray}$$

The latter is precisely the counit of the adjunction (4.3), which we have shown to be an equivalence. ◻

4.3.1

We need a preliminary result, which is of interest in its own right.

Proposition 4.3.2. Let $f:X\rightarrow Y$ be a map in $\mathsf{Sch}_{\text{lfp}}^{{<}\infty }$ . Then the $(\mathfrak{D}(X),\mathbb{H}(Y))$ -linear functor

(4.6) $$\begin{eqnarray}\operatorname{IndCoh}(X)\underset{\mathbb{H}(X)}{\otimes }\mathbb{H}_{X\rightarrow Y}\longrightarrow \operatorname{IndCoh}(Y_{X}^{\wedge }),\end{eqnarray}$$

obtained as the composition

$$\begin{eqnarray}\displaystyle \operatorname{IndCoh}(X)\underset{\mathbb{H}(X)}{\otimes }\mathbb{H}_{X\rightarrow Y} & \longrightarrow & \displaystyle \operatorname{IndCoh}(X)\underset{\mathbb{I}^{\wedge }(X)}{\otimes }\mathbb{I}_{X\rightarrow Y}^{\wedge }\nonumber\\ \displaystyle & \xrightarrow[{}]{\simeq } & \displaystyle \operatorname{IndCoh}(Y_{X}^{\wedge }),\nonumber\end{eqnarray}$$

is an equivalence of categories.

Proof. The source category is compactly generated by objects of the form $[C_{X},(\text{}^{\prime }f)_{\ast }^{\operatorname{IndCoh}}(\unicode[STIX]{x1D714}_{X})]$ for $C_{X}\in \operatorname{Coh}(X)$ . Hence, it is clear that the functor in question (let us denote it by $\unicode[STIX]{x1D719}$ ) admits a continuous and conservative right adjoint: indeed, $\unicode[STIX]{x1D719}$ sends

$$\begin{eqnarray}[C_{X},(\text{}^{\prime }f)_{\ast }^{\operatorname{IndCoh}}(\unicode[STIX]{x1D714}_{X})]{\rightsquigarrow}(\text{}^{\prime }f)_{\ast }^{\operatorname{IndCoh}}(C_{X}),\end{eqnarray}$$

whence it preserves compactness and generates the target under colimits. It remains to show that $\unicode[STIX]{x1D719}$ is fully faithful on objects of the form $[C_{X},(\text{}^{\prime }f)_{\ast }^{\operatorname{IndCoh}}(\unicode[STIX]{x1D714}_{X})]$ . The nil-isomorphism $\unicode[STIX]{x1D6FD}:(X\times X)_{X}^{\wedge }\rightarrow (X\times Y)_{X}^{\wedge }$ induces the adjunction

Observe that both functors are $\operatorname{IndCoh}((X\times X)_{X}^{\wedge })$ -linear and preserve the $\operatorname{IndCoh}_{0}$ -subcategories. To conclude the proof, just note that $(\text{}^{\prime }f)_{\ast }^{\operatorname{IndCoh}}(\unicode[STIX]{x1D714}_{X})$ is the image of the unit of $\mathbb{H}(X)$ under $\unicode[STIX]{x1D6FD}_{\ast }^{\operatorname{IndCoh}}$ , and use the above adjunction.◻

Corollary 4.3.3. For $f:X\rightarrow Y$ as above and ${\mathcal{C}}$ a right $\mathbb{I}^{\wedge }(X)$ -module, the natural functor

$$\begin{eqnarray}{\mathcal{C}}\underset{\mathbb{H}(X)}{\otimes }\mathbb{H}_{X\rightarrow Y}\longrightarrow {\mathcal{C}}\underset{\mathbb{I}^{\wedge }(X)}{\otimes }\mathbb{I}_{X\rightarrow Y}^{\wedge }\end{eqnarray}$$

is an equivalence.

Proof. It suffices to prove the assertion for ${\mathcal{C}}=\mathbb{I}^{\wedge }(X)$ , viewed as a right module over itself. Thanks to the right $\mathbb{I}^{\wedge }(X)$ -linear equivalence

$$\begin{eqnarray}\mathbb{I}^{\wedge }(X)\simeq \operatorname{IndCoh}(X)\underset{\mathfrak{D}(X)}{\otimes }\operatorname{IndCoh}(X),\end{eqnarray}$$

the assertion reduces to the proposition above. ◻

Theorem 4.3.4. The lax $(\infty ,2)$ -functor

$$\begin{eqnarray}\mathbb{H}:\mathsf{Sch}_{\text{lfp}}^{{<}\infty }\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat}),\end{eqnarray}$$

obtained by restricting $\mathbb{H}^{\text{geom}}$ to schemes, is strict.

Proof. Let $U\rightarrow X\rightarrow Y$ be a string in $\mathsf{Sch}_{\text{lfp}}^{{<}\infty }$ . We need to prove that the convolution functor

(4.7) $$\begin{eqnarray}\mathbb{H}_{U\rightarrow X}\underset{\mathbb{H}(X)}{\otimes }\mathbb{H}_{X\rightarrow Y}\longrightarrow \mathbb{I}_{U\rightarrow Y}^{\wedge }\end{eqnarray}$$

is an equivalence onto the subcategory $\mathbb{H}_{U\rightarrow Y}\subseteq \mathbb{I}_{U\rightarrow Y}^{\wedge }$ . One easily checks that the essential image of the functor is indeed $\mathbb{H}_{U\rightarrow Y}$ , whence it remains to prove fully faithfulness. By construction, (4.7) factors as the composition

$$\begin{eqnarray}\mathbb{H}_{U\rightarrow X}\underset{\mathbb{H}(X)}{\otimes }\mathbb{H}_{X\rightarrow Y}\longrightarrow \mathbb{I}_{U\rightarrow X}^{\wedge }\underset{\mathbb{H}(X)}{\otimes }\mathbb{H}_{X\rightarrow Y}\longrightarrow \mathbb{I}_{U\rightarrow Y}^{\wedge }.\end{eqnarray}$$

Now the first arrow is obviously fully faithful, while the second one is an equivalence by the above corollary. ◻

4.4.1

Let $\mathbb{A}$ and $\mathbb{B}$ be two coefficient systems. Consider the following pieces of data:

1. – for each $S\in \mathsf{Aff}$ , a monoidal functor $\mathbb{A}(S)\rightarrow \mathbb{B}(S)$ ;

2. – for each $S\rightarrow T$ , an $(\mathbb{A}(S),\mathbb{A}(T))$ -linear functor

   (4.8) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{S\rightarrow T}:\mathbb{A}_{S\rightarrow T}\longrightarrow \mathbb{B}_{S\rightarrow T}\end{eqnarray}$$
   that induces an $(\mathbb{A}(S),\mathbb{B}(T))$ -equivalence $\mathbb{A}_{S\rightarrow T}\otimes _{\mathbb{A}(T)}\mathbb{B}(T)\rightarrow \mathbb{B}_{S\rightarrow T}$ ;

3. – natural higher compatibilities with respect to strings of affine schemes.

These data give rise to a morphism $\mathbb{A}\rightarrow \mathbb{B}$ .

4.4.2

It is easy to see that the morphism $\mathbb{Q}\rightarrow \mathbb{H}$ (defined on $\mathsf{Aff}_{\text{lfp}}^{{<}\infty }$ ) falls under this rubric. Indeed, we just need to verify that the tautological $(\operatorname{QCoh}(S),\mathbb{H}(T))$ -linear functor

(4.9) $$\begin{eqnarray}\operatorname{QCoh}(S)\underset{\operatorname{QCoh}(T)}{\otimes }\mathbb{H}(T)\longrightarrow \mathbb{H}_{S\rightarrow T}\end{eqnarray}$$

is an equivalence, for any $S\rightarrow T$ in $\mathsf{Aff}_{\text{lfp}}^{{<}\infty }$ . This has been proven in [Reference BeraldoBer17b] in greater generality.

5.1.1

We say that $\mathbb{A}$ satisfies the right Beck–Chevalley condition if the two requirements of §§ 5.1.2 and 5.1.5 are met.

5.1.2 The first requirement

We ask that, for any arrow $S\rightarrow T$ in $\mathsf{Aff}_{\text{type}}$ , the $(\mathbb{A}(S),\mathbb{A}(T))$ -bimodule $\mathbb{A}_{S\rightarrow T}$ be right dualizable; see § 2.1.2 for our conventions. Let us denote by $\mathbb{A}_{T\leftarrow S}$ such right dual.

5.1.3

Assume now that $\mathbb{A}$ satisfies the above requirement, so that the bimodules $\mathbb{A}_{?\leftarrow ?}$ are defined. Before formulating the second requirement, we need to fix some notation. For a commutative (but not necessarily cartesian) diagram

(5.3)

in $\mathsf{Aff}_{\text{type}}$ , define

$$\begin{eqnarray}\mathbb{A}_{S\leftarrow U\rightarrow T}:=\mathbb{A}_{S\leftarrow U}\underset{\mathbb{A}(U)}{\otimes }\mathbb{A}_{U\rightarrow T};\quad \mathbb{A}_{S\rightarrow V\leftarrow T}:=\mathbb{A}_{S\rightarrow V}\underset{\mathbb{A}(V)}{\otimes }\mathbb{A}_{V\leftarrow T}.\end{eqnarray}$$

Denote by $\mathit{u}\!\operatorname{ -}\!\text{type}$ the largest class of arrows in $\mathsf{Aff}_{\text{type}}$ that makes $\mathsf{Corr}(\mathsf{Aff}_{\text{type}})_{\operatorname{all};\mathit{u}\!\operatorname{-}\!\text{type}}$ well defined.Footnote ⁷ Namely, an arrow $S\rightarrow T$ in $\mathsf{Aff}_{\text{type}}$ belongs to $\mathit{u}\!\operatorname{ -}\!\text{type}$ if, for any $T^{\prime }\rightarrow T$ in $\mathsf{Aff}_{\text{type}}$ , the scheme $S\times _{T}T^{\prime }$ belongs to $\mathsf{Aff}_{\text{type}}$ .

5.1.4

Consider a commutative diagram like (5.3). The resulting commutative diagram

in $\mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ gives rise, by changing the vertical arrows with their right duals, to a lax commutative diagram

In other words, any commutative diagram (5.3) yields a canonical $(\mathbb{A}(S),\mathbb{A}(T))$ -linear functor

(5.4) $$\begin{eqnarray}\mathbb{A}_{S\rightarrow V\leftarrow T}\longrightarrow \mathbb{A}_{S\leftarrow U\rightarrow T}.\end{eqnarray}$$

5.1.5 The second requirement

In particular, for $S\rightarrow V\in \mathit{u}\!\operatorname{ -}\!\text{type}$ and $T\rightarrow V$ arbitrary, we have

(5.5) $$\begin{eqnarray}\mathbb{A}_{S\rightarrow V\leftarrow T}\longrightarrow \mathbb{A}_{S\leftarrow S\times _{V}T\rightarrow T}\end{eqnarray}$$

and we require that such functor be an equivalence.

5.1.6

Let us now explain what the right Beck–Chevalley condition is good for. Tautologically, if $\mathbb{A}$ satisfies the right Beck–Chevalley condition, the assignment

(5.6) $$\begin{eqnarray}S{\rightsquigarrow}\mathbb{A}(S),\quad [S\leftarrow U\rightarrow T]{\rightsquigarrow}\mathbb{A}_{S\leftarrow U\rightarrow T}\end{eqnarray}$$

extends to a functor

$$\begin{eqnarray}\mathsf{Corr}(\mathsf{Aff}_{\text{type}})_{\operatorname{all};\mathit{u}\!\operatorname{-}\!\text{type}}\longrightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat}).\end{eqnarray}$$

Further, thanks to [Reference Gaitsgory and RozenblyumGR17, ch. V.1, Theorem 3.2.2], the latter automatically extends further to an $(\infty ,2)$ -functor

$$\begin{eqnarray}\mathbb{A}^{\operatorname{R-BC}}:\mathsf{Corr}(\mathsf{Aff}_{\text{type}})_{\operatorname{ all};\mathit{u}\!\operatorname{-}\!\text{type}}^{\mathit{u}\!\operatorname{-}\!\text{type},2-\operatorname{op}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat}).\end{eqnarray}$$

Thus, for $\mathbb{A}$ satisfying the right Beck–Chevalley condition, the corresponding sheaf theory $\left.\mathsf{ShvCat}^{\mathbb{A}}\right|_{\mathsf{Aff}_{\text{type}}^{\operatorname{op}}}$ admits $\ast$ -pushforwards (defined to be right adjoint to pullbacks). Moreover, these pushforwards satisfy base-change against pullbacks along the appropriate fibre squares.

5.1.7

The definition of left Beck–Chevalley condition for $\mathbb{A}$ is totally symmetric: each $\mathbb{A}_{S\rightarrow T}$ must admit a left dual $\mathbb{A}_{T\leftarrow S}^{L}$ and, for any cartesian diagram (5.3) with $T\rightarrow V$ in $\mathit{u}\!\operatorname{ -}\!\text{type}$ , the structure functor

$$\begin{eqnarray}\mathbb{A}_{S\leftarrow U}^{L}\underset{\mathbb{A}(U)}{\otimes }\mathbb{A}_{U\rightarrow T}\longrightarrow \mathbb{A}_{S\rightarrow V}\underset{\mathbb{A}(V)}{\otimes }\mathbb{A}_{V\leftarrow T}^{L}\end{eqnarray}$$

must be an equivalence. Thus, if $\mathbb{A}$ satisfies the left Beck–Chevalley condition, the sheaf theory $\mathsf{ShvCat}^{\mathbb{A}}|_{\mathsf{Aff}_{\text{type}}^{\operatorname{op}}}$ admits $!$ -pushforwards (defined to be left adjoint to pullbacks), again, satisfying base-change against pullbacks along the appropriate fibre squares.

5.1.8

A coefficient system $\mathbb{A}$ is said to be ambidextrous if it satisfies the right Beck–Chevalley condition and, for any $S\rightarrow T\in \mathsf{Aff}_{\text{type}}$ , the $(\mathbb{A}(T),\mathbb{A}(S))$ -bimodule $\mathbb{A}_{S\rightarrow T}$ is ambidextrous (see § 2.1.2 for the definition). Any ambidextrous $\mathbb{A}$ automatically satisfies the left Beck–Chevalley condition as well. Thus, for $\mathbb{A}$ ambidextrous, we obtain two extensions of $\mathbb{A}$ ,

$$\begin{eqnarray}\displaystyle & \displaystyle \mathbb{A}^{\operatorname{R-BC}}:\mathsf{Corr}(\mathsf{Aff}_{\text{type}})_{\mathit{u}\!\operatorname{-}\!\text{type};\operatorname{all}}^{\mathit{u}\!\operatorname{-}\!\text{type},2-\operatorname{op}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \mathbb{A}^{\operatorname{L-BC}}:\mathsf{Corr}(\mathsf{Aff}_{\text{type}})_{\operatorname{all};\mathit{u}\!\operatorname{-}\!\text{type}}^{\mathit{u}\!\operatorname{-}\!\text{type}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat}), & \displaystyle \nonumber\end{eqnarray}$$

that are exchanged by duality.

5.1.9

Let us spell out these pieces of structure in more detail. First, up to switching vertical and horizontal arrows in $\mathbb{A}^{\operatorname{R-BC}}$ (see Remark 3.3.3), the two $(\infty ,2)$ -functors $\mathbb{A}^{\operatorname{R-BC}},\mathbb{A}^{\operatorname{L-BC}}$ have a common underlying $(\infty ,1)$ -functor

$$\begin{eqnarray}\displaystyle & \displaystyle \mathbb{A}^{\mathsf{Corr}}:\mathsf{Corr}(\mathsf{Aff}_{\text{type}})_{\operatorname{all};\mathit{u}\!\operatorname{-}\!\text{type}}\longrightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat}) & \displaystyle \nonumber\\ \displaystyle & \displaystyle [S\leftarrow U\rightarrow T]{\rightsquigarrow}\mathbb{A}_{S\leftarrow U\rightarrow T}:=\mathbb{A}_{S\leftarrow U}\underset{\mathbb{A}(U)}{\otimes }\mathbb{A}_{U\rightarrow T}. & \displaystyle \nonumber\end{eqnarray}$$

Secondly, the two enhancements of $\mathbb{A}^{\mathsf{Corr}}$ to $\mathbb{A}^{\operatorname{L-BC}}$ and $\mathbb{A}^{\operatorname{R-BC}}$ amount to the following data: for $U^{\prime }\rightarrow U$ of $\mathit{u}\!\operatorname{ -}\!\text{type}$ over $S\times T$ , there are two mutually dual structure functors $\mathbb{A}_{S\leftarrow U^{\prime }\rightarrow T}\rightleftarrows \mathbb{A}_{S\leftarrow U\rightarrow T}$ , compatible in $U$ in the natural way. Such enhancements will be used in §§ 6.3.1 and 6.3.2 to construct the two kinds of pushforwards in the setting of $\mathsf{ShvCat}^{\mathbb{H}}$ on stacks.

5.1.10 Easy examples

It is obvious that $\mathbb{Q}$ and $\mathbb{D}$ are ambidextrous. For instance, for the former,

$$\begin{eqnarray}\mathbb{Q}^{\mathsf{Corr}}:\mathsf{Corr}(\mathsf{Aff})_{\operatorname{all};\operatorname{all}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat})\end{eqnarray}$$

is defined on $1$ -arrows by $\mathbb{Q}_{S\leftarrow U\rightarrow T}\simeq \operatorname{QCoh}(U)$ , the latter equipped with its obvious $(\operatorname{QCoh}(S),\operatorname{QCoh}(T))$ -bimodule structure. The two mutually dual structure functors $\mathbb{Q}_{S\leftarrow U^{\prime }\rightarrow T}\rightleftarrows \mathbb{Q}_{S\leftarrow U\rightarrow T}$ are simply the pullback and pushforward functors along $U^{\prime }\rightarrow U$ .

We leave it as an exercise to show that the coefficient system $\mathbb{S}^{\prime }$ responsible for singular support is ambidextrous: it extends to a functor out of $\mathsf{Corr}(\mathsf{Aff}_{\text{q-smooth}})_{\operatorname{all};\text{smooth}}^{\text{smooth}}$ .

5.1.11

Let us now turn to $\mathbb{I}^{\wedge }$ . We have the following result, which will later help us understand base-change for $\mathbb{H}$ .

Proposition 5.1.12. The functor $\mathbb{I}^{\wedge }:\mathsf{Sch}_{\text{aft}}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ satisfies the right Beck–Chevalley condition, so that it extends to an $(\infty ,2)$ -functor

(5.7) $$\begin{eqnarray}(\mathbb{I}^{\wedge })^{\operatorname{R-BC}}:\mathsf{Corr}(\mathsf{Sch}_{\text{aft}})_{\operatorname{all};\operatorname{all}}^{\operatorname{all},2-\operatorname{op}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat}).\end{eqnarray}$$

Proof. We start by setting up some notation. For $X\rightarrow Y$ in $\mathsf{Sch}_{\text{aft}}$ , consider the maps

$$\begin{eqnarray}\unicode[STIX]{x1D701}:(X\times X)_{X}^{\wedge }\simeq X\times _{X_{\operatorname{dR}}}X\longrightarrow (X\times X)_{X\times _{Y}X}^{\wedge }\simeq X\times _{Y_{\operatorname{dR}}}X,\end{eqnarray}$$

(5.8) $$\begin{eqnarray}\unicode[STIX]{x1D702}:(Y\times Y)_{X}^{\wedge }\simeq Y\times _{Y_{\operatorname{dR}}}X_{\operatorname{dR}}\times _{Y_{\operatorname{dR}}}Y\longrightarrow (Y\times Y)_{Y}^{\wedge }\simeq Y\times _{Y_{\operatorname{dR}}}Y,\end{eqnarray}$$

where $\unicode[STIX]{x1D701}$ is induced by $\unicode[STIX]{x1D6E5}_{X/Y}:X\rightarrow X\times _{Y}X$ . With the help of Lemma 4.2.1, one can easily check that the functors

$$\begin{eqnarray}\unicode[STIX]{x1D701}^{!}:\mathbb{I}_{X\rightarrow Y}^{\wedge }\underset{\mathbb{I}^{\wedge }(Y)}{\otimes }\mathbb{I}_{Y\leftarrow X}^{\wedge }\longrightarrow \mathbb{I}^{\wedge }(X),\quad \unicode[STIX]{x1D702}^{!}:\mathbb{I}^{\wedge }(Y)\longrightarrow \mathbb{I}_{Y\leftarrow X}^{\wedge }\underset{\mathbb{I}^{\wedge }(X)}{\otimes }\mathbb{I}_{X\rightarrow Y}^{\wedge }\end{eqnarray}$$

exhibit $\mathbb{I}_{Y\leftarrow X}^{\wedge }:=\operatorname{IndCoh}(Y\times _{Y_{\operatorname{dR}}}X)$ as the right dual of the $(\mathbb{I}^{\wedge }(X),\mathbb{I}^{\wedge }(Y))$ -bimodule $\mathbb{I}_{X\rightarrow Y}^{\wedge }$ .

Now let

(5.9)

be a commutative square in $\mathsf{Sch}_{\text{aft}}$ . By Lemma 4.2.1, one easily gets equivalences

$$\begin{eqnarray}\mathbb{I}_{S\leftarrow U\rightarrow T}^{\wedge }\simeq \operatorname{IndCoh}(S\times _{S_{\operatorname{dR}}}U_{\operatorname{dR}}\times _{T_{\operatorname{dR}}}T),\quad \mathbb{I}_{S\rightarrow V\leftarrow T}^{\wedge }\simeq \operatorname{IndCoh}(S\times _{V_{\operatorname{dR}}}T),\end{eqnarray}$$

compatible with the natural $(\mathbb{I}^{\wedge }(S),\mathbb{I}^{\wedge }(T))$ -bimodule structures on both sides. Further, the structure arrow induced by the right Beck–Chevalley condition

$$\begin{eqnarray}\mathbb{I}_{S\rightarrow V\leftarrow T}^{\wedge }\longrightarrow \mathbb{I}_{S\leftarrow U\rightarrow T}^{\wedge }\end{eqnarray}$$

is the $!$ -pullback functor along the natural map $U_{\operatorname{dR}}\rightarrow (S\times _{V}T)_{\operatorname{dR}}$ , whence it is an equivalence whenever the square is nil-cartesian (i.e. cartesian at the level of reduced schemes).◻

5.2.1

Observe that, for any $S\in \mathsf{Aff}_{\text{lfp}}^{{<}\infty }$ , the monoidal category $\mathbb{H}(S)$ is rigid and compactly generated. Recall now the definition of $1_{\mathbb{H}(S)}^{\text{fake}}\in \mathbb{H}(S)^{\ast }$ and the notion of very rigid monoidal category; see § 2.1.5.

Proof. It suffices to show that $1_{\mathbb{H}(S)}^{\text{fake}}\in \mathbb{H}(S)^{\ast }$ admits a lift through the forgetful functor

$$\begin{eqnarray}\mathsf{Fun}_{\mathbb{H}(S)\otimes \mathbb{H}(S)^{\operatorname{rev}}}\big(\mathbb{H}(S),\mathbb{H}(S)^{\ast }\big)\longrightarrow \mathbb{H}(S)^{\ast }.\end{eqnarray}$$

Recall from [Reference BeraldoBer17b] that the functor

$$\begin{eqnarray}\mathfrak{D}(S)\xrightarrow[{}]{\mathsf{oblv}_{L}}\operatorname{QCoh}(S)\xrightarrow[{}]{\unicode[STIX]{x1D6F6}_{S}}\operatorname{IndCoh}(S)\xrightarrow[{}]{\text{}^{\prime }\unicode[STIX]{x1D6E5}_{\ast }^{\operatorname{IndCoh}}}\mathbb{H}(S)\end{eqnarray}$$

factors as the composition

$$\begin{eqnarray}\mathfrak{D}(S)\longrightarrow \mathsf{Fun}_{\mathbb{H}(S)\otimes \mathbb{H}(S)^{\operatorname{rev}}}\big(\mathbb{H}(S),\mathbb{H}(S)\big)\longrightarrow \mathbb{H}(S),\end{eqnarray}$$

where the DG category in the middle is by definition the Drinfeld centre of $\mathbb{H}(S)$ . A variation of the argument there shows that

$$\begin{eqnarray}\mathfrak{D}(S)\xrightarrow[{}]{\mathsf{oblv}_{L}}\operatorname{QCoh}(S)\xrightarrow[{}]{\unicode[STIX]{x1D6EF}_{S}}\operatorname{IndCoh}(S)\xrightarrow[{}]{\text{}^{\prime }\unicode[STIX]{x1D6E5}_{\ast }^{\operatorname{IndCoh}}}\mathbb{H}(S)^{\ast }\end{eqnarray}$$

factors as the composition

$$\begin{eqnarray}\mathfrak{D}(S)\longrightarrow \mathsf{Fun}_{\mathbb{H}(S)\otimes \mathbb{H}(S)^{\operatorname{rev}}}\big(\mathbb{H}(S),\mathbb{H}(S)^{\ast }\big)\longrightarrow \mathbb{H}(S)^{\ast }.\end{eqnarray}$$

Finally, one computes $1_{\mathbb{H}(S)}^{\text{fake}}\in \mathbb{H}(S)^{\ast }$ explicitly: it is readily checked that

$$\begin{eqnarray}1_{\mathbb{H}(S)}^{\text{fake}}\simeq \unicode[STIX]{x1D6E5}_{\ast }^{\operatorname{IndCoh}}(\unicode[STIX]{x1D6EF}_{S}({\mathcal{O}}_{S})),\end{eqnarray}$$

a fact that concludes the proof. ◻

5.2.3

Coupling this with Corollary 2.1.7, we obtain that each bimodule $\mathbb{H}_{S\rightarrow T}$ is ambidextrous: moreover, its left and right duals are canonically identified with $(\mathbb{H}_{S\rightarrow T})^{\ast }$ .

Let us now determine the right dual to $\mathbb{H}_{S\rightarrow T}$ explicitly. By the above, we already know what the DG category underlying $\mathbb{H}_{T\leftarrow S}:=(\mathbb{H}_{S\rightarrow T})^{R}$ must be: it is the dual of the DG category $\operatorname{IndCoh}_{0}((S\times T)_{S}^{\wedge })$ . The latter is self-dual as a plain DG category, so we are just searching for the correct $(\mathbb{H}(T),\mathbb{H}(S))$ -bimodule structure on $\operatorname{IndCoh}_{0}((S\times T)_{S}^{\wedge })$ .

We claim that $\mathbb{H}_{T\leftarrow S}$ is equivalent to $\operatorname{IndCoh}_{0}((T\times S)_{S}^{\wedge })$ , equipped with the obvious $(\mathbb{H}(T),\mathbb{H}(S))$ -bimodule structure. We will establish this fact directly, by constructing the evaluation and coevaluation that make $\operatorname{IndCoh}_{0}((T\times S)_{S}^{\wedge })$ right dual to $\mathbb{H}_{S\rightarrow T}$ .

Lemma 5.2.4. For $S\rightarrow T$ a map in $\mathsf{Aff}_{\text{lfp}}^{{<}\infty }$ , the natural functor

$$\begin{eqnarray}\mathbb{H}_{S\rightarrow T}\underset{\mathbb{H}(T)}{\otimes }\operatorname{IndCoh}_{0}((T\times S)_{S}^{\wedge })\longrightarrow \mathbb{I}_{S\rightarrow T}^{\wedge }\underset{\mathbb{I}^{\wedge }(T)}{\otimes }\mathbb{I}_{T\leftarrow S}^{\wedge }\simeq \operatorname{IndCoh}(S\times _{T_{\operatorname{dR}}}S)\xrightarrow[{}]{\unicode[STIX]{x1D701}^{!}}\mathbb{I}^{\wedge }(S)\end{eqnarray}$$

lands into the full subcategory $\mathbb{H}(S)\subseteq \mathbb{I}^{\wedge }(S)$ .

Proof. We will use the commutative diagram

with cartesian square. The DG category

$$\begin{eqnarray}\mathbb{H}_{S\rightarrow T}\underset{\mathbb{H}(T)}{\otimes }\operatorname{IndCoh}_{0}((T\times S)_{S}^{\wedge })\end{eqnarray}$$

is generated by a single canonical compact object, which is sent by our functor to $\unicode[STIX]{x1D701}^{!}\circ \unicode[STIX]{x1D709}_{\ast }^{\operatorname{IndCoh}}(\unicode[STIX]{x1D714}_{S\times _{T}S})\in \mathbb{I}^{\wedge }(S)$ . Hence, it suffices to show that the object

$$\begin{eqnarray}(\widetilde{\unicode[STIX]{x1D6E5}}_{S/T})^{!}\circ \unicode[STIX]{x1D709}_{\ast }^{\operatorname{IndCoh}}(\unicode[STIX]{x1D714}_{S\times _{T}S})\simeq \unicode[STIX]{x1D70B}_{\ast }^{\operatorname{IndCoh}}\circ \unicode[STIX]{x1D70B}^{!}(\unicode[STIX]{x1D714}_{S})\end{eqnarray}$$

belongs to the image of $\unicode[STIX]{x1D6F6}_{S}:\operatorname{QCoh}(S){\hookrightarrow}\operatorname{IndCoh}(S)$ . This is clear: $\unicode[STIX]{x1D70B}_{\ast }^{\operatorname{IndCoh}}\unicode[STIX]{x1D70B}^{!}$ is equivalent as a functor to the universal envelope of the Lie algebroid $\mathbb{T}_{S/S\times T}\rightarrow \mathbb{T}_{S}$ , and by assumption $\mathbb{T}_{S/S\times T}$ belongs to $\unicode[STIX]{x1D6F6}_{S}(\mathsf{Perf}(S))$ . We conclude as in [Reference Arinkin and GaitsgoryAG18, Proposition 3.2.3].◻

5.2.5

Hence, we have constructed an $(\mathbb{H}(S),\mathbb{H}(S))$ -linear functor

(5.10) $$\begin{eqnarray}\mathbb{H}_{S\rightarrow T}\underset{\mathbb{H}(T)}{\otimes }\operatorname{IndCoh}_{0}((T\times S)_{S}^{\wedge })\longrightarrow \mathbb{H}(S),\end{eqnarray}$$

which will be our evaluation. To construct the coevaluation, we need another lemma.

Lemma 5.2.6. For a diagram $S\leftarrow U\rightarrow T$ in $\mathsf{Aff}_{\text{lfp}}^{{<}\infty }$ , the functor

$$\begin{eqnarray}\displaystyle \operatorname{IndCoh}_{0}((S\times U)_{U}^{\wedge })\underset{\mathbb{H}(U)}{\otimes }\mathbb{H}_{U\rightarrow T} & \rightarrow & \displaystyle \mathbb{I}_{S\leftarrow U}^{\wedge }\underset{\mathbb{I}^{\wedge }(U)}{\otimes }\mathbb{I}_{U\rightarrow T}^{\wedge }\nonumber\\ \displaystyle & \xrightarrow[{}]{\simeq } & \displaystyle \operatorname{IndCoh}(S\times _{S_{\operatorname{dR}}}U_{\operatorname{dR}}\times _{T_{\operatorname{dR}}}T)\simeq \operatorname{IndCoh}((S\times T)_{U}^{\wedge })\nonumber\end{eqnarray}$$

is an equivalence onto the subcategory $\operatorname{IndCoh}_{0}((S\times T)_{U}^{\wedge })\subseteq \operatorname{IndCoh}((S\times T)_{U}^{\wedge })$ .

Proof. Denote by $\unicode[STIX]{x1D719}:U\rightarrow S\times T$ and by $^{\prime }\unicode[STIX]{x1D719}:U\rightarrow (S\times T)_{U}^{\wedge }$ the obvious maps. The source DG category is compactly generated by a single canonical object. Base-change along the pullback square

shows that such object is sent to $^{\prime }\unicode[STIX]{x1D719}_{\ast }^{\operatorname{IndCoh}}(\unicode[STIX]{x1D714}_{U})\in \operatorname{IndCoh}((S\times T)_{U}^{\wedge })$ , which is a compact generator of $\operatorname{IndCoh}_{0}((S\times T)_{U}^{\wedge })$ . All that remains is to show that the functor

$$\begin{eqnarray}\operatorname{IndCoh}_{0}((S\times U)_{U}^{\wedge })\underset{\mathbb{H}(U)}{\otimes }\mathbb{H}_{U\rightarrow T}\longrightarrow \mathbb{I}_{S\leftarrow U}^{\wedge }\underset{\mathbb{I}^{\wedge }(U)}{\otimes }\mathbb{I}_{U\rightarrow T}^{\wedge }\end{eqnarray}$$

is fully faithful. This is evident: the functor in question arises as the composition

$$\begin{eqnarray}\operatorname{IndCoh}_{0}((S\times U)_{U}^{\wedge })\underset{\mathbb{H}(U)}{\otimes }\mathbb{H}_{U\rightarrow T}{\hookrightarrow}\mathbb{I}_{S\leftarrow U}^{\wedge }\underset{\mathbb{H}(U)}{\otimes }\mathbb{H}_{U\rightarrow T}\xrightarrow[{}]{\simeq }\mathbb{I}_{S\leftarrow U}^{\wedge }\underset{\mathbb{I}^{\wedge }(U)}{\otimes }\mathbb{I}_{U\rightarrow T}^{\wedge },\end{eqnarray}$$

where the second arrow is an equivalence thanks to Corollary 4.3.3. ◻

5.2.7

We now use $\unicode[STIX]{x1D702}^{!}:\operatorname{IndCoh}((T\times T)_{T}^{\wedge })\longrightarrow \operatorname{IndCoh}((T\times T)_{S}^{\wedge })$ as in (5.8), together with the equivalence

$$\begin{eqnarray}\unicode[STIX]{x1D703}:\operatorname{IndCoh}_{0}((T\times S)_{S}^{\wedge })\underset{\mathbb{H}(S)}{\otimes }\mathbb{H}_{S\rightarrow T}\rightarrow \operatorname{IndCoh}_{0}((T\times T)_{S}^{\wedge })\end{eqnarray}$$

of the above lemma, to construct the functor

(5.11) $$\begin{eqnarray}\mathbb{H}(T)\xrightarrow[{}]{\unicode[STIX]{x1D702}^{!}}\operatorname{IndCoh}_{0}((T\times T)_{S}^{\wedge })\xrightarrow[{}]{\unicode[STIX]{x1D703}^{-1}}\operatorname{IndCoh}_{0}((T\times S)_{S}^{\wedge })\underset{\mathbb{H}(S)}{\otimes }\mathbb{H}_{S\rightarrow T}.\end{eqnarray}$$

As the next proposition shows, this is the coevaluation we were looking for.

5.2.9

Henceforth, we will freely use the $(\mathbb{H}(T),\mathbb{H}(S))$ -linear equivalence $\mathbb{H}_{T\leftarrow S}\simeq \operatorname{IndCoh}_{0}((T\times S)_{S}^{\wedge })$ . We are finally ready to settle the ambidexterity of the coefficient system $\mathbb{H}$ .

Half of the proof of this theorem has been done in Lemma 5.2.6. All that remains is to add the following statement.

Lemma 5.2.11. Let $S\rightarrow V\leftarrow T$ be a diagram in $\mathsf{Aff}_{\text{lfp}}^{{<}\infty }$ , with either $S\rightarrow V$ or $T\rightarrow V$ bounded.Footnote ⁸ Then the functor

$$\begin{eqnarray}\mathbb{H}_{S\rightarrow V}\underset{\mathbb{H}(V)}{\otimes }\mathbb{H}_{V\leftarrow T}\longrightarrow \mathbb{I}_{S\rightarrow V}^{\wedge }\underset{\mathbb{I}^{\wedge }(V)}{\otimes }\mathbb{I}_{V\leftarrow T}^{\wedge }\xrightarrow[{}]{\simeq }\operatorname{IndCoh}(S\times _{V_{\operatorname{dR}}}T)\end{eqnarray}$$

is an equivalence onto the subcategory

$$\begin{eqnarray}\operatorname{IndCoh}_{0}((S\times T)_{S\times _{V}T}^{\wedge })\subseteq \operatorname{IndCoh}(S\times _{V_{\operatorname{dR}}}T).\end{eqnarray}$$

Proof. Let $\unicode[STIX]{x1D709}:S\times _{V}T\rightarrow (S\times T)_{S\times _{V}T}^{\wedge }\simeq S\times _{V_{\operatorname{dR}}}T$ be the canonical map. As before, $\mathbb{H}_{S\rightarrow V}\underset{\mathbb{H}(V)}{\otimes }\mathbb{H}_{V\leftarrow T}$ is compactly generated by its canonical object. Now, the functor in question sends such object to $\unicode[STIX]{x1D709}_{\ast }^{\operatorname{IndCoh}}(\unicode[STIX]{x1D714}_{S\times _{V}T})$ , which is a compact generator of $\operatorname{IndCoh}_{0}((S\times T)_{S\times _{V}T}^{\wedge })$ . Hence, all that remains is to verify that the functor

$$\begin{eqnarray}\mathbb{H}_{S\rightarrow V}\underset{\mathbb{H}(V)}{\otimes }\mathbb{H}_{V\leftarrow T}\longrightarrow \mathbb{I}_{S\rightarrow V}^{\wedge }\underset{\mathbb{I}^{\wedge }(V)}{\otimes }\mathbb{I}_{V\leftarrow T}^{\wedge }\end{eqnarray}$$

is fully faithful. Assume that $S\rightarrow V$ is bounded; the argument for the other case is symmetric. We have the following sequence of left $\operatorname{QCoh}(S)$ -linear fully faithful functors:

$$\begin{eqnarray}\displaystyle \mathbb{H}_{S\rightarrow V}\underset{\mathbb{H}(V)}{\otimes }\mathbb{H}_{V\leftarrow T} & \simeq & \displaystyle \operatorname{QCoh}(S)\underset{\operatorname{QCoh}(V)}{\otimes }\mathbb{H}_{V\leftarrow T}\nonumber\\ \displaystyle & {\hookrightarrow} & \displaystyle \operatorname{QCoh}(S)\underset{\operatorname{QCoh}(V)}{\otimes }\mathbb{I}_{V\leftarrow T}^{\wedge }\nonumber\\ \displaystyle & \simeq & \displaystyle \operatorname{QCoh}(S)\underset{\operatorname{QCoh}(V)}{\otimes }\operatorname{IndCoh}(V)\underset{\mathfrak{D}(V)}{\otimes }\operatorname{IndCoh}(T).\nonumber\end{eqnarray}$$

To conclude, recall [Reference GaitsgoryGai13, Proposition 4.4.2] that the tautological functor $\operatorname{QCoh}(S)\otimes _{\operatorname{QCoh}(V)}\operatorname{IndCoh}(V)\rightarrow \operatorname{IndCoh}(S)$ is fully faithful whenever $S\rightarrow V$ is bounded.◻

5.2.12

Following the template of § 5.1.9, let us summarize the consequences of the ambidexterity of $\mathbb{H}$ . First, we obtain that $\mathbb{H}$ extends to a functor

$$\begin{eqnarray}\mathbb{H}^{\mathsf{Corr}}:\mathsf{Corr}(\mathsf{Aff}_{\text{lfp}}^{{<}\infty })_{\operatorname{all};\text{bdd}}\longrightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat}),\end{eqnarray}$$

which has been shown to send

$$\begin{eqnarray}[S\leftarrow U\rightarrow T]{\rightsquigarrow}\mathbb{H}_{S\leftarrow U\rightarrow T}:=\mathbb{H}_{S\leftarrow U}\underset{\mathbb{H}(U)}{\otimes }\mathbb{H}_{U\rightarrow T}\simeq \operatorname{IndCoh}_{0}((S\times T)_{U}^{\wedge }).\end{eqnarray}$$

In other words, $\mathbb{H}^{\mathsf{Corr}}$ coincides with the restriction of $\mathbb{H}^{\text{geom}}$ on $\mathsf{Corr}(\mathsf{Aff}_{\text{lfp}}^{{<}\infty })_{\operatorname{all};\text{bdd}}$ . Therefore, we have the following result.

5.2.14

Secondly, $\mathbb{H}^{\mathsf{Corr}}$ admits two extensions to $(\infty ,2)$ -functors,

$$\begin{eqnarray}\mathbb{H}^{\operatorname{R-BC}}:\mathsf{Corr}(\mathsf{Aff}_{\text{lfp}}^{{<}\infty })_{\operatorname{all};\text{bdd}}^{\text{bdd},2-\operatorname{op}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat})\end{eqnarray}$$

and

$$\begin{eqnarray}\mathbb{H}^{\operatorname{L-BC}}:\mathsf{Corr}(\mathsf{Aff}_{\text{lfp}}^{{<}\infty })_{\operatorname{all};\text{bdd}}^{\text{bdd}}\longrightarrow \mathsf{ALG}^{\text{bimod}}(\mathsf{DGCat}),\end{eqnarray}$$

described as follows. To a $2$ -morphism

$$\begin{eqnarray}[S\leftarrow U^{\prime }\rightarrow T]\rightarrow [S\leftarrow U\rightarrow T],\end{eqnarray}$$

induced by $U^{\prime }\rightarrow U$ bounded, $\mathbb{H}^{\operatorname{R-BC}}$ assigns the $!$ -pullback

$$\begin{eqnarray}\operatorname{IndCoh}_{0}((S\times T)_{U}^{\wedge })\longrightarrow \operatorname{IndCoh}_{0}((S\times T)_{U^{\prime }}^{\wedge }),\end{eqnarray}$$

while the $\mathbb{H}^{\operatorname{L-BC}}$ assigns the dual $(\ast ,0)$ -pushforward

$$\begin{eqnarray}\operatorname{IndCoh}_{0}((S\times T)_{U^{\prime }}^{\wedge })\longrightarrow \operatorname{IndCoh}_{0}((S\times T)_{U}^{\wedge }),\end{eqnarray}$$

which is well defined thanks to boundedness; see Theorem 3.1.6.

6.1.1

Objects of

$$\begin{eqnarray}\mathsf{ShvCat}^{\mathbb{H}}({\mathcal{Y}})\simeq \lim _{S\in (\mathsf{Aff}_{\text{lfp}}^{{<}\infty })_{/{\mathcal{Y}}}}\mathbb{H}(S)\operatorname{ -}\mathbf{mod}\end{eqnarray}$$

will be often represented simply by ${\mathcal{C}}\simeq \{{{\mathcal{C}}_{S}\}}_{S\in (\mathsf{Aff}_{\text{lfp}}^{{<}\infty })_{/{\mathcal{Y}}}}$ , leaving the coherent system of compatibilities $\mathbb{H}_{S\rightarrow T}\otimes _{\mathbb{H}(T)}{\mathcal{C}}_{T}\simeq {\mathcal{C}}_{S}$ implicit. For any $f:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ in $\mathsf{Stk}_{\text{lfp}}^{{<}\infty }$ , denote by $f^{\ast ,\mathbb{H}}$ the structure functor. Explicitly (and tautologically), $f^{\ast ,\mathbb{H}}$ sends