Section 4. (0NZM) · ScalingStacks

4. Tensor products and integral transforms

In this section, we study the relation between the geometry and algebra of perfect stacks. We begin with some basic properties of tensor products of $\infty$ -categories. Then we prove (Theorem 4.7) that the $\infty$ -category of sheaves on a product of perfect stacks, and more generally a derived fiber product, is given by the tensor product of the $\infty$ -categories of sheaves on the factors. This implies that the $\infty$ -category of sheaves on a perfect stack is self-dual, which in turn allows us to describe $\infty$ -categories of functors by $\infty$ -categories of integral kernels (Corollary 4.10). Finally, we extend this last result to a relative setting where the base is an arbitrary derived stack with affine diagonal.

4.1. Tensor products of $\infty$ -categories

In this section, we consider some properties of tensor products of $\infty$ -categories that will be used in what follows. First we prove (Proposition 4.1) that $\infty$ -categories of modules over associative algebra objects are dualizable, with duals given by modules for the opposite algebra, and that tensor product of algebras induces tensor products on module categories. We then discuss the tensor product of small stable categories, and its compatibility with passing to the corresponding presentable stable $\infty$ -categories of $\operatorname{Ind}$ -objects.

4.1.1. Algebras and modules

Recall the tensor product of presentable stable $\infty$ -categories developed in [L4], [L5]. Namely, the $\infty$ -category $\mathcal{P}r^{\rm L}$ of presentable $\infty$ -categories (with morphisms given by left adjoints) carries a natural symmetric monoidal tensor product that preserves stable objects.

Let $\mathcal{C}$ be a monoidal $\infty$ -category. For two presentable $\infty$ -categories $\mathcal{M}$ and $\mathcal{M}^{\prime}$ left tensored over $\mathcal{C}$ , we denote by $\Fun^{\rm L}_{\mathcal{C}}(\mathcal{M},\mathcal{M}^{\prime})$ the $\infty$ -category of left adjoints from $\mathcal{M}$ to $\mathcal{M}^{\prime}$ that preserve the tensor over $\mathcal{C}$ .

Similarly, the opposite category of $\mathcal{P}r^{\rm L}$ is the $\infty$ -category $\mathcal{P}r^{\rm R}$ of presentable $\infty$ -categories with morphisms given by right adjoints. For two presentable $\infty$ -categories $\mathcal{M}$ and $\mathcal{M}^{\prime}$ left cotensored over $\mathcal{C}$ , we denote by $\Fun^{\rm R}_{\mathcal{C}}(\mathcal{M},\mathcal{M}^{\prime})$ the $\infty$ -category of right adjoints from $\mathcal{M}$ to $\mathcal{M}^{\prime}$ that preserve the cotensor over $\mathcal{C}$ .

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Proposition 4.1. Let $\mathcal{C}$ be a stable presentable symmetric monoidal $\infty$ -category, and $A\in\mathcal{C}$ an associative algebra object.

(1)

For any $\mathcal{C}$ -module $\mathcal{M}$ , there is a canonical equivalence of $\infty$ -categories

$\Mod_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathcal{M}\simeq\Mod_{A}(\mathcal{M}).$
(2)

For $A^{\prime}\in\mathcal{C}$ a second associative algebra, there is a canonical equivalence of $\infty$ -categories

$\Mod_{A\otimes A^{\prime}}(\mathcal{C})\simeq\Mod_{A}(\mathcal{C})\otimes_{\mathcal{C}}\Mod_{A^{\prime}}(\mathcal{C}).$
(3)

The $\infty$ -category of modules $\Mod_{A}(\mathcal{C})$ is dualizable as a $\mathcal{C}$ -module with dual given by the $\infty$ -category of modules $\Mod_{A^{\rm op}}(\mathcal{C})$ over the opposite algebra.

The proof will depend on the following two lemmas:

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Lemma 4.2. Let $\mathcal{M}$ and $\mathcal{M}^{\prime}$ be stable presentable $\infty$ -categories that are left tensored and cotensored over $\mathcal{C}$ . Let $G:\mathcal{M}^{\prime}\rightarrow\mathcal{M}$ be a right adjoint that is tensored and cotensored over $\mathcal{C}$ . Assume further that $G$ is colimit preserving. Then $G$ is conservative if the induced functor $\Fun_{\mathcal{C}}^{\rm R}(\mathcal{D},\mathcal{M}^{\prime})\rightarrow\Fun_{\mathcal{C}}^{\rm R}(\mathcal{D},\mathcal{M})$ is conservative for any $\mathcal{D}$ .

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Proof. Suppose $G$ is not conservative. Then to prove the lemma, it suffices to exhibit a presentable $\infty$ -category $\mathcal{D}$ also cotensored over $\mathcal{C}$ and a nontrivial right adjoint $j:\mathcal{D}\rightarrow\mathcal{M}^{\prime}$ cotensored over $\mathcal{C}$ such that $j\circ G$ is trivial.

Define $\mathcal{D}$ to be the full $\infty$ -subcategory of $\mathcal{M}^{\prime}$ of $G$ -acyclic objects, that is, objects $m\in\mathcal{M}$ such that $G(m)$ is trivial. Our first task is to show that $\mathcal{D}$ is indeed presentable.

Observe that $\mathcal{D}$ is equivalent to the fiber product $\mathcal{D}\simeq 0\times_{\mathcal{M}}\mathcal{M}^{\prime}$ , where the limit is computed in the $\infty$ -category $\rm Cat_{\infty}$ of $\infty$ -categories. Recall by [L2, Proposition 5.5.3.13], the natural functor $\mathcal{P}r^{\rm L}\rightarrow{\rm Cat}_{\infty}$ preserves limits. Furthermore, the forgetful functor $\mathrm{Mod}_{\mathcal{C}}(\mathcal{P}r^{\rm L})\to\mathcal{P}r^{\rm L}$ also preserves limits since it has a left adjoint (given by induction).

Since the functor $G$ preserves colimits and is $\mathcal{C}$ -linear, we may regard it as a morphism in $\mathrm{Mod}_{\mathcal{C}}(\mathcal{P}r^{\rm L})$ . Thus $\mathcal{D}$ can be computed as a limit in $\mathrm{Mod}_{\mathcal{C}}(\mathcal{P}r^{\rm L})$ , and so can be regarded as an object of $\mathcal{P}r^{\rm L}$ . In other words, $\mathcal{D}$ is presentable and furthermore tensored over $\mathcal{C}$ . Finally, since $\mathcal{D}$ is tensored over $\mathcal{C}$ , it is automatically cotensored as well.

Now it remains to show that the inclusion $j:\mathcal{D}\rightarrow\mathcal{M}^{\prime}$ is indeed a right adjoint and cotensored over $\mathcal{C}$ . Since $j$ preserves all limits and colimits (and in particular $\kappa$ -filtered colimits), the adjoint functor theorem applies. Finally, since $G$ is cotensored over $\mathcal{C}$ , $j$ is as well. ∎

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Lemma 4.3. Let $\mathcal{M}$ be a stable presentable $\infty$ -category which is left tensored and cotensored over $\mathcal{C}$ , and let $A$ be an associative algebra in $\mathcal{C}$ . Then the forgetful functor $G:\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathcal{M}\to\mathcal{M}$ is conservative.

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Proof. Observe that for any $\mathcal{D}$ tensored over $\mathcal{C}$ , the pullback

\Fun^{\rm L}_{\mathcal{C}}(\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathcal{M},\mathcal{D})\rightarrow\Fun^{\rm L}_{\mathcal{C}}(\mathcal{M},\mathcal{D})

induced by the induction $F:\mathcal{M}\to\mathrm{Mod}_{A}\otimes_{\mathcal{C}}\mathcal{M}$ is conservative. In other words, if a functor out of $\mathrm{Mod}_{A}\times\mathcal{M}$ (which preserves colimits in each variable) is trivial when restricted to $\mathcal{M}$ , then it is necessarily trivial.

Consequently, switching to opposite categories, we have that the corresponding functor

\Fun^{\rm R}_{\mathcal{C}}(\mathcal{D},\mathrm{Mod}_{A}\otimes\mathcal{M})\rightarrow\Fun^{\rm R}_{\mathcal{C}}(\mathcal{D},\mathcal{M})

induced by the forgetful functor $G:\mathrm{Mod}_{A}\otimes_{\mathcal{C}}\mathcal{M}\to\mathcal{M}$ is conservative.

Now we can apply Lemma 4.2 with $\mathcal{M}^{\prime}=\mathrm{Mod}_{A}\otimes\mathcal{M}$ to obtain that $G$ is conservative. ∎

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Proof of Proposition 4.1. We first prove that $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathcal{M}$ is equivalent to $\mathrm{Mod}_{A}(\mathcal{M})$ by the natural evaluation functor. Consider the adjunction

where $F(-)=A\otimes-$ is the induction, and $G$ is the forgetful functor.

The above adjunction induces an adjunction

and thus a functor to modules over the monad $T=(G\otimes{\rm id})\circ(F\otimes{\rm id})$ acting on $\mathcal{M}$ . The functor underlying $T$ is given by tensoring with $A$ , so we also have an equivalence $\mathrm{Mod}_{T}(\mathcal{M})\simeq\mathrm{Mod}_{A}(\mathcal{M})$ .

By its universal characterization, the functor $G\otimes{\rm id}$ is colimit preserving. Note as well that $G$ and hence $G\otimes{\rm id}$ is also $\mathcal{C}$ -linear (or in other words, the adjunction satisfies an analogue of the projection formula). Thus it follows from Lemma 4.3 that $G\otimes{\rm id}$ is also conservative. Thus $G\otimes{\rm id}$ satisfies the monadic Barr-Beck conditions, and we obtain the desired equivalence $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathcal{M}\simeq\mathrm{Mod}_{A}(\mathcal{M})$ .

Next, we can apply this to the instance where $\mathcal{M}$ is the $\infty$ -category of left modules over another associative algebra $A^{\prime}$ to conclude that there is a natural equivalence $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A^{\prime}}(\mathcal{C})\simeq\mathrm{Mod}_{A}(\mathrm{Mod}_{A^{\prime}}(\mathcal{C}))$ . We now have a chain of adjunctions

in which the composite $G^{\prime}\circ G^{\prime\prime}$ is colimit preserving and conservative, and hence satisfies the monadic Barr-Beck conditions.

Furthermore, the above adjunction naturally extends to a diagram in which the cycle of left adjoints (denoted by bowed arrows), and hence also the cycle of right adjoints (denoted by straight arrows), commute

Here $F(-)=A_{1}\otimes A_{2}\otimes-$ is the induction, $G$ is the forgetful functor, $f$ is the natural functor factoring through $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A^{\prime}}(\mathcal{C})$ , and $g$ is its right adjoint. From this diagram, we obtain a morphism of monads

Now the underlying functors of the monads $GF(-)$ and $G^{\prime}G^{\prime\prime}F^{\prime\prime}F^{\prime}(-)$ are both equivalent to the tensor $A\otimes A^{\prime}\otimes(-)$ , so the above morphism of monads is an equivalence. Thus we obtain the promised equivalence $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A^{\prime}}(\mathcal{C})\simeq\mathrm{Mod}_{A}(\mathrm{Mod}_{A^{\prime}}(\mathcal{C}))\simeq\mathrm{Mod}_{A\otimes A^{\prime}}(\mathcal{C})$ .

Finally, we show that the $\infty$ -category of left $A$ -modules $\mathrm{Mod}_{A}(\mathcal{C})$ is a dualizable $\mathcal{C}$ -module by directly exhibiting the $\infty$ -category of right $A$ -modules $\mathrm{Mod}_{A^{\rm op}}(\mathcal{C})$ as its dual. The trace map is given by the two-sided bar construction

\tau:\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A^{\rm op}}(\mathcal{C})\to\mathcal{C}\qquad M,N\mapsto M\otimes_{A}N

The unit map is given by the induction

u:\mathcal{C}\to\mathrm{Mod}_{A^{\rm op}}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A}(\mathcal{C})\simeq\mathrm{Mod}_{A^{\rm op}\otimes A}(\mathcal{C})\qquad c\mapsto A\otimes c

where we regard $A\otimes c$ as an $A$ -bimodule.

One can verify directly that the composition

is equivalent to the identity. First, $({\rm id}\otimes u)(M)$ is equivalent to $A\otimes M$ regarded as an $A\otimes A^{\rm op}\otimes A$ -module, and second, $(\tau\otimes{\rm id})(A\otimes M)$ is equivalent to $A\otimes_{A}M\simeq M$ . ∎

4.1.2. Small stable categories

We have been working with the symmetric monoidal structure on the $\infty$ -category $\mathcal{P}r^{\rm L}$ of presentable $\infty$ -categories as developed in [L4], [L5]. We will also need the tensor product of small stable idempotent complete $\infty$ -categories, in particular, the $\infty$ -categories of compact objects in presentable stable $\infty$ -categories.

Let ${st}$ be the full $\infty$ -subcategory of the $\infty$ -category of stable categories (with morphisms exact functors) consisting of those $\infty$ -categories that are idempotent complete. Recall that an $\infty$ -category $\mathcal{C}$ is idempotent complete if the essential image of the Yoneda embedding $\mathcal{C}\rightarrow\mathcal{P}(\mathcal{C})$ is closed under retracts.

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Proposition 4.4. The $\infty$ -category ${st}$ carries a symmetric monoidal structure characterized by the property that for $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{D}\in{st}$ , the $\infty$ -category of exact functors $\Fun_{{st}}(\mathcal{C}_{1}\otimes\mathcal{C}_{2},\mathcal{D})$ is equivalent to the full $\infty$ -subcategory of all functors $\mathcal{C}_{1}\times\mathcal{C}_{2}\to\mathcal{D}$ that preserve finite colimits in $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ separately. Furthermore, passing to the corresponding stable presentable $\infty$ -categories of $\operatorname{Ind}$ -objects is naturally a symmetric monoidal functor.

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Proof. For $\mathcal{C}_{1},\mathcal{C}_{2}\in{st}$ , we define their tensor product by

\mathcal{C}_{1}\otimes\mathcal{C}_{2}=(\operatorname{Ind}(\mathcal{C}_{1})\otimes\operatorname{Ind}(\mathcal{C}_{2}))^{c}

where the tensor product of the right hand side is calculated in the $\infty$ -category $\mathcal{P}r^{\rm L}$ of presentable $\infty$ -categories (with morphisms left adjoints), and the superscript ^c denotes the full $\infty$ -subcategory of compact objects of a presentable $\infty$ -category. Since $\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2}$ is idempotent complete and retracts of compact objects are compact, $(\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2})^{c}$ is idempotent complete as well. Thus the tensor product $\mathcal{C}_{1}\otimes\mathcal{C}_{2}$ is indeed an object of ${st}$ .

For $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{D}\in{st}$ , let $\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})$ be the full $\infty$ -subcategory of functors $\mathcal{C}_{1}\times\mathcal{C}_{2}\to\mathcal{D}$ that preserve finite colimits in $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ separately. We claim that For $\mathcal{C}_{1},\mathcal{C}_{2}\in{st}$ , the tensor product $\mathcal{C}_{1}\otimes\mathcal{C}_{2}$ corepresents the functor $\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},-)$ in the sense that for any $\mathcal{D}\in{st}$ , there is a canonical equivalence

\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})\simeq\Fun_{{st}}(\mathcal{C}_{1}\otimes\mathcal{C}_{2},\mathcal{D}).

As a consequence, the associativity and symmetry of the tensor product $\mathcal{C}_{1}\otimes\mathcal{C}_{2}$ will immediately follow from the analogous properties of $\Fun^{\prime}$ .

For $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{D}\in\mathcal{P}r^{\rm L}$ , let $\Fun^{L\times L}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})$ be the full $\infty$ -subcategory of functors $\mathcal{C}_{1}\times\mathcal{C}_{2}\to\mathcal{D}$ that preserve colimits in $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ separately. To prove the claim, observe that the inclusion $\mathcal{D}\rightarrow\operatorname{Ind}\mathcal{D}$ induces a fully faithful functor

\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})\rightarrow\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\operatorname{Ind}\mathcal{D})\simeq\Fun^{L\times L}(\operatorname{Ind}\mathcal{C}_{1}\times\operatorname{Ind}\mathcal{C}_{2},\operatorname{Ind}\mathcal{D})

Its essential image consists of functors that preserve compact objects. By definition of the monoidal structure on the $\infty$ -category $\mathcal{P}r^{\rm L}$ of presentable $\infty$ -categories, we have a further equivalence

\Fun^{L\times L}(\operatorname{Ind}\mathcal{C}_{1}\times\operatorname{Ind}\mathcal{C}_{2},\operatorname{Ind}\mathcal{D})\simeq\Fun^{\rm L}(\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2},\operatorname{Ind}\mathcal{D}).

Since the compact objects of $\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2}$ are generated by finite colimits of external products of compacts objects, we obtain an equivalence between $\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})$ and the full $\infty$ -subcategory of $\Fun^{\rm L}(\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2},\operatorname{Ind}\mathcal{D})$ consisting of functors that preserve compact objects. In other words, we have the asserted equivalence that characterizes the tensor product

\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})\simeq\Fun_{{st}}((\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2})^{c},\mathcal{D}).

Finally, the assertion that the functor $\operatorname{Ind}:{st}\to\mathcal{P}r^{\rm L}$ is symmetric monoidal is immediate from the constructions and the natural equivalence $\operatorname{Ind}(\mathcal{C}^{c})\simeq\mathcal{C}$ , for $\mathcal{C}\in\mathcal{P}r^{\rm L}$ . ∎

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Remark 4.5. Given small stable idempotent complete $\infty$ -categories $\mathcal{C}_{1},\mathcal{C}_{2}$ , by construction their tensor product $\mathcal{C}_{1}\otimes\mathcal{C}_{2}$ is again a small stable idempotent complete $\infty$ -category. Though it is possible to consider other versions of a tensor product on small stable $\infty$ -categories that need not preserve idempotent complete $\infty$ -categories, our approach builds it in from the beginning.

4.2. Sheaves on fiber products

In this section, we study the $\infty$ -category of sheaves on the derived fiber product of perfect stacks. The main technical result is that it is equivalent to the tensor product of the $\infty$ -categories of sheaves on the factors (Theorem 4.7). The proof involves first showing that an analogous assertion holds for $\infty$ -categories of perfect complexes. The remainder of the section is devoted to collecting corollaries of the main techinical result.

Recall that for a perfect stack $X$ , compact (equivalently, perfect or dualizable) objects in the $\infty$ -category $\qc(X)$ form a small stable idempotent complete $\infty$ -category $\qc(X)^{c}$ , and there is canonical equivalence $\qc(X)\simeq\operatorname{Ind}\qc(X)^{c}$ . Recall as well the tensor product of small stable idempotent complete $\infty$ -categories, and that the functor $\operatorname{Ind}$ is symmetric monoidal.

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Proposition 4.6. Let $X_{1},X_{2}$ be perfect stacks. Then external tensor product defines an equivalence

\boxtimes:\qc(X_{1})^{c}\otimes\qc(X_{2})^{c}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\qc(X_{1}\times X_{2})^{c}

In other words, the $\infty$ -category of perfect complexes on the product is the (small stable idempotent complete) tensor product of the $\infty$ -categories of perfect complexes on the factors.

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Proof. Set $X=X_{1}\times X_{2}$ . By Proposition 3.24, we know that the external product takes compact objects to compact objects, and $\qc(X)^{c}$ is generated by external products.

Thus it suffices to verify that for $M_{i},N_{i}\in\qc(X_{i})^{c}$ we have an equivalence

\Hom_{X}(M_{1}\boxtimes M_{2},N_{1}\boxtimes N_{2})\simeq\Hom_{X_{1}}(M_{1},N_{1})\otimes\Hom_{X_{2}}(M_{2},N_{2}).

Using the fact that each $M_{i}$ is dualizable and $p_{2}$ satisfies the projection formula (since it is perfect), we calculate

$\displaystyle\Hom_{X}(p_{1}^{}M_{1}\otimes p_{2}^{}M_{2},p_{1}^{}N_{1}\otimes p_{2}^{}N_{2})$	$\displaystyle\simeq$	$\displaystyle\Gamma(X,p_{1}^{}M_{1}^{\vee}\otimes p_{1}^{}N_{1}\otimes p_{2}^{}M_{2}^{\vee}\otimes p_{2}^{}N_{2})$
	$\displaystyle\simeq$	$\displaystyle\Gamma(X_{2},(p_{2})_{}(p_{1}^{}{\mathcal{H}om}_{X_{1}}(M_{1},N_{2})\otimes p_{2}^{*}{\mathcal{H}om}_{X_{2}}(M_{2},N_{2})))$
	$\displaystyle\simeq$	$\displaystyle\Gamma(X_{2},\Hom_{X_{1}}(M_{1},N_{1})\otimes{\mathcal{H}om}_{X_{2}}(M_{2},N_{2}))$
	$\displaystyle\simeq$	$\displaystyle\Hom_{X_{1}}(M_{1},N_{1})\otimes\Hom_{X_{2}}(M_{2},N_{2})$

∎

We now prove our main theorem which identifies $\infty$ -categories of sheaves on fiber products algebraically. The proof relies on the following consequence [L2, Corollary 5.5.3.4] of the $\infty$ -categorical adjoint functor theorem: there is a canonical equivalence

({\mathcal{P}r}^{\rm L})^{op}\simeq{\mathcal{P}r}^{\rm R}

between the opposite of the $\infty$ -category $\mathcal{P}r^{\rm L}$ of presentable $\infty$ -categories with morphisms left adjoints and the $\infty$ -category $\mathcal{P}r^{\rm R}$ of presentable $\infty$ -categories with morphisms right adjoints. In other words, we can reverse diagrams of presentable $\infty$ -categories, in which the functors are all left adjoints, by passing to the corresponding right adjoints.

We will also use the fact that the calculation of small limits of presentable $\infty$ -categories is independent of context. Namely, by [L2, Proposition 5.5.3.13, Theorem 5.5.3.18], the forgetful functors from ${\mathcal{P}r}^{\rm L}$ , ${\mathcal{P}r}^{\rm R}$ to all $\infty$ -categories preserves small limits. In particular, given a small diagram of both left and right adjoints, the universal maps from the limit to the terms of the diagram are also both left and right adjoints.

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Theorem 4.7. Let $X_{1}$ , $X_{2}$ , $Y$ be perfect stacks with maps $p_{1}:X_{1}\to Y$ , $p_{2}:X_{2}\to Y$ . Then there is a canonical equivalence

\qc(X_{1})\otimes_{\qc(Y)}\qc(X_{2})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\qc(X_{1}\times_{Y}X_{2}).

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Proof. To begin, consider the case $Y=\Spec k$ , so $X_{1}\times_{Y}X_{2}=X_{1}\times X_{2}$ . By Proposition 4.6 and the fact that $\operatorname{Ind}:{st}\to\mathcal{P}r^{\rm L}$ is symmetric monoidal, the external product functor provides an equivalence $\boxtimes:\qc(X_{1})\otimes\qc(X_{2})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\qc(X_{1}\times X_{2})$ .

To begin the case of a general perfect stack $Y$ , consider the augmented cosimplicial diagram

with the obvious maps constructed from the given maps $p_{1},p_{2}$ .

Applying the (contravariant) functor $\qc$ , we obtain an augmented simplicial $\infty$ -category

with structure maps given by pullbacks. By the absolute case of the theorem when $Y=\Spec k$ , if we forget the augmentation, we obtain the simplicial $\infty$ -category with simplices

\qc(X_{1})\otimes\qc(Y)\otimes\cdots\otimes\qc(Y)\otimes\qc(X_{2})

and structure maps given by tensor contractions. This is precisely the two-sided bar construction [L4, 4.5] whose geometric realization, by definition [L5, 5], calculates the tensor product of $\qc(Y)$ -modules $\qc(X_{1})\otimes_{\qc(Y)}\qc(X_{2})$ . Furthermore, the augmentation provides the natural map

which we will prove is an equivalence.

The above geometric realization is a colimit in $\mathcal{P}r^{\rm L}$ , and hence (as observed prior to the statement of the theorem) may be evaluated as a limit in the opposite category $\mathcal{P}r^{\rm R}$ . Thus we find that $\qc(X_{1})\otimes_{\qc(Y)}\qc(X_{2})$ is also the totalization of the cosimplicial $\infty$ -category

with structure maps given by pushforwards. (We note for future reference that these structure maps are pushforwards along affine morphisms, hence are also colimit preserving, i.e., left adjoints.)

In particular, pushforward along the augmentation provides a natural functor

which is an equivalence if and only if $\tilde{\pi}^{*}$ is an equivalence.

To summarize some of the above structure, we have a diagram of commuting left (lower arrows) and right (upper arrows) adjoints

where $\tau_{*}$ is the universal map from the totalization to the zero cosimplices, and likewise, $\tau^{*}$ is the universal map from the zero simplices to the geometric realization. Thus we obtain a map of monads

acting on $\qc(X_{1}\times X_{2})$ .

The geometric pushforward $\pi_{*}$ is conservative and preserves colimits since $\pi$ is affine. Hence by the Barr-Beck theorem, we have a canonical equivalence

\qc(X_{1}\times_{Y}X_{2})\simeq\Mod_{T_{geom}}(\qc(X_{1}\times X_{2})).

We also claim that the universal map $\tau_{*}$ is conservative and preserves colimits. For the first assertion, recall that $\tau_{*}$ is nothing more than the forgetful map from the totalization to the zero cosimplices. Since the $\infty$ -categories involved are all stable, evaluating conservatism of a functor is equivalent to determining if nonzero objects are sent to zero. But an object sent to zero in the zeroth cosimplices is sent to zero in all cosimplices, and hence is equivalent to the zero object in the limit $\infty$ -category.

To see $\tau_{*}$ preserves colimits, recall that the structure maps of our cosimplicial diagram are both right and left adjoints. It then follows (as observed prior to the statement of the theorem) that the totalization may be evaluated equivalently back in the category $\mathcal{P}r^{\rm L}$ . In particular, the universal functor $\tau_{*}$ is a morphism in $\mathcal{P}r^{\rm L}$ , and hence a left adjoint and so preserves colimits.

We may now apply the Barr-Beck theorem, giving a canonical equivalence

\qc(X_{1})\otimes_{\qc(Y)}\qc(X_{2})\simeq\Mod_{T_{alg}}(\qc(X_{1}\times X_{2})).

Thus it remains to show that the above morphism of monads is an equivalence. It is a straightforward diagram chase to check that the monad $T_{alg}=\tau_{*}\tau^{*}$ is nothing more than the composition $\pi_{1}^{*}\pi_{0*}$ of the geometric functors associated to the initial cosimplicial maps

Thus by base change, it is equivalent to the monad $T_{geom}=\pi_{*}\pi^{*}$ . This concludes the proof of the theorem. ∎

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Corollary 4.8. For $\pi:X\to Y$ any map of perfect stacks, $\qc(X)$ is self-dual as a $\qc(Y)$ -module.

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Proof. By Theorem 4.7, we have a canonical factorization $\qc(X\times_{Y}X)\simeq\qc(X)\otimes_{\qc(Y)}\qc(X)$ . Using this identification, we can define the unit and trace by the correspondences $u=\Delta_{*}\pi^{*}:\qc(Y)\to\qc(X\times_{Y}X)$ and $\tau=\pi_{*}\Delta^{*}:\qc(X\times_{Y}X)\to\qc(Y)$ , where $\Delta:X\to X\times_{Y}X$ is the relative diagonal. We need to check that the following composition is the identity:

The argument is a chase in the following diagram (with Cartesian square):

Applying base change and identities for compositions, we have equivalences of functors

$\displaystyle(\operatorname{id}\otimes\tau)\circ(u\otimes\operatorname{id})$	$\displaystyle=$	$\displaystyle\pi_{1}({\rm id}_{1}\times\Delta_{23})^{}(\Delta_{12}\times{\rm id}_{3})_{}\pi^{}_{2}$
	$\displaystyle\simeq$	$\displaystyle\pi_{1}\Delta_{}\Delta^{}\pi^{}_{2}$
	$\displaystyle\simeq$	$\displaystyle{\rm id}_{\qc(X)}$

∎

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Remark 4.9. The above argument also shows that $\qc(X)^{c}$ is dualizable over perfect $k$ -modules $\qc(\Spec k)^{c}$ when $X$ is smooth and proper: smoothness is required for the unit in $\qc(X\times X)$ to be perfect and properness is required for the trace to land in perfect $k$ -modules.

For a derived stack $X$ , let $\mathcal{M},\mathcal{M}^{\prime}$ be stable presentable $\qc(X)$ -modules. To reduce notation, we write $\Fun_{X}(\mathcal{M},\mathcal{M}^{\prime})$ for the stable presentable $\infty$ -category of $\qc(X)$ -linear colimit preserving functors $\mathcal{M}\to\mathcal{M}^{\prime}$ .

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Corollary 4.10. Let $X,X^{\prime}$ and $Y$ be perfect stacks with maps $X\rightarrow Y\leftarrow X^{\prime}$ . Then there is a natural equivalence of $\infty$ -categories

\qc(X\times_{Y}X^{\prime})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Fun_{Y}(\qc(X),\qc({X^{\prime}})).

In other words, the $\infty$ -category of integral kernels is equivalent to the $\infty$ -category of functors.

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Proof. The statement is an immediate consequence of Theorem 4.7 and the fact that $\qc(X)$ is self-dual (Corollary 4.8). It implies that internal hom of $\qc(Y)$ -modules out of $\qc(X)$ is calculated by tensoring with $\qc(X)^{\vee}\simeq\qc(X)$ . ∎

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Remark 4.11. The equivalence of Corollary 4.10 is naturally monoidal in the following sense. For perfect stacks $X,X^{\prime},X^{\prime\prime}$ mapping to a perfect stack $Y$ , there is a convolution map

\qc(X\times_{Y}X^{\prime})\otimes\qc(X^{\prime}\times_{Y}X^{\prime\prime})\to\qc(X\times_{Y}X^{\prime\prime})

given by pulling back and pushing forward with respect to the triple product $X\times_{Y}X^{\prime}\times_{Y}X^{\prime\prime}$ (see Section 5.2). On the other hand, we have a composition map

\Fun_{Y}(\qc(X),\qc(X^{\prime}))\otimes\Fun_{Y}(\qc(X^{\prime}),\qc(X^{\prime\prime}))\to\Fun_{Y}(\qc(X),\qc(X^{\prime\prime})),

and the equivalence of the theorem intertwines these composition maps.

Finally, for a finite simplicial set $\Sigma$ , we would like to compare the formation of mapping stacks $X^{\Sigma}=\Map(\Sigma,X)$ (this can be viewed as the cotensoring of stacks over simplicial sets) with the formation of tensor products of $\infty$ -categories $\mathcal{C}\otimes\Sigma$ (the tensoring of symmetric monoidal $\infty$ -categories over simplicial sets). By the notation $\mathcal{C}\otimes\Sigma$ , we mean the geometric realization of the simplicial $\infty$ -category given by the constant assignment of $\mathcal{C}$ to each simplex of the simpicial set $\Sigma$ .

0NY5

Corollary 4.12. Let $X$ be a perfect stack, and let $\Sigma$ be a finite simplicial set. Then there is canonical equivalence

\qc(X^{\Sigma})\simeq\qc(X)\otimes\Sigma.

In other words, the $\infty$ -category of sheaves on the mapping stack is calculated as the tensor product of the $\infty$ -categories of sheaves on the simplices.

0NY6

Proof. We calculate $\qc(X^{\Sigma})$ by induction on the simplices as an iterated fiber product of copies of $X$ over perfect stacks (by Proposition 3.24), and apply Theorem 4.7 at each stage. ∎

4.3. General base stacks

In this section, we extend Corollary 4.10 on integral transforms to a relative setting where the base is allowed to be an arbitrary derived stack with affine diagonal (not necessarily perfect). In order to describe integral transforms relative to such a base, we will utilize the simple behavior of $\infty$ -categories of sheaves under affine base change.

0NY7

Proposition 4.13. Let $Y$ be a derived stack with affine diagonal, and let $f:\Spec A\rightarrow Y$ be an affine over $Y$ . Then $\Mod_{A}$ is a self-dual $\qc(Y)$ -module. In particular, for any $X\to Y$ there is a canonical equivalence

\qc(X\times_{Y}\Spec A)\simeq\qc(X)\otimes_{\qc(Y)}\Mod_{A}.

0NY8

Proof. Since $Y$ has affine diagonal, the map $f:\Spec A\rightarrow Y$ is a relative affine, which implies that $f_{*}$ is colimit preserving and conservative. Hence $f_{*}$ satisfies the monadic Barr-Beck criteria [L4, Theorem 3.4.5], implying that the natural map $\Mod_{A}\rightarrow\Mod_{f_{*}f^{*}}(\qc(Y))$ is an equivalence. By the projection formula, the monad $f_{*}f^{*}(-)$ is equivalent to the functor $f_{*}A\otimes(-)$ , with monad structure given by the algebra structure on $A$ . As a consequence, we see that $\Mod_{A}$ is equivalent to $\Mod_{f_{*}A}(\qc(Y))$ . Now Proposition 4.1 gives that $\Mod_{f_{*}A}(\qc(Y))$ is self-dual as a $\qc(Y)$ -module.

Since the $\infty$ -category $\Mod_{A}$ is a dualizable $\qc(Y)$ -module it follows that the functor

{(-)\otimes_{\qc(Y)}\Mod_{A}}

commutes with limits of $\qc(Y)$ -module categories. Thus we have equivalences

\qc(X)\otimes_{\qc(Y)}\Mod_{A}\simeq(\lim_{A^{\prime}\in\it{Aff}/X}\Mod_{A^{\prime}})\otimes_{\qc(Y)}\Mod_{A}\simeq\lim_{A^{\prime}\in\it{Aff}/X}(\Mod_{A^{\prime}}\otimes_{\qc(Y)}\Mod_{A}).

Another application of Proposition 4.1 implies the following equivalence

\lim_{A^{\prime}\in\it{Aff}/X}(\Mod_{A^{\prime}}\otimes_{\qc(Y)}\Mod_{A})\simeq\lim_{A^{\prime}\in\it{Aff}/X}(\Mod_{f^{\prime}_{*}A^{\prime}\otimes f_{*}A}(\qc(Y))).

Using that the map $f^{\prime}\times f:\Spec A^{\prime}\times_{Y}\Spec A\rightarrow Y$ is affine and that $(f^{\prime}\times f)_{*}(A^{\prime}\boxtimes A)\simeq f^{\prime}_{*}A^{\prime}\otimes f_{*}A$ , we obtain that the above is further equivalent to

\lim_{A^{\prime}\in\it{Aff}/X}(\qc(\Spec A^{\prime}\times_{Y}\Spec A))\simeq\qc(\colim_{A^{\prime}\in\it{Aff}/X}(\Spec A^{\prime}\times_{Y}\Spec A))

\simeq\qc((\colim_{A^{\prime}\in\it{Aff}/X}\Spec A^{\prime})\times_{Y}\Spec A)\simeq\qc(X\times_{Y}\Spec A)

Here we have used that $\qc(-)$ sends all colimits to limits, and that $(-)\times_{Y}\Spec A$ commutes with colimits (since it is the left adjoint to the mapping stack over $Y$ ). ∎

We now show that in the general setting where the base is an arbitrary derived stack with affine diagonal, functors continue to be given by integral kernels.

0NY9

Theorem 4.14. Let $f:X\rightarrow Y$ be a perfect map of derived stacks with affine diagonal, and let $g:X^{\prime}\rightarrow Y$ be an arbitrary map of derived stacks. Then there is a natural map $\qc({X\times_{Y}X^{\prime}})\rightarrow\Fun_{Y}(\qc(X),\qc({X^{\prime}}))$ that is an equivalence of $\infty$ -categories.

0NYA

Proof. Consider the Cartesian diagram

We define a functor $\qc({X\times_{Y}X^{\prime}})\rightarrow\Fun(\qc(X),\qc({X^{\prime}}))$ by sending a quasi-coherent sheaf $M\in\qc(X\times_{Y}X^{\prime})$ to the functor $\tilde{f}_{*}(M\otimes\tilde{g}^{*}-)$ . This is a colimit preserving functor, since $\tilde{f}$ is perfect. Furthermore, using the projection formula for the map $\tilde{f}$ , this functor naturally admits the extra structure of $\qc(Y)$ -linearity as follows

\tilde{f}_{*}(M\otimes\tilde{g}^{*}-)\otimes g^{*}V\simeq\tilde{f}_{*}(M\otimes\tilde{g}^{*}(-)\otimes\tilde{f}^{*}g^{*}V)

\simeq\tilde{f}_{*}(M\otimes\tilde{g}^{*}(-)\otimes\tilde{g}^{*}f^{*}V)\simeq\tilde{f}_{*}(M\otimes\tilde{g}^{*}(-\otimes f^{*}V)),

for any $V\in\qc(Y)$ . Therefore, we in fact obtain a functor $\qc(X\times_{Y}X^{\prime})\rightarrow\Fun_{Y}(\qc(X),\qc(X^{\prime}))$ , and the rest of this proof will be devoted to showing it is an equivalence.

Recall that the $\infty$ -category of quasi-coherent sheaves on $X^{\prime}$ is given by the limit

\qc({X^{\prime}})\simeq\lim_{\it{Aff}/X^{\prime}}\Mod_{A}.

By working locally in the target, this provides a description of the $\infty$ -category of $\qc(Y)$ -linear functors with values in $\qc({X^{\prime}})$ as the limit

\Fun_{Y}(\qc(X),\qc({X^{\prime}}))\simeq\Fun_{Y}(\qc(X),\lim_{\it{Aff}/X^{\prime}}\Mod_{A})\simeq\lim_{\it{Aff}/X^{\prime}}\Fun_{Y}(\qc(X),\Mod_{A})

Likewise, we have a description of the $\infty$ -category of quasi-coherent sheaves on the fiber product $X\times_{Y}X^{\prime}$ as a limit

\qc({X\times_{Y}X^{\prime}})\simeq\lim_{\it{Aff}/X^{\prime}}\qc({X\times_{Y}\Spec A}).

This follows from the fact that the functor $\qc(-)$ takes colimits to limits, and the fiber product functor $X\times_{Y}(-)$ commutes with all colimits (because it has a right adjoint).

Now one can analyze the functor $\qc({X\times_{Y}X^{\prime}})\rightarrow\Fun_{Y}(\qc(X),\qc(X^{\prime}))$ by considering the terms in the above two limits. That is, to prove the theorem, it suffices to prove it locally in the target $X^{\prime}$ : for any $\Spec A\rightarrow X^{\prime}$ , we must show that the functor

\qc({X\times_{Y}\Spec A})\rightarrow\Fun_{Y}(\qc(X),\Mod_{A})

is an equivalence.

We will prove this in two steps. First, we will deal with case that the base $Y$ is affine. Afterward, we will use this case to deal with a general base $Y$ .

So assume for the time being that $Y=\Spec B$ . Then $X$ is a perfect stack over $B$ , and by Corollary 4.8, $\qc(X)$ is a self-dual $\Mod_{B}$ -module. Thus we have equivalences

\Fun_{B}(\qc(X),\Mod_{A})\simeq\Fun_{B}(\Mod_{B},\qc(X)^{\vee}\otimes_{B}\Mod_{A})\simeq\qc(X)\otimes_{B}\Mod_{A}.

By Proposition 4.13 we know that the functor $\qc(-)$ takes affine base change to tensor product of $\infty$ -categories. Therefore we have an equivalence

\qc({X\times_{B}\Spec A})\simeq\qc(X)\otimes_{B}\Mod_{A}.

Putting together the above equivalences, we conclude that we have equivalences

\qc({X\times_{B}\Spec A})\simeq\qc(X)\otimes_{B}\Mod_{A}\simeq\Fun_{B}(\qc(X),\Mod_{A}),

This proves the theorem when $Y$ is affine.

Working locally in the base $Y$ , we will now use the above discussion to prove the theorem in general. As above, since $\qc(-)$ and fiber products behave well with respect to colimits we can calculate the $\infty$ -category of quasi-coherent sheaves on the fiber product $X\times_{Y}\Spec A$ as the limit

\qc(X\times_{Y}\Spec A)\simeq\lim_{B\in\it{Aff}/Y}\qc(X\times_{Y}\Spec B\times_{Y}\Spec A).

To calculate the $\infty$ -category of functors, we use the following: by Proposition 4.13, the $\infty$ -category $\Mod_{A}$ is a dualizable $\qc(Y)$ -module and hence the functor $\Mod_{A}\otimes_{\qc(Y)}(-)$ commutes with all limits. Therefore we have an equivalence

\Mod_{A}\simeq\Mod_{A}\otimes_{\qc(Y)}(\lim_{B\in\it{Aff}/Y}\Mod_{B})\simeq\lim_{B\in\it{Aff}/Y}\Mod_{A}\otimes_{\qc(Y)}\Mod_{B},

and so in particular we obtain equivalences

\Fun_{Y}(\qc(X),\Mod_{A})\simeq\Fun_{Y}(\qc({X}),\lim_{B\in\it{Aff}/Y}\Mod_{A}\otimes_{\qc(Y)}\Mod_{B})

\simeq\lim_{B\in\it{Aff}/Y}\Fun_{Y}(\qc({X}),\Mod_{A}\otimes_{\qc(Y)}\Mod_{B}).

By the adjunction between induction and restriction, and a repeated application of Proposition 4.13, we also have equivalences

\Fun_{Y}(\qc({X}),\Mod_{A}\otimes_{\qc(Y)}\Mod_{B})\simeq\Fun_{B}(\qc(X)\otimes_{\qc(Y)}\Mod_{B},\Mod_{A}\otimes_{\qc(Y)}\Mod_{B}).

\simeq\Fun_{B}(\qc({X\times_{Y}\Spec B}),\qc({\Spec A\times_{Y}\Spec B}))

Finally, by the above discussion and the affine case of the theorem with base $\Spec B$ , we obtain the following chain of equivalences

\qc({X\times_{Y}\Spec A})\simeq\lim_{B\in\it{Aff}/Y}\qc({X\times_{Y}\Spec B\times_{Y}\Spec A})

\simeq\lim_{B\in\it{Aff}/Y}\Fun_{B}(\qc({X\times_{Y}B}),\qc({A\times_{Y}\Spec B}))\simeq\Fun_{Y}(\qc(X),\Mod_{A}).

Since we previously reduced the theorem to the case when $X^{\prime}=\Spec A$ , this completes the proof. ∎

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5