ScalingStacks

The $\infty$-category ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St}}}$ of presentable stable $\infty$-categories is a closed symmetric monoidal $\infty$-category with product $\otimes$ and internal mapping object given by the presentable stable $\infty$-category $\mathrm{Fun}^{\mathrm{L}}({\mathcal{A}},{\mathcal{B}})$ of colimit-preserving functors [53, 6.3.1.14, 6.3.1.17]. Following [8, §4.1.2], we can then define the tensor product on small idempotent-complete stable $\infty$-categories as

$${\mathcal{C}}\widehat{\otimes}{\mathcal{D}}=(\Ind({\mathcal{C}})\otimes\Ind({\mathcal{D}}))^{\omega}.$$

The tensor product of idempotent-complete small stable $\infty$-categories is characterized by the universal property that maps out of ${\mathcal{A}}\otimes{\mathcal{B}}$ correspond to maps out of the product ${\mathcal{A}}\times{\mathcal{B}}$ which preserve finite colimits in each variable [8, 4.4]. If ${\mathcal{A}}$ and ${\mathcal{B}}$ are arbitrary small stable $\infty$-categories, then we set ${\mathcal{A}}\widehat{\otimes}{\mathcal{B}}:=\Idem({\mathcal{A}})\widehat{\otimes}\Idem({\mathcal{B}})$.

More precisely, we can define $\Cat_{\infty}^{\perf}$ as a symmetric monoidal $\infty$-category as follows. Let ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St,cg}}}$ denote the full subcategory of ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St}}}$ on the compactly-generated stable $\infty$-categories. The criterion of [53, 2.2.1.2] implies that ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St,cg}}}$ is a symmetric monoidal subcategory of ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St}}}$; the tensor product of compactly-generated stable $\infty$-categories is itself compactly-generated, as is the unit ${\mathcal{S}}_{\infty}\simeq\Ind({\mathcal{S}}_{\infty}^{\omega})$.

For a small stable idempotent-complete $\infty$-category ${\mathcal{A}}$ and a presentable $\infty$-category ${\mathcal{B}}$, $\mathrm{Fun}^{\ex}$ and $\mathrm{Fun}^{\mathrm{L}}$ are related by the formula

$$\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{B}})\simeq\mathrm{Fun}^{\mathrm{L}}(\Ind({\mathcal{A}}),{\mathcal{B}}),$$

which follows from [52, 5.3.5.10] and the fact that functors which preserve filtered colimits and finite colimits preserve all colimits. Note that

$$\Ind\colon\Cat_{\infty}^{\perf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St}}}$$

factors through the full subcategory ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St,cg}}}$ by definition. This gives an equivalence of $\infty$-categories between $\Cat_{\infty}^{\perf}$ and the subcategory ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St}}}_{\omega}$ of ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St}}}$ whose objects are the compactly-generated stable $\infty$-categories and whose maps

$$\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{B}})\simeq\mathrm{Fun}^{\mathrm{L}}_{\omega}(\Ind({\mathcal{A}}),\Ind({\mathcal{B}}))\subset\mathrm{Fun}^{\mathrm{L}}(\Ind({\mathcal{A}}),\Ind({\mathcal{B}})),$$

are the full subcategory of the colimit-preserving functors $\Ind({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind({\mathcal{B}})$ which preserve compact objects [52, 5.5.7.10]. We regard $\Cat_{\infty}^{\perf}$ as a symmetric monoidal $\infty$-category via this equivalence. The observation of [53, 6.3.1.17] implies that $\Cat_{\infty}^{\perf}$ is closed. Hence we have the following result.

Given a small stable idempotent-complete $\infty$-category ${\mathcal{A}}$, we have the $\infty$-category of ${\mathcal{A}}$-modules, given by the compactly-generated stable $\infty$-category $\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$. The stable Yoneda embedding provides an exact functor [52, 5.3.5.2]

$${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty}).$$

Provided ${\mathcal{A}}$ is idempotent-complete,  [52, 5.4.2.4] tells us that the essential image of the Yoneda embedding is precisely the $\infty$-category of compact ${\mathcal{A}}$-modules

$${\mathcal{A}}\simeq\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})^{\omega}.$$

Moreover, we know that if ${\mathcal{A}}$ is an arbitrary small stable $\infty$-category, then the Yoneda map ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})^{\omega}$ models the idempotent-completion of ${\mathcal{A}}$.

We use the preceding results to characterize $\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{B}})$ in terms of a certain subcategory of $\mathrm{Fun}^{\mathrm{L}}({\mathcal{A}}\widehat{\otimes}{\mathcal{B}}^{\op},{\mathcal{S}}_{\infty})$, the $\infty$-category of ${\mathcal{A}}$-${\mathcal{B}}$-bimodules. Specifically, the Yoneda embedding ${\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex}({\mathcal{B}}^{\op},{\mathcal{S}}_{\infty})$ provides the following composite

	$\displaystyle\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{B}})$	$\displaystyle\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex}({\mathcal{A}},\mathrm{Fun}^{\ex}({\mathcal{B}}^{\op},{\mathcal{S}}_{\infty}))$
		$\displaystyle\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\mathrm{L}}(\Ind({\mathcal{A}}),\mathrm{Fun}^{\mathrm{L}}(\Ind({\mathcal{B}}^{\op}),{\mathcal{S}}_{\infty}))$
		$\displaystyle\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\mathrm{L}}(\Ind({\mathcal{A}})\otimes\Ind({\mathcal{B}}^{\op}),{\mathcal{S}}_{\infty})$
		$\displaystyle\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex}({\mathcal{A}}\widehat{\otimes}{\mathcal{B}}^{\op},{\mathcal{S}}_{\infty}),$

which exhibits $\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{B}})$ as a full subcategory of ${\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{B}}$-modules.

We have the following useful corollary, which is the analogue of a characterization originally written down by Toën [80]. For each object $a\in{\mathcal{A}}$, we have a map of small idempotent-complete stable $\infty$-categories ${\mathcal{S}}_{\infty}^{\omega}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}$ given by sending $\mathbb{S}\in{\mathcal{S}}_{\infty}^{\omega}$ to $a\in{\mathcal{A}}$. Since ${\mathcal{S}}_{\infty}^{\omega}$ is the unit of the tensor $\widehat{\otimes}$, we obtain a restriction map

$$\nu_{a}\colon\mathrm{Fun}^{\ex}({\mathcal{A}}\widehat{\otimes}{\mathcal{B}}^{\op},{\mathcal{S}}_{\infty})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex}({\mathcal{B}}^{\op},{\mathcal{S}}_{\infty}).$$

If the image of an element of $\mathrm{Fun}^{\ex}({\mathcal{A}}\widehat{\otimes}{\mathcal{B}}^{\op},{\mathcal{S}}_{\infty})$ under $\nu_{a}$ is compact for every $a\in{\mathcal{A}}$, we will say that the element is right-compact.

Our final goal in this section is to characterize the dualizable objects of $\Cat_{\infty}^{\perf}$. To do so, we need to introduce certain smallness conditions on small stable $\infty$-categories.

Note that a small stable $\infty$-category ${\mathcal{A}}$ is proper if and only if its idempotent-completion $\Idem({\mathcal{A}})$ is proper, as retracts of compact objects are compact.

We will typically only be interested in smoothness and properness of small stable $\infty$-categories which are also idempotent-complete. This is because these are the situations which arise in algebra and geometry, e.g. when ${\mathcal{A}}$ is the stable $\infty$-category of perfect modules for a ring spectrum or perfect complexes for a scheme, and such categories are always idempotent complete. Conversely (as we will show in section 4) any idempotent-complete small stable $\infty$-category ${\mathcal{A}}$ is equivalent to the stable $\infty$-category of perfect modules for some spectral category.

We now recall the definitions of dualizability in symmetric monoidal $\infty$-categories from [53, §4.2.5]. The salient fact here is that dualizability can be detected in the (symmetric monoidal) homotopy category:

In other words, an object $A$ of ${\mathcal{C}}$ is dualizable if there exists an object $DA$ together with an evaluation map $\epsilon:A\otimes DA\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}1$ and a coevaluation map $\delta:1\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}DA\otimes A$ such that the composites

$$A\simeq A\otimes 1\overset{A\otimes\delta}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}A\otimes DA\otimes A\overset{\epsilon\otimes A}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}1\otimes A\simeq A$$

and

$$DA\simeq 1\otimes DA\overset{\delta\otimes DA}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}DA\otimes A\otimes DA\overset{DA\otimes\epsilon}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}DA\otimes 1\simeq DA$$

are the respective identities in $\Ho({\mathcal{C}})$. The object $DA$ is called the dual of $A$, and is unique up to equivalence in ${\mathcal{C}}$.

Recall that the discussion preceding theorem 3.1 above identifies $\Cat_{\infty}^{\perf}$ as a symmetric monoidal subcategory of ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St,cg}}}$; in particular, the functor $\Ind$ preserves dualizable objects. Next, observe that ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St,cg}}}$ is a rigid symmetric monoidal category; that is, all objects in ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St,cg}}}$ are dualizable. This is because, for ${\mathcal{A}}\in\Cat_{\infty}^{\perf}$,

$$\Ind({\mathcal{A}}^{\op})\simeq\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{S}}_{\infty})\simeq\mathrm{Fun}^{\mathrm{L}}(\Ind({\mathcal{A}}),{\mathcal{S}}_{\infty})$$

is the dual of ${\mathcal{A}}$ and the coevaluation map

$${\mathcal{S}}_{\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind({\mathcal{A}}^{\op})\otimes\Ind({\mathcal{A}})\simeq\Ind({\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{A}})\simeq\mathrm{Fun}^{\ex}({\mathcal{A}}\widehat{\otimes}{\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$$

is given by formation of mapping spectra in ${\mathcal{A}}^{\op}$.

In analogy with the situation for dg-categories [21, §4], this allow us to obtain the following characterization of the dualizable objects.