Subsection 3.1. (0NRR) · ScalingStacks

3.1. Tensor products of stable $\infty$ -categories

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.

0NJW

Theorem 3.1. The $\infty$ -category of small idempotent-complete stable $\infty$ -categories is a closed symmetric monoidal category with respect to $\widehat{\otimes}$ . The unit is the $\infty$ -category ${\mathcal{S}}_{\infty}^{\omega}$ of compact spectra and the internal mapping object is given for small idempotent-complete stable $\infty$ -categories ${\mathcal{A}}$ and ${\mathcal{B}}$ by $\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{B}})$ .

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}).

0NJX

Proposition 3.2. For any small stable $\infty$ -category ${\mathcal{A}}$ , the stable Yoneda embedding

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

induces an equivalence $\Ind({\mathcal{A}})\simeq\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ .

0NJY

Proof. Clearly $\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ admits filtered colimits, as the filtered colimit of finite colimit preserving functors itself preserves finite colimits. This gives a map $\Ind({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ which is evidently fully faithful since, using the fact that the usual Yoneda embedding is fully faithful and that mapping spaces between representables in $\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ are computed as the limit

\lim_{n}\Omega^{n}\map(a,\Sigma^{n}b)\simeq\lim_{n}\map(a,\Omega^{n}\Sigma^{n}b)\simeq\map(a,b).

To show that this map is also essentially surjective, we must show that any exact functor $f\colon{\mathcal{A}}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ is ind-representable. Consider the pullback

where the right vertical map is the stable Yoneda embedding. We claim that the $\infty$ -category ${\mathcal{A}}_{/f}$ is filtered: to see this, let $K$ be a finite simplicial set and $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}_{/f}$ a functor. Since both ${\mathcal{A}}$ and $\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})_{/f}$ admit finite colimits and both functors to $\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ preserve finite colimits, we may extend $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}_{/f}$ to a colimit diagram $K^{\triangleright}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}_{/f}$ . In particular, this gives a cone on $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}_{/f}$ , which shows that ${\mathcal{A}}_{/f}$ is a filtered $\infty$ -category. Finally, since filtered colimits in $\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ are computed pointwise, it follows that $f$ is a colimit of the diagram ${\mathcal{A}}_{/f}\longrightarrow\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ , which is to say that it is ind-representable. ∎

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.

0NJZ

Corollary 3.3. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be small stable idempotent-complete $\infty$ -categories. There is an equivalence of small stable idempotent-complete $\infty$ -categories between $\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{B}})$ and $\infty$ -category of right-compact ${\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{B}}$ -modules.

0NK0

Proof. Since the Yoneda embedding is fully faithful, it suffices to look at the essential image of the composite. The image of $\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{B}})$ in $\mathrm{Fun}^{\ex}({\mathcal{A}}\widehat{\otimes}{\mathcal{B}}^{\op},{\mathcal{S}}_{\infty})$ under $\nu_{a}$ is identified with the image of $\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{B}})$ in $\mathrm{Fun}^{\ex}({\mathcal{A}},\mathrm{Fun}^{\ex}({\mathcal{B}}^{\op},{\mathcal{S}}_{\infty}))$ under the corresponding map ${\mathcal{S}}_{\infty}^{\omega}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}$ . Since this lies inside the image of $B$ inside $\mathrm{Fun}^{\ex}({\mathcal{B}}^{\op},{\mathcal{S}}_{\infty})$ under the Yoneda embedding, the result follows. ∎

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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3.1. Tensor products of stable ∞\infty-categories

3.1. Tensor products of stable $\infty$ -categories