Subsection 3.3. (0NRT) · ScalingStacks

3.3. Dualizability

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:

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Definition 3.6. Let ${\mathcal{C}}^{\otimes}$ be a symmetric monoidal $\infty$ -category. An object of the underlying $\infty$ -category ${\mathcal{C}}$ of ${\mathcal{C}}^{\otimes}$ is said to be dualizable if it is dualizable as an object of the symmetric monoidal homotopy category of ${\mathcal{C}}^{\otimes}$ .

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.

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Theorem 3.7. An idempotent-complete small stable $\infty$ -category ${\mathcal{A}}$ is dualizable (as an object of the symmetric monoidal $\infty$ -category $\Cat_{\infty}^{\perf}$ of idempotent-complete small stable $\infty$ -categories) if and only if ${\mathcal{A}}$ is smooth and proper. Moreover, the dual of a dualizable object ${\mathcal{A}}$ is its opposite $\infty$ -category ${\mathcal{A}}^{\op}$ .

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Proof. By the proceeding discussion, $\Ind({\mathcal{A}})$ is a dualizable object of ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St,cg}}}$ with dual $\Ind({\mathcal{A}}^{\op})$ . Thus the dual of ${\mathcal{A}}$ in $\Cat_{\infty}^{\perf}$ is ${\mathcal{A}}^{\op}$ , and ${\mathcal{A}}$ is dualizable in $\Cat_{\infty}^{\perf}\simeq{{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St}}}_{\omega}$ if and only if the evaluation and coevaluation maps lie in the subcategory ${{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St}}}_{\omega}\subset{{\mathcal{P}\mathrm{r}}^{\mathrm{L}}_{\mathrm{St,cg}}}$ . But the evaluation map

\Ind({\mathcal{A}}\widehat{\otimes}{\mathcal{A}}^{\op})\simeq\Ind({\mathcal{A}})\otimes\Ind({\mathcal{A}})^{*}\longrightarrow\Ind({\mathcal{S}}_{\infty}^{\omega})\simeq{\mathcal{S}}_{\infty}

is induced by the mapping spectrum functor $\mathrm{Map}_{\mathcal{A}}\colon{\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{A}}\longrightarrow{\mathcal{S}}_{\infty}$ in ${\mathcal{A}}$ ; dually, the coevaluation map

{\mathcal{S}}_{\infty}\simeq\Ind({\mathcal{S}}_{\infty}^{\omega})\longrightarrow\Ind({\mathcal{A}}^{\op})\otimes\Ind({\mathcal{A}})\simeq\Ind({\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{A}})

given by the map ${\mathcal{S}}_{\infty}\longrightarrow\Ind({\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{A}})$ which classifies ${\mathcal{A}}$ as an ${\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{A}}$ -module. Hence the evaluation map lies in this subcategory if and only if the mapping spectra ${\mathcal{A}}(a,b)$ in $\mathcal{A}$ are compact, and the coevaulation map lies in this subcategory if and only if ${\mathcal{A}}$ is a compact ${\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{A}}$ -module. Therefore, by definition, ${\mathcal{A}}$ is a dualizable object of $\Cat_{\infty}^{\perf}$ if and only if ${\mathcal{A}}$ is smooth and proper. ∎

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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