ScalingStacks

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. ∎