ScalingStacks

Definition 3.4. A small stable $\infty$-category ${\mathcal{A}}$ is proper if, for all pairs of objects $a$ and $b$ of ${\mathcal{A}}$, the mapping spectrum ${\mathcal{A}}(a,b)$ (recall definition 2.15) is compact.

Definition 3.5. A small stable $\infty$-category ${\mathcal{A}}$ is smooth if it is perfect as an ${\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{A}}$-module (i.e., in the smallest subcategory of $\mathrm{Fun}^{\ex}({\mathcal{A}}\widehat{\otimes}{\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ generated by the representables under finite colimits and retracts). If ${\mathcal{A}}$ is idempotent-complete, we may equivalently require that ${\mathcal{A}}$ is a representable ${\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{A}}$-module: since ${\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{A}}$ is an idempotent-complete stable $\infty$-category, it is closed under finite colimits and retracts, and so any perfect ${\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{A}}$-module is representable.