Dualizable presentable $\infty$-categories

Abstract

We prove that for any presentably symmetric monoidal $\infty$-category $\mathcal{V}$, the $\infty$-category ${\mathbf{Mod}}_{\mathcal{V}}({\mathbf{Pr}}^{\mathrm{L}}{)}^{\mathrm{dbl}}$ of dualizable presentable $\mathcal{V}$-modules and internal left adjoints between them is itself presentable. Along the way, we survey formal properties of these dualizable $\mathcal{V}$-modules. We pay close attention to the case of the $\infty$-category of spectra, where we survey the foundational properties of ``compact morphisms''.

80 pages; comments welcome!