Dualizable presentable $\infty$-categories

TL;DR

This paper demonstrates that the $ abla$-category of dualizable presentable modules over a symmetric monoidal $ abla$-category is itself presentable, providing foundational insights especially for the $ abla$-category of spectra.

Contribution

It establishes the presentability of the $ abla$-category of dualizable modules over any presentably symmetric monoidal $ abla$-category, including the case of spectra.

Findings

The $ abla$-category of dualizable modules is presentable.

Survey of formal properties of dualizable $ abla$-modules.

Analysis of compact morphisms in the $ abla$-category of spectra.

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''.