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Dualizable presentable $\infty$-categories

Maxime Ramzi

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''.

Dualizable presentable $\infty$-categories

Abstract

We prove that for any presentably symmetric monoidal -category , the -category of dualizable presentable -modules and internal left adjoints between them is itself presentable. Along the way, we survey formal properties of these dualizable -modules. We pay close attention to the case of the -category of spectra, where we survey the foundational properties of ``compact morphisms''.

Paper Structure

This paper contains 36 sections, 134 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Continuous $K$-theory
  4. Brauer groups
  5. Outline
  6. Sectionwise outline
  7. Conventions and notations
  8. Acknowledgements
  9. Preliminaries
  10. Presentable $\mathcal{V}$-modules
  11. Internal left adjoints
  12. Characterizations of dualizability
  13. An adjunction
  14. Compact assembly
  15. Compact maps
  16. ...and 21 more sections

Key Result

Theorem 1

Let $\mathcal{V}$ be a presentably symmetric monoidal $\infty$-category. The $\infty$-category $\mathbf{Mod}(\mathcal{V})^{\mathrm{dbl}}$ of dualizable presentable $\mathcal{V}$-modules and internal left adjoints between them is itself presentableIt is crucial to restrict the morphisms, otherwise th

Theorems & Definitions (360)

  • Theorem 1
  • Remark 1.2
  • Theorem 2
  • Remark 1.3
  • Remark 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Remark 1.7
  • Theorem 2.1: heine
  • Theorem 2.3: heine
  • ...and 350 more