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Monoidal Relative Categories Model Monoidal $\infty$-Categories

Kensuke Arakawa

Abstract

We prove that the homotopy theory of monoidal relative categories is equivalent to that of monoidal $\infty$-categories, and likewise in the symmetric monoidal setting. As an application, we give a concise and complete proof of the fact that every presentably monoidal or presentably symmetric monoidal $\infty$-category is presented by a monoidal or symmetric monoidal model category, which, in the monoidal case, was sketched by Lurie, and in the symmetric monoidal case, was proved by Nikolaus--Sagave.

Monoidal Relative Categories Model Monoidal $\infty$-Categories

Abstract

We prove that the homotopy theory of monoidal relative categories is equivalent to that of monoidal -categories, and likewise in the symmetric monoidal setting. As an application, we give a concise and complete proof of the fact that every presentably monoidal or presentably symmetric monoidal -category is presented by a monoidal or symmetric monoidal model category, which, in the monoidal case, was sketched by Lurie, and in the symmetric monoidal case, was proved by Nikolaus--Sagave.
Paper Structure (10 sections, 15 theorems, 53 equations)

This paper contains 10 sections, 15 theorems, 53 equations.

Table of Contents

  1. Variations of Mandell's Constructions
  2. The Functor Fact
  3. The Functor Perm
  4. Proof of Theorem \ref{['thm:Man10']}
  5. Comparing $\operatorname{Perm}\circ\operatorname{Fact}\left(\mathcal{C}\right)$ with $\mathcal{C}$
  6. Comparing $\operatorname{Fact}\circ\operatorname{Perm}\left(F\right)$ with $F$
  7. Main Result
  8. Application
  9. Relative Categories
  10. Relative Cartesian Fibrations

Key Result

Theorem 1

Monoidal localization determines an equivalence of $\infty$-categories where $\sf{MonRelCat}[{\rm DK}^{-1}]$ denotes the localization of the category of monoidal relative categories at the monoidal functors that are DK-equivalences, and $\mathcal{M}\sf{on}\mathcal{C}\sf{at}_{\infty}$ denotes the $\infty$-category of monoidal $\infty$-categories. A similar result holds

Theorems & Definitions (43)

  • Theorem 1: Theorem \ref{['thm:main']}
  • Remark 2
  • Corollary 3: Theorem \ref{['thm:NS17']}
  • Remark 4
  • Definition 7
  • Remark 8
  • Definition 9
  • Definition 11
  • Definition 12
  • Definition 15
  • ...and 33 more