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On the equivalence of two approaches to multiplicative homotopy theories

Kensuke Arakawa

Abstract

We study the relation of two frameworks for multiplicative homotopy theories: Presentably symmetric monoidal $\infty$-categories and combinatorial symmetric monoidal model categories. Our main theorem establishes an equivalence of their homotopy theories. As consequences, we solve Pavlov's conjecture and obtain a solution to a special case of Hovey's 10th problem. We also prove several variations of the main theorem, such as an analog for non-symmetric monoidal semi-model categories.

On the equivalence of two approaches to multiplicative homotopy theories

Abstract

We study the relation of two frameworks for multiplicative homotopy theories: Presentably symmetric monoidal -categories and combinatorial symmetric monoidal model categories. Our main theorem establishes an equivalence of their homotopy theories. As consequences, we solve Pavlov's conjecture and obtain a solution to a special case of Hovey's 10th problem. We also prove several variations of the main theorem, such as an analog for non-symmetric monoidal semi-model categories.
Paper Structure (22 sections, 55 theorems, 76 equations, 1 table)

This paper contains 22 sections, 55 theorems, 76 equations, 1 table.

Table of Contents

  1. Monoidal model categories and their variations
  2. Plain case
  3. Enriched case
  4. Semi-model-categorical case
  5. Main result
  6. Proof of Proposition \ref{['prop:main_L']}
  7. Proof of Proposition \ref{['prop:main_flat']}
  8. Relative-categorical multiplicative Gabriel--Ulmer duality
  9. Strongly $\kappa$-combinatorial model categories
  10. Proof of Proposition \ref{['prop:main_flat']}
  11. Proof of Proposition \ref{['prop:main_U']}
  12. Symmetric cubical sets
  13. Proof of Proposition \ref{['prop:main_U']}
  14. Variations of the main theorem
  15. Review of semi-model categories
  16. ...and 7 more sections

Key Result

Theorem A

The $\infty$-category of presentably symmetric monoidal $\infty$-categories is the localization of the category of combinatorial symmetric monoidal model categories and symmetric monoidal left Quillen functors. Moreover, the maximal sub $\infty$-groupoid of the former is the localization of the subc

Theorems & Definitions (129)

  • Theorem A: Theorems \ref{['thm:main']} and \ref{['thm:var_SM']}
  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • proof
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Proposition 1.8
  • ...and 119 more