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Classification Diagrams of Marked Simplicial Sets

Kensuke Arakawa

Abstract

We prove that the classification diagram functor from the category of marked simplicial sets to the category of bisimplicial sets carries cartesian equivalences to Rezk equivalences. As a corollary, we obtain Mazel-Gee's theorem on localizations of relative $\infty$-categories.

Classification Diagrams of Marked Simplicial Sets

Abstract

We prove that the classification diagram functor from the category of marked simplicial sets to the category of bisimplicial sets carries cartesian equivalences to Rezk equivalences. As a corollary, we obtain Mazel-Gee's theorem on localizations of relative -categories.
Paper Structure (14 sections, 28 theorems, 31 equations)

This paper contains 14 sections, 28 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Localizations of $\infty$-Categories
  3. Mazel-Gee's Localization Theorem
  4. What This Paper is About
  5. Outline of the Paper
  6. A Remark on Marked Complete Segal Spaces
  7. A Model Structure for Marked Complete Segal Spaces
  8. Identifying of Fibrant Objects
  9. The Marked Complete Segal Space Model structure is Simplicial
  10. Identifying Fibrations of Fibrant Objects
  11. The Marked Complete Segal Space Model Structure is Cartesian
  12. Proof of Theorem \ref{['thm:mCSS_model_structure']}
  13. Relating Various Model Structures
  14. Main Result

Key Result

Theorem 1.2

Let $\mathcal{C}$ and $\mathcal{D}$ be $\infty$-categories, let $\mathcal{W}\subset\mathcal{C}$ be a subcategory containing all equivalences, and let $f:\mathcal{C}\to\mathcal{D}$ be a functor which carries every morphism in $\mathcal{W}$ to an equivalence. The following conditions are equivalent:

Theorems & Definitions (69)

  • Definition 1.1
  • Theorem 1.2: Mazel-Gee's Localization Theorem MR4045352
  • Corollary 1.3
  • Example 1.4
  • Proposition 1.5
  • Theorem 1.7: Theorem \ref{['thm:main']}
  • Definition 2.1
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • ...and 59 more