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The complicial model of $(\infty,ω)$-categories

Félix Loubaton

Abstract

It has been conjectured since the 1980s that Verity's $n$-complicial sets were a model for $(\infty,n)$-categories. This text is dedicated to providing a positive answer to this conjecture. The proof of this result relies on a thorough study of Gray operations in (strict) $ω$-categories, in complicial sets, and in enriched (stratified) Segal precategories.

The complicial model of $(\infty,ω)$-categories

Abstract

It has been conjectured since the 1980s that Verity's -complicial sets were a model for -categories. This text is dedicated to providing a positive answer to this conjecture. The proof of this result relies on a thorough study of Gray operations in (strict) -categories, in complicial sets, and in enriched (stratified) Segal precategories.
Paper Structure (47 sections, 206 theorems, 383 equations)

This paper contains 47 sections, 206 theorems, 383 equations.

Table of Contents

  1. Chapter 1.
  2. Chapter 2.
  3. Chapter 3.
  4. $(0,\omega)$-Categories and presheaves on $\Theta$
  5. Basic constructions
  6. $(0,\omega)$-Categories
  7. The category $\Theta$
  8. The link between presheaves on $\Theta$ and on $\Delta[\Theta]$
  9. Gray Operations
  10. Recollection on Steiner theory
  11. $2$-Polygraphs and presheaves on $\Theta_2$
  12. Gray operations on augmented directed complexes
  13. Gray operations on $(0,\omega)$-categories
  14. Gray tensor product of simplicial sets
  15. Study of complicial sets
  16. ...and 32 more sections

Key Result

Theorem 1

In the category of $(0,\omega)$-categories, there exists an isomorphism, natural in $A$, between $[A,1]\otimes[1]$ and the colimit of the following diagram \begin{tikzcd} {[1]\vee[A,1]} & {[A\otimes\{0\},1]} & { [A\otimes[1],1]} & {[A\otimes\{1\},1]} & {[A,1]\vee[1]} \arrow["\triangledown"', from=

Theorems & Definitions (523)

  • Theorem : \ref{['theo:appendice formula for otimes']}
  • Theorem : \ref{['theo:appendice formula for star']}
  • Theorem : \ref{['theo:interval_first_formula']}
  • Theorem : \ref{['theo:cyl_formula']}
  • Theorem : \ref{['theo:criterion_to_be_linked_to_identity']}
  • Theorem : \ref{['theo:complicialGray module']}
  • Theorem : \ref{['theo:theorem model 1']}
  • Theorem : \ref{['theo:equivalence avec theta']}
  • Theorem : \ref{['theo:appendice formula for otimes']}
  • Theorem : \ref{['theo:appendice formula for star']}
  • ...and 513 more