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A strengthened $(\infty, n)$-categorical pasting theorem

Clémence Chanavat

Abstract

We extend Campion's pasting theorem for $(\infty, n)$-categories to a larger class of polygraphs, called the directed complexes with frame-acyclic molecules. It follows, for instance, that this pasting theorem applies to any polygraph presented by a semi-simplicial set. We also set up a comparison between directed complexes and Henry's regular polygraphs, and show that they coincide up to dimension $3$. As a corollary of our main results, the pasting theorem also applies to the class of regular $3$-polygraphs.

A strengthened $(\infty, n)$-categorical pasting theorem

Abstract

We extend Campion's pasting theorem for -categories to a larger class of polygraphs, called the directed complexes with frame-acyclic molecules. It follows, for instance, that this pasting theorem applies to any polygraph presented by a semi-simplicial set. We also set up a comparison between directed complexes and Henry's regular polygraphs, and show that they coincide up to dimension . As a corollary of our main results, the pasting theorem also applies to the class of regular -polygraphs.
Paper Structure (12 sections, 46 theorems, 56 equations)

This paper contains 12 sections, 46 theorems, 56 equations.

Table of Contents

  1. Directed complexes and their strict functors
  2. Molecules and directed complexes
  3. Strict functors of directed complexes
  4. Comparison with regular polygraphs
  5. Poset of subdivisions
  6. Layering of molecules
  7. Generalities on thetas
  8. Contractibility of the poset of subdivisions
  9. The pasting theorem
  10. Recollection on (infty, n)-categories
  11. A pasting theorem for frame-acyclic directed complexes
  12. Applications

Key Result

Theorem 1

Let $X$ be a directed complex with frame-acyclic molecules. Then ${\mathit{Mol}{/}\raisebox{-2pt}{$X$}}$ is a polygraph and a homotopy polygraph.

Theorems & Definitions (130)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Conjecture
  • Theorem
  • Example 1.3
  • Remark 1.7
  • Remark 1.8
  • Example 1.9
  • ...and 120 more