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Homotopy theory of stricter $n$-categories

Clémence Chanavat

TL;DR

The paper introduces stricter $oldsymbol{ ext{n}}$-categories by enforcing higher exchange laws via Hadzihasanovic's regular directed complexes, and develops foundational tools such as the Gray product and suspension. It constructs a folk model structure on stricter $oldsymbol{ ext{n}}$-categories and demonstrates that this structure is right transferred from the diagrammatic $( orall\,n)$-model structure through a diagrammatic nerve, connecting strict models to diagrammatic set models of $( orall\,n)$-categories. A key result is that the walking equivalence in stricter $oldsymbol{ ext{ω}}$-categories aligns with the stricter polygraph generated by walking equivalences in diagrammatic sets, illuminating how these two formalisms mirror each other in the appropriate dimension range. The work lays groundwork for comparing the diagrammatic and standard models of $( orall\,n)$-categories (notably for $n\, ext{≤}\,3$) and establishes a coherent homotopy-theoretic framework for stricter higher categories, with a strong emphasis on functoriality, monoidality, and transfer of model structures.

Abstract

We make strict $n$-categories even stricter by requiring they satisfy higher exchange laws governed by Hadzihasanovic's theory of regular directed complexes. We study the first properties of stricter $n$-categories, in particular, we define the Gray product, and prove stability under suspension, which is non-trivial. After reviewing and briefly expanding the theory diagrammatic sets and their associated model structures for $(\infty, n)$-categories, we construct a folk model structure on stricter $n$-categories, show that the walking equivalence coincides with the stricter polygraph generated by the walking equivalence in diagrammatic sets, and finally, that the folk model structure on stricter $n$-categories is right transferred from the diagrammatic model structure along a nerve construction.

Homotopy theory of stricter $n$-categories

TL;DR

The paper introduces stricter -categories by enforcing higher exchange laws via Hadzihasanovic's regular directed complexes, and develops foundational tools such as the Gray product and suspension. It constructs a folk model structure on stricter -categories and demonstrates that this structure is right transferred from the diagrammatic -model structure through a diagrammatic nerve, connecting strict models to diagrammatic set models of -categories. A key result is that the walking equivalence in stricter -categories aligns with the stricter polygraph generated by walking equivalences in diagrammatic sets, illuminating how these two formalisms mirror each other in the appropriate dimension range. The work lays groundwork for comparing the diagrammatic and standard models of -categories (notably for ) and establishes a coherent homotopy-theoretic framework for stricter higher categories, with a strong emphasis on functoriality, monoidality, and transfer of model structures.

Abstract

We make strict -categories even stricter by requiring they satisfy higher exchange laws governed by Hadzihasanovic's theory of regular directed complexes. We study the first properties of stricter -categories, in particular, we define the Gray product, and prove stability under suspension, which is non-trivial. After reviewing and briefly expanding the theory diagrammatic sets and their associated model structures for -categories, we construct a folk model structure on stricter -categories, show that the walking equivalence coincides with the stricter polygraph generated by the walking equivalence in diagrammatic sets, and finally, that the folk model structure on stricter -categories is right transferred from the diagrammatic model structure along a nerve construction.

Paper Structure

This paper contains 14 sections, 60 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Stricter ω-categories
  3. Definitions and properties
  4. Gray product of stricter ω-categories
  5. Strict and stricter categories
  6. Suspension of stricter ω-categories
  7. Diagrammatic sets
  8. Diagrams in a diagrammatic set
  9. Degenerate diagrams and equivalences
  10. Model structure for diagrammatic (∞, n)-categories
  11. Homotopy theory of stricter ω-categories
  12. Folk model structure on stricter ω-categories
  13. Coherent walking equivalence
  14. Right transfer from the diagrammatic model structures

Key Result

Theorem 1

Let $n \in \mathbb{N} \cup \left\{ {\omega} \right\}$. There exists a model structure, called the folk model structure, on the category ${n\mathbf{Cat}}^>$ of stricter $n$- categories. This model structure is right transferred along both:

Theorems & Definitions (143)

  • Theorem
  • Conjecture
  • Conjecture
  • Remark 2.3
  • Lemma 2.5
  • proof
  • Remark 2.7
  • Remark 2.15
  • Remark 2.16
  • Lemma 2.17
  • ...and 133 more