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On the Unicity of the Homotopy Theory of Higher Categories

Clark Barwick, Christopher Schommer-Pries

TL;DR

The paper develops a rigorous, axiomatic foundation for the homotopy theory of $(\infty,n)$-categories, proving a unicity result: the moduli space Thy$_{(\infty,n)}$ is canonically $B(\mathbb{Z}/2)^n$ and any two theories satisfying the axioms are equivalent up to the $n$ independent oppositions. Central to the approach is the construction of a colossal model Cat$_{(\infty,n)}$ that satisfies the axioms and serves as a universal reference; together with versality, this yields equivalences with all other theories. The authors then verify that major models (Rezk’s complete Segal $\Theta_n$-spaces, $n$-fold complete Segal spaces, and relatedSegal-type formalisms) satisfy the axioms and are equivalent to the colossal model up to the $(\mathbb{Z}/2)^n$-action, implying a robust unification of the homotopy theory of higher categories. The paper also provides a recognition principle for presentations and discusses consequences for model categories and Quillen equivalences, establishing a coherent, universal landscape for comparing constructions of $(\infty,n)$-categories across frameworks.

Abstract

We axiomatise the theory of $(\infty,n)$-categories. We prove that the space of theories of $(\infty,n)$-categories is a $B(\mathbb{Z}/2)^n$. We prove that Rezk's complete Segal $Θ_n$-spaces, Simpson and Tamsamani's Segal $n$-categories, the first author's $n$-fold complete Segal spaces, Kan and the first author's $n$-relative categories, and complete Segal space objects in any model of $(\infty,n-1)$-categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of $(\mathbb{Z}/2)^n$.

On the Unicity of the Homotopy Theory of Higher Categories

TL;DR

The paper develops a rigorous, axiomatic foundation for the homotopy theory of -categories, proving a unicity result: the moduli space Thy is canonically and any two theories satisfying the axioms are equivalent up to the independent oppositions. Central to the approach is the construction of a colossal model Cat that satisfies the axioms and serves as a universal reference; together with versality, this yields equivalences with all other theories. The authors then verify that major models (Rezk’s complete Segal -spaces, -fold complete Segal spaces, and relatedSegal-type formalisms) satisfy the axioms and are equivalent to the colossal model up to the -action, implying a robust unification of the homotopy theory of higher categories. The paper also provides a recognition principle for presentations and discusses consequences for model categories and Quillen equivalences, establishing a coherent, universal landscape for comparing constructions of -categories across frameworks.

Abstract

We axiomatise the theory of -categories. We prove that the space of theories of -categories is a . We prove that Rezk's complete Segal -spaces, Simpson and Tamsamani's Segal -categories, the first author's -fold complete Segal spaces, Kan and the first author's -relative categories, and complete Segal space objects in any model of -categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of .

Paper Structure

This paper contains 18 sections, 52 theorems, 148 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on gaunt $n$-categories
  3. Strict $n$-categories
  4. Gaunt $n$-categories
  5. Endomorphisms and Automorphisms of $\mathop{\mathrm{Gaunt}}\nolimits_n$
  6. Correspondences and the category $\mathop{\mathrm{Gaunt}}\nolimits_n$
  7. The categories $\Theta_n$ and $\Upsilon_n$
  8. The moduli space of theories of $(\infty,n)$-categories
  9. The Unicity Theorem
  10. A colossal model of $(\infty,n)$-categories
  11. The connectedness of the space of theories
  12. The loopspace of the space of theories
  13. A recognition principle for categorical presentations
  14. Presentations of $(\infty,n)$-categories
  15. Strict $n$-categories as presheaves of sets
  16. ...and 3 more sections

Key Result

Theorem 1.1

The moduli space $\mathop{\mathrm{Thy}}\nolimits_{(\infty,n)}$ of theories of $(\infty,n)$-categories is a $B(\mathbb{Z}/2)^n$.

Figures (1)

  • Figure 1: A graphical depiction of a typical map (shown in red) $\delta: [p] \to [m] \times_{[1]} [n]$.

Theorems & Definitions (138)

  • Theorem 1.1: Unicity
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Proposition 2.7
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 128 more