On the Unicity of the Homotopy Theory of Higher Categories

TL;DR

This paper axiomatizes the theory of $( abla,n)$-categories and proves their equivalence across various models, showing the space of such theories is a $B( ext{Z}/2)^n$, thus establishing a unicity result.

Contribution

It introduces axioms for $( abla,n)$-categories and demonstrates the equivalence of multiple existing models, with the space of theories characterized as a $B( ext{Z}/2)^n$.

Findings

All considered models satisfy the axioms.

Theories are equivalent up to an action of $( ext{Z}/2)^n$.

The space of theories is a $B( ext{Z}/2)^n$.

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 $\Theta_n$-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 $(\mathbb{Z}/2)^n$.