On the Unicity of the Homotopy Theory of Higher Categories

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

47 pages. Revised in response to a referee report

References in corpus (2)

• Higher Topos Theory
• n-Relative Categories

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