On the unicity of the theory of higher categories

Barwick, Clark; Schommer-Pries, Christopher

doi:10.1090/jams/972

On the unicity of the theory of higher categories
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by Clark Barwick and Christopher Schommer-Pries

J. Amer. Math. Soc. 34 (2021), 1011-1058

DOI: https://doi.org/10.1090/jams/972

Published electronically: April 20, 2021

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

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