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arXiv · 1112.0040

On the Unicity of the Homotopy Theory of Higher Categories

Clark Barwick·Christopher Schommer-Pries
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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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Clark Barwick, Christopher Schommer-Pries. 2011-11-30. On the Unicity of the Homotopy Theory of Higher Categories. https://arxiv.org/abs/1112.0040

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