# On the Unicity of the Homotopy Theory of Higher Categories

@article{Barwick2011OnTU, title={On the Unicity of the Homotopy Theory of Higher Categories}, author={Clark Barwick and Christopher J. Schommer-Pries}, journal={arXiv: Algebraic Topology}, year={2011} }

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…

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