On the unicity of the theory of higher categories

We axiomatise the theory of ( ∞ , n ) (\infty ,n) -categories. We prove that the space of theories of ( ∞ , n ) (\infty ,n) -categories is a B ( Z / 2 ) n B(\mathbb {Z}/2)^n . We prove that Rezk’s complete Segal Θ n \Theta _n spaces, Simpson and Tamsamani’s Segal n n -categories, the first author’s n n -fold complete Segal spaces, Kan and the first author’s n n -relative categories, and complete Segal space objects in any model of ( ∞ , n − 1 ) (\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 ( Z / 2 ) n (\mathbb {Z}/2)^n .