A cartesian presentation of weak $n$–categories

Abstract

We propose a notion of weak $\left(n+k,n\right)$–category, which we call $\left(n+k,n\right)$–$\Theta$–spaces. The $\left(n+k,n\right)$–$\Theta$–spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal’s category ${\Theta }_n$. This notion is a generalization of that of complete Segal spaces (which are precisely the $\left(\infty ,1\right)$–$\Theta$–spaces). Our main result is that the above model category is cartesian.