A cartesian presentation of weak n-categories

TL;DR

This paper introduces (n+k,n)-Theta-spaces as a new model for weak n-categories, generalizing complete Segal spaces, and proves that the associated model category is cartesian, facilitating compositional reasoning.

Contribution

It defines a new class of weak n-categories called (n+k,n)-Theta-spaces and proves the model category structure on presheaves over Theta_n is cartesian, extending existing models.

Findings

The (n+k,n)-Theta-spaces precisely model weak (n+k,n)-categories.

The model category of presheaves on Theta_n is shown to be cartesian.

This framework generalizes complete Segal spaces to higher dimensions.

Abstract

We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-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 (infty,1)-Theta-spaces). Our main result is that the above model category is cartesian.