Higher quasi-categories vs higher Rezk spaces

TL;DR

This paper introduces n-quasi-categories as fibrant objects in a model category on presheaves over Joyal's n-cell category, establishing their equivalence with Rezk tz spaces to model (, n)-categories.

Contribution

It defines n-quasi-categories via a modified approach inspired by Cisinski and Joyal and proves their equivalence with Rezk tz spaces, generalizing known results for n=1.

Findings

Constructed two Quillen equivalences between n-quasi-categories and Rezk tz spaces.

Showed n-quasi-categories model (, n)-categories.

Recovered known equivalences for n=1 between quasi-categories and complete Segal spaces.

Abstract

We introduce a notion of n-quasi-categories as fibrant objects of a model category structure on presheaves on Joyal's n-cell category \Theta_n. Our definition comes from an idea of Cisinski and Joyal. However, we show that this idea has to be slightly modified to get a reasonable notion. We construct two Quillen equivalences between the model category of n-quasi-categories and the model category of Rezk \Theta_n-spaces showing that n-quasi-categories are a model for (\infty, n)-categories. For n = 1, we recover the two Quillen equivalences defined by Joyal and Tierney between quasi-categories and complete Segal spaces.