On truncated quasi-categories

TL;DR

This paper studies the model structure of n-truncated quasi-categories, establishing their construction via Bousfield localization and demonstrating Quillen equivalences with related models.

Contribution

It constructs the model structure for n-truncated quasi-categories as a Bousfield localization and proves Quillen equivalences with other models like Rezk's $(n,1)$-$\Theta$-spaces.

Findings

Model structure for n-truncated quasi-categories established.

Bousfield localization used to construct the model.

Quillen equivalences with categories and Rezk's spaces proved.

Abstract

For each $n \geq -1$, a quasi-category is said to be $n$-truncated if its hom-spaces are $(n-1)$-types. In this paper we study the model structure for $n$-truncated quasi-categories, which we prove can be constructed as the Bousfield localisation of Joyal's model structure for quasi-categories with respect to the boundary inclusion of the $(n+2)$-simplex. Furthermore, we prove the expected Quillen equivalences between categories and $1$-truncated quasi-categories and between $n$-truncated quasi-categories and Rezk's $(n,1)$-$\Theta$-spaces.