Groupoidal and truncated $n$-quasi-categories

TL;DR

This paper introduces new models for $n$-quasi-categories that are groupoidal and truncated, establishing their properties within specific model structures and relating them to spaces and homotopy types.

Contribution

It defines groupoidal and truncated $n$-quasi-categories and proves their equivalence to spaces and homotopy $n$-types via Quillen equivalences.

Findings

Established model structures for groupoidal and truncated $n$-quasi-categories.

Proved Quillen equivalences with spaces and homotopy types.

Constructed a cylinder object for $n$-quasi-categories.

Abstract

We define groupoidal and $(n+k)$-truncated $n$-quasi-categories, which are the translation to the world of $n$-quasi-categories of groupoidal and truncated $(\infty, n)$-$\Theta$-spaces defined by Rezk. We show that these objects are the fibrant objects of model structures on the category of presheaves on $\Theta_n$ obtained by localisation of Ara's model structure for $n$-quasi-categories. Furthermore, we prove that the inclusion $\Delta \to \Theta_n$ induces a Quillen equivalence between the model structure for groupoidal (resp. and $n$-truncated) $n$-quasi-categories and the Kan-Quillen model structure for spaces (resp. homotopy $n$-types) on simplicial sets. To get to these results, we also construct a cylinder object for $n$-quasi-categories.