A Quillen adjunction between globular and complicial approaches to $(\infty,n)$-categories

TL;DR

This paper establishes a Quillen adjunction linking two models of $( abla,n)$-categories, enhancing the understanding of their compatibility and providing a bridge between globular and complicial approaches.

Contribution

It proves the compatibility of suspension and complicial nerve constructions and constructs a Quillen pair connecting Rezk's and Verity's models of $( abla,n)$-categories.

Findings

Proves compatibility between suspension and complicial nerve.

Constructs a Quillen pair between two models of $( abla,n)$-categories.

Bridges globular and complicial approaches to higher categories.

Abstract

We prove the compatibility between the suspension construction and the complicial nerve of $\omega$-categories. As a motivating application, we produce a Quillen pair between the models of $(\infty,n)$-categories given by Rezk's complete Segal $\Theta_n$-spaces and Verity's $n$-complicial sets.