Comparison of models for $(\infty, n)$-categories, I

TL;DR

This paper develops and compares various models for $( abla, n)$-categories, establishing model structures and Quillen equivalences to unify different approaches in higher category theory.

Contribution

It introduces model structures for categories modeling $( abla, n)$-categories and proves their Quillen equivalences, extending the known results for $( abla, 1)$-categories.

Findings

Established model structures for $( abla, n)$-categories

Proved Quillen equivalences between different models

Unified the framework for higher $( abla, n)$-categories

Abstract

While many different models for $(\infty,1)$-categories are currently being used, it is known that they are Quillen equivalent to one another. Several higher-order analogues of them are being developed as models for $(\infty, n)$-categories. In this paper, we establish model structures for some naturally arising categories of objects which should be thought of as $(\infty,n)$-categories. Furthermore, we establish Quillen equivalences between them.