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

Abstract

While many different models for $\left(\infty ,1\right)$–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 $\left(\infty ,n\right)$–categories. In this paper, we establish model structures for some naturally arising categories of objects which should be thought of as $\left(\infty ,n\right)$–categories. Furthermore, we establish Quillen equivalences between them.