Comparison of models for (∞, n)-categories, I

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