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

Abstract

Abstract. 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 (∞, 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.

A survey of models for (∞, n)-categories

“…Let us now show that we have the desired model categories. In the case in which we enrich in the complete Segal space model structure CSS, a full proof applying this strategy is given by the author and Rezk in [14]. Essentially the same proof technique can be used to obtain a similar result for enriching in SeCat c .…”

“…Theorem 6.11. [15] There is a model structure CSS(CSS) on the category of bisimplical spaces in which the fibrant objects are precisely the 2-fold complete Segal spaces.…”

“…Theorem 6.18. [14], [15] (1) The bisimplicial nerve functor induces a Quillen equivalence CSS − Cat ⇆ SeCat f (CSS).…”

On the equivalence between $\Theta _{n}$-spaces and iterated Segal spaces

“…They also prove that these axioms are satisfied in the case of n-fold Segal spaces and n -spaces, which implies that these two models are equivalent. Another comparison, which relates the two models directly in the setting of model categories, has been given more recently by Bergner and Rezk [BR14].…”

Higher Quasi-Categories vs Higher Rezk Spaces

“…We will call these objects Rezk Θ n -spaces in this paper. Another model based on Rezk Θ n -spaces has been introduced by Bergner and Rezk in [17]. A second generalization of complete Segal spaces called n-fold Segal spaces has been introduced by Barwick (see Section 12 of [8]).…”