Abstract. In this paper we put a cofibrantly generated model category structure on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories.
In this paper we give a summary of the comparisons between different definitions of so-called (∞, 1)-categories, which are considered to be models for ∞-categories whose n-morphisms are all invertible for n > 1. They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories. The four different structures, all of which are equivalent, are simplicial categories, Segal categories, complete Segal spaces, and quasi-categories.
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.
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