Modeling $(\infty ,1)$-categories with Segal spaces

Abstract

In this paper, we construct a model structure for $(\infty ,1)$-categories on the category of simplicial spaces, whose fibrant objects are the Segal spaces. In particular, we show that it is Quillen equivalent to the models of $(\infty ,1)$-categories given by complete Segal spaces and Segal categories. We furthermore prove that this model structure has desirable properties: it is cartesian closed and left proper. As applications, we get a simple description of the inclusion of categories into $(\infty ,1)$-categories and of homotopy limits of $(\infty ,1)$-categories.

21 pages; final version to appear in Bulletin of the London Mathematical Society