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

TL;DR

This paper develops a new model structure for $( abla,1)$-categories using Segal spaces, establishing equivalences with existing models and demonstrating desirable categorical properties.

Contribution

It introduces a novel model structure for $( abla,1)$-categories on simplicial spaces, proving its equivalence to known models and its cartesian closed, left proper nature.

Findings

Model structure for $( abla,1)$-categories constructed

Quillen equivalence with existing models established

Model structure shown to be cartesian closed and left proper

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.