Modeling $(\infty,1)$-categories with Segal spaces
Lyne Moser, Joost Nuiten
TL;DR
The paper addresses constructing a workable model for (∞,1)-categories using Segal spaces. It introduces a cofibrantly generated model structure on simplicial spaces, the categorical model structure, whose fibrant objects are Segal spaces and whose weak equivalences are Dwyer–Kan equivalences, with cofibrations reflecting a discreteness condition on the object space. The main result is a Quillen equivalence between this structure and the complete Segal space model, placing it between Segal categories and complete Segal spaces. The model is cartesian closed and left proper, enabling straightforward descriptions of inclusions of ordinary categories into (∞,1)-categories and the computation of homotopy limits of (∞,1)-categories.
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.