Cartesian Fibrations of Complete Segal Spaces

TL;DR

This paper develops a new framework for Cartesian fibrations within complete Segal spaces, providing model structures and tools to better understand presheaves valued in higher categories.

Contribution

It introduces a model structure for fibrations of simplicial spaces, enabling new constructions of Cartesian fibrations via complete Segal spaces.

Findings

Established a model structure for fibrations of simplicial spaces.

Provided criteria to recognize fibrations and weak equivalences.

Enabled new methods to construct Cartesian fibrations.

Abstract

Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in $(\infty,1)$-category theory to study presheaves valued in $(\infty,1)$-categories. In this work we define and study fibrations modeling presheaves valued in simplicial spaces and their localizations. This includes defining a model structure for these fibrations and giving effective tools to recognize its fibrations and weak equivalences. This in particular gives us a new method to construct Cartesian fibrations via complete Segal spaces. In addition to that, it allows us to define and study fibrations modeling presheaves of Segal spaces.