Cartesian Fibrations of Complete Segal Spaces

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.