A Model for the Higher Category of Higher Categories

Abstract

We use fibrations of complete Segal spaces to construct four complete Segal spaces: Reedy fibrant simplicial spaces, Segal spaces, complete Segal spaces, and spaces. Moreover, we show each one comes with a universal fibration that classifies Reedy left fibrations, Segal coCartesian fibrations, coCartesian fibrations and left fibrations and prove these are representable fibrations. Finally, we use equivalences between quasi-categories and complete Segal spaces to present analogous constructions using fibrations of quasi-categories.