A MODEL FOR THE HIGHER CATEGORY OF HIGHER CATEGORIES

We use fibrations of complete Segal spaces as introduced in [Ras22, Ras23a] 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 in the sense of [Ras22]. Finally, we use equivalences between quasi-categories and complete Segal spaces constructed in [JT07, Ras21a] to present analogous constructions using fibrations of quasi-categories. As part of establishing the results, we also develop a theory of minimal Reedy fibrations for elegant Reedy categories, which can be of independent interest.