A Model for the Higher Category of Higher Categories

TL;DR

This paper constructs a hierarchy of models for higher categories using fibrations of complete Segal spaces and quasi-categories, establishing universal fibrations and their representability.

Contribution

It introduces a comprehensive framework connecting various models of higher categories via fibrations and demonstrates their equivalences.

Findings

Construction of four complete Segal spaces with universal fibrations

Proof that these fibrations are representable

Establishment of equivalences between models using fibrations

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.