Cartesian Fibrations and Representability

Abstract

We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in $\infty$-category theory. In particular, we give a construction of representable Cartesian fibrations using over-categories and prove the Yoneda lemma for representable Cartesian fibration, which generalizes the established Yoneda lemma for right fibrations. We then use the theory of Cartesian fibrations to study complete Segal objects internal to an $\infty$-category. Concretely, we prove the {\it fundamental theorem of complete Segal objects}, which characterizes equivalences of complete Segal objects. Finally we give two application of the results. First, we present a method to construct Segal objects and second we study the representability of the universal coCartesian fibration.