Cartesian fibrations and representability

We use the complete Segal approach to the theory of Cartesian fibrations we developed previously 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 fibrations, 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 fundamental theorem of complete Segal objects, which characterizes equivalences of complete Segal objects. Finally we give two applications of the results. First, we present a method to construct Segal objects and second we study the representability of the universal co-Cartesian fibration.

Keywords

higher category theory, Cartesian fibration, internal ∞-category, complete Segal object

2010 Mathematics Subject Classification

18F20

Full Text (PDF format)

Received 25 February 2021

Received revised 30 July 2021

Accepted 31 July 2021

Published 10 August 2022