Quasi-Categories vs. Segal Spaces: Cartesian Edition

TL;DR

This paper establishes the equivalence of four different models for Cartesian fibrations and the Cartesian model structure using Quillen equivalences, unifying various approaches in higher category theory.

Contribution

It proves that four distinct definitions of Cartesian fibrations and their model structures are all Quillen equivalent, connecting quasi-categories and Segal spaces through known equivalences.

Findings

All four definitions of Cartesian fibrations are Quillen equivalent.

The equivalences are established via known Quillen equivalences between quasi-categories and Segal spaces.

The work unifies different models in higher category theory for Cartesian fibrations.

Abstract

We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets, on bisimplicial spaces, on bisimplicial sets, on marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal-Tierney and the straightening construction due to Lurie.