Quasi-categories vs. Segal spaces: Cartesian edition

Abstract

We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent:

1. 1.

   On marked simplicial sets (due to Lurie [31]),

2. 2.

   On bisimplicial spaces (due to deBrito [12]),

3. 3.

   On bisimplicial sets,

4. 4.

   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.