A Chevalley criterion for cartesian arrows
Emily Riehl – 20 January 2021
In "Fibrations and Yoneda's lemma in a 2-category" Ross Street proves a "Chevalley criterion" which gives two internal characterizations of fibrations in terms of the presence of certain right adjoints. In "Fibrations and Yoneda's lemma in an ∞-cosmos" we extend this to a characterization of cartesian fibrations between ∞-categories. We recently observed that cartesian arrows can be characterized similarly in terms of the presence of certain "relative" right adjoints, aka, absolute right lifting diagrams. The classical characterization of cartesian fibrations is a more-or-less immediate corollary, by considering the generic cartesian lift of the universal arrow. This is joint work with Dominic Verity.