Two-sided cartesian fibrations of synthetic $(\infty ,1)$-categories

Abstract

Within the framework of Riehl-Shulman's synthetic $(\infty ,1)$-category theory, we present a theory of two-sided cartesian fibrations. Central results are several characterizations of the two-sidedness condition à la Chevalley, Gray, Street, and Riehl-Verity, a two-sided Yoneda Lemma, as well as the proof of several closure properties. Along the way, we also define and investigate a notion of fibered or sliced fibration which is used later to develop the two-sided case in a modular fashion. We also briefly discuss discrete two-sided cartesian fibrations in this setting, corresponding to $(\infty ,1)$-distributors. The systematics of our definitions and results closely follows Riehl-Verity's $\infty$-cosmos theory, but formulated internally to Riehl-Shulman's simplicial extension of homotopy type theory. All the constructions and proofs in this framework are by design invariant under homotopy equivalence. Semantically, the synthetic $(\infty ,1)$-categories correspond to internal $(\infty ,1)$-categories implemented as Rezk objects in an arbitrary given $(\infty ,1)$-topos.

arXiv comment76 pages. This text is essentially Chapter 5 and Appendices A and B from author's PhD thesis arXiv:2202.13132. Updated as per referee comments