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

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.