Within the framework of Riehl-Shulman's synthetic (∞, 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 (∞, 1)-distributors.The systematics of our definitions and results closely follows Riehl-Verity's ∞-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 (∞, 1)-categories correspond to internal (∞, 1)-categories implemented as Rezk objects in an arbitrary given (∞, 1)-topos.
We show that the extension types occurring in Riehl-Shulman's work on synthetic (∞, 1)-categories can be interpreted in the intended semantics in a way so that they are strictly stable under substitution. The splitting method used here is due to Voevodsky in 2009. It was later generalized by Lumsdaine-Warren to the method of local universes.
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