Two-sided cartesian fibrations of synthetic (∞,1)-categories
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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.
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