We study cocartesian fibrations in the setting of the synthetic (∞, 1)-category theory developed in simplicial type theory introduced by Riehl and Shulman. Our development culminates in a Yoneda Lemma for cocartesian fibrations.

Casts and crusts of the tympanic membrane were removed from patients with previous otitis media. Casts are formed of adherent keratinocytes in a matrix of dried exudate. Detachment from the drumhead leaves the underlying epidermis intact. Casts are defined as those crusts that have detached spontaneously from the surface of the drum. A mechanism based on epithelial proliferation induced by inflammation within the middle ear, with subsequent keratinocyte retention and thickening, is proposed.

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We give structural results about bifibrations of (internal) (∞, 1)categories with internal sums. This includes a higher version of Moens' Theorem, characterizing cartesian bifibrations with extensive aka stable and disjoint internal sums over lex bases as Artin gluings of lex functors.We also treat a generalized version of Moens' Theorem due to Streicher which does not require the Beck-Chevalley condition.Furthermore, we show that also in this setting the Moens fibrations can be characterized via a condition due to Zawadowski.Our account overall follows Streicher's presentation of fibered category theory à la Bénabou, generalizing the results to the internal, higher-categorical case, formulated in a synthetic setting.Namely, we work inside simplicial homotopy type theory, which has been introduced by Riehl and Shulman as a logical system to reason about internal (∞, 1)-categories, interpreted as Rezk objects in any given Grothendieck-Rezk-Lurie (∞, 1)-topos.

As observed by various people recently the topos sSet of simplicial sets appears as essential subtopos of a topos cSet of cubical sets, namely presheaves over the category FL of finite lattices and monotone maps between them. The latter is a variant of the cubical model of type theory due to Cohen et al. for the purpose of providing a model for a variant of type theory which validates Voevodsky's Univalence Axiom and has computational meaning.One easily shows that sheaves, i.e. simplicial sets, are closed under most of the type theoretic operations of cSet. Finally, we construct in cSet a fibrant univalent universe for those types that are sheaves.