Synthetic fibered $(\infty,1)$-category theory

Abstract

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.

A Formal Logic for Formal Category Theory (Extended Version)

“…Our work in this paper fits broadly into a line of work on directed dependent type theories, a type theory where the identity type is interpreted as morphisms in a (possibly ∞-)category. In directed type theories based on a bisimplicial model [38,11,49,48], morphism types are defined using an interval object, like in cubical type theory [8,15,5,4], and universal properties like "morphism induction" are an internally definable property of certain types. Other type theories [35,1] define morphism types via an induction principle, corresponding to the lifting properties of certain kinds of fibrations of categories.…”

Strict stability of extension types

“…Indeed, continuing the program of [RS17], some parts of synthetic (∞, 1)-category theory have recently in simplicial HoTT been developed in [CRS18,BW21,Wei22,Mar22], and in [WL20] within a (bi-)cubical variant of the theory. Many of these developments are crucially inspired by results from Riehl-Verity's ∞-cosmos theory, a very general and powerful account to model-independent (∞, n)-category theory itself not formulated within type theory.…”

Cocartesian fibrations and straightening internal to an $\infty$-topos

“…Buchholtz and Weinberger [BW21] developed the theory of cocartesian fibrations in the framework of synthetic ∞-category theory, which is an extension of homotopy type theory that makes it possible to study higher categories from a type theoretic point of view. As homotopy type theory admits semantics in arbitrary ∞-topoi, their results recover most of what is covered in § 3.…”