A Synthetic Perspective on $(\infty,1)$-Category Theory: Fibrational and Semantic Aspects

Abstract

Reasoning about weak higher categorical structures constitutes a challenging task, even to the experts. One principal reason is that the language of set theory is not invariant under the weaker notions of equivalence at play, such as homotopy equivalence. From this point of view, it is natural to ask for a different foundational setting which more natively supports these notions. Our work takes up on suggestions in the original article arXiv:1705.07442 by Riehl--Shulman to further develop synthetic $(\infty,1)$-category theory in simplicial homotopy type theory, including in particular the study of cocartesian fibrations. Together with a collection of analytic results, notably due to Riehl--Verity and Rasekh, it follows that our type-theoretic account constitutes a synthetic theory of fibrations of internal $(\infty,1)$-categories, w.r.t. to an arbitrary Grothendieck--Rezk--Lurie-$(\infty,1)$-topos via Shulman's major result arXiv:1904.07004 about strictification of univalent universes.