Strict stability of extension types

Abstract

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.

Two-sided cartesian fibrations of synthetic $(\infty,1)$-categories

“…By the semantics of simplicial HoTT [48,56,74] in (∞, 1)-toposes of the form E op for an (∞, 1)-topos E, the synthetic (∞, 1)-categories in our theory are interpreted as internal (∞, 1)-categories in E, i.e. Rezk objects.…”