Strict stability of extension types

Abstract

The theory of $(\infty,1)$-categories can be developed synthetically in an augmentation of homotopy type theory introduced by Riehl--Shulman. Central to their development is an additional type forming operation called extensions. The original article sketches the semantics of this formal system, explaining how the simplicial homotopy theory can be used to reason about $(\infty,1)$-categories presented using the Segal space model. However, they leave it open to demonstrate the strict stability of extension types. We prove this using the splitting method of Voevodsky, later generalized by Lumsdaine--Warren to local universes. The practical upshot is that this system has semantics in simplicial objects of an $\infty$-topos, and thus can be used to prove theorems about internal $\infty$-categories in the sense of Martini--Wolf.