Science
Strict stability of extension types
Key Points
Announce Type: replace-cross 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.
arXiv:2203.07194v5 Announce Type: replace-cross
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.