Canonicity and homotopy canonicity for cubical type theory

Abstract

Cubical type theory provides a constructive justification of homotopy type theory and satisfies canonicity: every natural number is convertible to a numeral. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices. In this paper we show by a sconing argument that if we remove these equations for the path lifting operation from the system, we still retain homotopy canonicity: every natural number is path equal to a numeral.

All $(\infty,1)$-toposes have strict univalent universes

“…This gives a model like that of [Coq12] in which every type belongs to a unique universe, which is the closest possible approximation to (non-cumulative) Russell-type universes obtainable in a natural model. Probably this construction could also be adapted to a semantic structure that models Russelltype universes more closely, such as the generalized algebraic theories of [Coq18,Ste19b,CHS19].…”