[HoTT] Looking for a reference that HITs are a strict extension of HoTT

[Jasper Hugunin <jasperh@cs.washington.edu>]

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Hello all,

Many ways of doing HoTT (Coq + Univalence Axiom, Cubical Type Theory) make
sense without including support for defining Higher Inductive Types. The
possibility of defining small, closed types which are not hsets (like the
circle) or have infinite h-level (like the 2-sphere, conjectured?) makes
constructing HITs from other types seem difficult, since all the type
formers except universes preserve h-level.

Does anyone know a proof that it is impossible to construct some HITs from
basic type formers (say 0, 1, 2, Sigma, Pi, W, and a hierarchy of univalent
universes U_n), up to equivalence?

- Jasper Hugunin

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