Hi Jasper,

here's an argument: Without HITs, it's consistent to assume that every type in U_n is an n-type (since, as you said, all type formers preserve h-level). But with HIT's, consider the type
  Sigma (k: Nat), S^k.
This is not a k-type for any k since the k-th fundamental group is nontrivial if you choose the base point correctly  (see Licata-Brunerie CPP 2013).

Remarks: 1. If we knew that S^2 is not a k-type for any k, then this would work as well for the second step, but as you said, we don't know so far whether this can be shown in HoTT.
2. For more general universe hierarchies than the one you use, for example indexed over omega+1 or indexed over any poset of arbitrary height, my argument won't work; I can't think of a proof for that situation off the top of my head.

Nicolai


On 07/09/18 04:56, Jasper Hugunin wrote:
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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