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

[Nicolai Kraus <nicolai.kraus@gmail.com>]

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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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