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

[Thorsten Altenkirch <Thorsten.Altenkirch@nottingham.ac.uk>]

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Don’t you need at least some sort of quotients? How do you define the Cauchy Reals otherwise?

Ok using resizing (not recommended) you can encode quotients (as in a topos).

However quotients are not enough.

In https://arxiv.org/abs/1705.07088,  Lumsdaine and Shulman Section 9 given an example based on a construction by Blass which shows that there are QITs (set truncated HITs) that are not definable using quotients.

I say “likely” because I think that their construction doesn’t allow for univalence. On the other hand I don’t see a way how to define their counterexample using univalence either.

Thorsten

From: <homotopytypetheory@googlegroups.com> on behalf of Jasper Hugunin <jasperh@cs.washington.edu>
Date: Friday, 7 September 2018 at 04:56
To: "homotopytypetheory@googlegroups.com" <homotopytypetheory@googlegroups.com>
Subject: [HoTT] Looking for a reference that HITs are a strict extension of HoTT

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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