* Re: [HoTT] Looking for a reference that HITs are a strict extension of HoTT
2018-09-07 3:56 [HoTT] Looking for a reference that HITs are a strict extension of HoTT Jasper Hugunin
@ 2018-09-07 6:14 ` Nicolai Kraus
2018-09-07 6:30 ` Nicolai Kraus
2018-09-07 12:38 ` Thorsten Altenkirch
1 sibling, 1 reply; 5+ messages in thread
From: Nicolai Kraus @ 2018-09-07 6:14 UTC (permalink / raw)
To: Jasper Hugunin, HomotopyTypeTheory
[-- Attachment #1: Type: text/plain, Size: 2238 bytes --]
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
>
> --
> You received this message because you are subscribed to the Google
> Groups "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send
> an email to HomotopyTypeTheory+unsubscribe@googlegroups.com
> <mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com>.
> For more options, visit https://groups.google.com/d/optout.
--
You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.
[-- Attachment #2: Type: text/html, Size: 3334 bytes --]
^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: [HoTT] Looking for a reference that HITs are a strict extension of HoTT
2018-09-07 3:56 [HoTT] Looking for a reference that HITs are a strict extension of HoTT Jasper Hugunin
2018-09-07 6:14 ` Nicolai Kraus
@ 2018-09-07 12:38 ` Thorsten Altenkirch
1 sibling, 0 replies; 5+ messages in thread
From: Thorsten Altenkirch @ 2018-09-07 12:38 UTC (permalink / raw)
To: Jasper Hugunin, homotopytypetheory
[-- Attachment #1: Type: text/plain, Size: 2714 bytes --]
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
--
You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com<mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com>.
For more options, visit https://groups.google.com/d/optout.
This message and any attachment are intended solely for the addressee
and may contain confidential information. If you have received this
message in error, please contact the sender and delete the email and
attachment.
Any views or opinions expressed by the author of this email do not
necessarily reflect the views of the University of Nottingham. Email
communications with the University of Nottingham may be monitored
where permitted by law.
--
You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.
[-- Attachment #2: Type: text/html, Size: 7506 bytes --]
^ permalink raw reply [flat|nested] 5+ messages in thread