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

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

[Jasper Hugunin]

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

[Nicolai Kraus]

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

[Nicolai Kraus]

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Small addition to my first remark:

On 07/09/18 07:14, Nicolai Kraus wrote:
> 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.

Since we don't need an internal argument, it should be possible to use 
S^2 together with Thierry's result in Christian's post
https://groups.google.com/forum/#!topic/homotopytypetheory/imPb56IqxOI
But this is only for CCHM type theory.
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.
>

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

[Nicolai Kraus]

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The addition from my second email didn't make sense, sorry. The first email 
with the easy argument should be correct.
Nicolai 

On Friday, September 7, 2018 at 7:30:23 AM UTC+1, Nicolai Kraus wrote:
>
> Small addition to my first remark:
>
> On 07/09/18 07:14, Nicolai Kraus wrote:
>
> 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. 
>
>
> Since we don't need an internal argument, it should be possible to use S^2 
> together with Thierry's result in Christian's post
> https://groups.google.com/forum/#!topic/homotopytypetheory/imPb56IqxOI
> But this is only for CCHM type theory.
> Nicolai
>

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

[Thorsten Altenkirch]

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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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