From: "Daniel R. Grayson" <danielrich...@gmail.com>
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Cc: danielrich...@gmail.com
Subject: Re: [HoTT] Re: Conjecture
Date: Mon, 3 Apr 2017 04:50:26 -0700 (PDT) [thread overview]
Message-ID: <2a6ce7aa-99c8-48a5-bd16-bbcff260b0b9@googlegroups.com> (raw)
In-Reply-To: <20eea0d3-b602-9e0d-fb14-89c0cacbf24e@gmail.com>
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Thanks, Nicolai and Favonia. It's a nice result, Nicolai!
On Monday, April 3, 2017 at 5:56:36 AM UTC-4, Nicolai Kraus wrote:
>
> Dan, this is one instance of my "general universal property of the
> propositional truncation", arXiv:1411.2682.
> In that paper I show that, if you fix a number n, then for a type Y of
> h-level n, the function type
> ||X|| -> Y
> is equivalent to the Sigma-type with the following components:
> (1) a function X -> Y
> (2) the condition that (1) is weakly constant
> (3) a coherence condition for (2)
> (4) a coherence condition for (3)
> ...
> (n) a coherence condition for (n-1)
>
> This can be presented as a natural transformation between semi-simplicial
> types.
> What you have formalized is the case n=3 (one direction).
> (In my presentation, I don't use the component "c x x = idpath (f x)"
> because it can be inferred,
> and if you go higher than 3, this component would make additional
> coherence conditions necessary.)
> Nicolai
>
> On 03/04/17 01:35, Daniel R. Grayson wrote:
>
> Here's a fact related to the current discussion, which I have formalized
> today. I would appreciate knowing
> whether it's already known. It gives a criterion for factoring through
> the propositional truncation
> when the target is of h-level 3.
>
> Definition squash_to_HLevel_3 {X : UU} {Y : HLevel 3}
> (f : X -> Y) (c : ∏ x x', f x = f x') :
> (∏ x, c x x = idpath (f x)) ->
> (∏ x x' x'', c x x'' = c x x' @ c x' x'') ->
> ∥ X ∥ -> Y.
>
> You may find it in WellOrderedSets.v
> <https://github.com/DanGrayson/UniMath/blob/well-ordering-2/UniMath/Combinatorics/WellOrderedSets.v> on
> one of my branches.
>
>
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prev parent reply other threads:[~2017-04-03 11:50 UTC|newest]
Thread overview: 22+ messages / expand[flat|nested] mbox.gz Atom feed top
2017-03-27 21:57 Conjecture Martin Escardo
2017-03-29 21:08 ` [HoTT] Conjecture Nicolai Kraus
2017-03-29 22:05 ` Martin Escardo
2017-03-30 10:59 ` Michael Shulman
2017-03-30 19:22 ` Egbert Rijke
2017-03-30 23:02 ` Nicolai Kraus
2017-03-30 22:49 ` Nicolai Kraus
2017-03-31 16:09 ` Martin Escardo
2017-04-05 19:37 ` Martin Escardo
2017-04-06 0:23 ` Jon Sterling
2017-04-06 5:55 ` Martin Escardo
2017-04-06 12:40 ` Vladimir Voevodsky
2017-04-06 13:50 ` Martin Escardo
[not found] ` <81c0782f-9287-4111-a4f1-01cb9c87c7e8@cs.bham.ac.uk>
2017-04-06 16:09 ` Martin Escardo
2017-04-06 11:52 ` Thomas Streicher
2017-04-07 9:49 ` Martin Escardo
2017-04-07 17:11 ` Michael Shulman
2017-04-07 18:10 ` Martin Escardo
2017-04-03 0:35 ` Conjecture Daniel R. Grayson
2017-04-03 2:20 ` [HoTT] Conjecture Favonia
2017-04-03 9:56 ` Nicolai Kraus
2017-04-03 11:50 ` Daniel R. Grayson [this message]
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