From: Nicolai Kraus <nicola...@gmail.com>
To: "Daniel R. Grayson" <danielrich...@gmail.com>,
Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Re: [HoTT] Re: Conjecture
Date: Mon, 3 Apr 2017 10:56:33 +0100 [thread overview]
Message-ID: <20eea0d3-b602-9e0d-fb14-89c0cacbf24e@gmail.com> (raw)
In-Reply-To: <3016af27-384d-4b95-9a04-5c803e08608e@googlegroups.com>
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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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next prev parent reply other threads:[~2017-04-03 9:56 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 [this message]
2017-04-03 11:50 ` Daniel R. Grayson
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