Discussion of Homotopy Type Theory and Univalent Foundations
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From: "Joyal, André" <joyal.andre@uqam.ca>
To: Michael Shulman <shulman@sandiego.edu>,
	Urs Schreiber <urs.schreiber@googlemail.com>
Cc: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: RE: [HoTT] What is knot in HOTT?
Date: Fri, 20 Jul 2018 15:17:19 +0000
Message-ID: <8C57894C7413F04A98DDF5629FEC90B147A4521B@Pli.gst.uqam.ca> (raw)
In-Reply-To: <CAOvivQyz+Vy7TCzSWhE8d9MEkWn5Vcuh1rNb8hxs=h=N=LwtnA@mail.gmail.com>

Of course, braids, knots and tangles can be constructed algebraically
using braided monoidal categories.

________________________________________
From: homotopytypetheory@googlegroups.com [homotopytypetheory@googlegroups.com] on behalf of Michael Shulman [shulman@sandiego.edu]
Sent: Friday, July 20, 2018 10:54 AM
To: Urs Schreiber
Cc: Homotopy Type Theory
Subject: Re: [HoTT] What is knot in HOTT?

It seems to me that especially if we want to construct *particular*
knots, we would need the smooth reals to at least be a ring and
probably to support trigonometric functions.

On Fri, Jul 20, 2018 at 6:45 AM, 'Urs Schreiber' via Homotopy Type
Theory <HomotopyTypeTheory@googlegroups.com> wrote:
>> Once we have the "smooth real numbers", wouldn't we just define S^1
>> and S^3 in terms of them as usual?  Or are you saying that the problem
>> is in characterizing the smooth reals inside differential cohesion?
>
> Yes.
>
> Possibly one could make progress by declaring shape to be homotopy
> localization at some type A^1 of which we only demand that it be
> homogeneous (as in Def. 4.8 in arxiv.org/abs/1806.05966) and then
> focus attention on A^n-manifolds (as in Def. 7.1).
>
> One could maybe declare that a smooth n-sphere to be an A^n-manifold
> whose shape is equivalent to Disc(S^n). Classically, this should work
> away from dimensions in which there are exotic spheres, hence in
> particular for the case n <= 3 of relevance in knot theory.
>
> Best wishes,
> urs
>
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  reply index

Thread overview: 16+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2018-07-19  5:18 José Manuel Rodriguez Caballero
2018-07-19  5:45 ` Egbert Rijke
2018-07-19  8:55   ` Ali Caglayan
2018-07-19 15:31     ` Michael Shulman
2018-07-20 10:27       ` 'Urs Schreiber' via Homotopy Type Theory
2018-07-20 13:32         ` Michael Shulman
2018-07-20 13:45           ` 'Urs Schreiber' via Homotopy Type Theory
2018-07-20 14:54             ` Michael Shulman
2018-07-20 15:17               ` Joyal, André [this message]
2018-07-20 16:40               ` 'Urs Schreiber' via Homotopy Type Theory
2018-07-20 16:42                 ` 'Urs Schreiber' via Homotopy Type Theory
2019-11-20 19:13     ` Ali Caglayan
2019-11-20 21:02       ` andré hirschowitz
2018-07-19 17:56   ` Daniel R. Grayson
2018-07-19 18:38     ` Egbert Rijke
2018-07-19 20:07       ` José Manuel Rodriguez Caballero

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Discussion of Homotopy Type Theory and Univalent Foundations

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