Discussion of Homotopy Type Theory and Univalent Foundations
 help / color / mirror / Atom feed
From: Michael Shulman <shulman@sandiego.edu>
To: 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 07:54:54 -0700	[thread overview]
Message-ID: <CAOvivQyz+Vy7TCzSWhE8d9MEkWn5Vcuh1rNb8hxs=h=N=LwtnA@mail.gmail.com> (raw)
In-Reply-To: <CA+Kbugc2hfcfc+bASYjVWSgyV8phC=rfT0rL4FAiN1gDf3RbEw@mail.gmail.com>

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
>
> --
> 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.

-- 
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.

  reply	other threads:[~2018-07-20 14:55 UTC|newest]

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 [this message]
2018-07-20 15:17               ` Joyal, André
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

Reply instructions:

You may reply publicly to this message via plain-text email
using any one of the following methods:

* Save the following mbox file, import it into your mail client,
  and reply-to-all from there: mbox

  Avoid top-posting and favor interleaved quoting:
  https://en.wikipedia.org/wiki/Posting_style#Interleaved_style

* Reply using the --to, --cc, and --in-reply-to
  switches of git-send-email(1):

  git send-email \
    --in-reply-to='CAOvivQyz+Vy7TCzSWhE8d9MEkWn5Vcuh1rNb8hxs=h=N=LwtnA@mail.gmail.com' \
    --to=shulman@sandiego.edu \
    --cc=HomotopyTypeTheory@googlegroups.com \
    --cc=urs.schreiber@googlemail.com \
    /path/to/YOUR_REPLY

  https://kernel.org/pub/software/scm/git/docs/git-send-email.html

* If your mail client supports setting the In-Reply-To header
  via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).