Discussion of Homotopy Type Theory and Univalent Foundations
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From: Michael Shulman <shulman@sandiego.edu>
To: Ali Caglayan <alizter@gmail.com>
Cc: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] What is knot in HOTT?
Date: Thu, 19 Jul 2018 08:31:48 -0700	[thread overview]
Message-ID: <CAOvivQykUDH9h4iJ1bz1uy1Uc7VLoRFEV4HyferLY+qbrNXORw@mail.gmail.com> (raw)
In-Reply-To: <74a010cb-0bf4-4d16-b72f-bf4eda0510ca@googlegroups.com>

This does seem like the kind of problem that Cohesive HoTT is designed
to solve.  You could define a knot in a purely topological way as a
map from the topological circle to the topological 3-sphere, fatten it
up into a tubular neighborhood to express the topological 3-sphere as
a pushout of the tubular neighborhood of the topological circle and a
complement, then apply the shape functor which preserves pushouts to
obtain the homotopy-theoretic data described by Egbert with the
homotopical 3-sphere as a pushout of the homotopical circle and some
other space.

On Thu, Jul 19, 2018 at 1:55 AM, Ali Caglayan <alizter@gmail.com> wrote:
> From what I have seen knot-theory has been very resistant to homotopy
> theoretic ideas (not classical ones). One 'clean' way of working with knot
> theory is to do so in the context of differential geometry. Cohesive HoTT
> supposedly can develop adequate differential geometry but at the moment it
> is very undeveloped.
>
> One of the difficulties with knot theory is that (classical) homotopy theory
> in HoTT isn't really done with real numbers and interval objects, which is
> needed if you want to define notions of ambient isotopy and such.
>
> I think the main difficulty here might be that HoTT is great at doing
> (synthetic) homotopy but not topology. The two can be easily confused. How
> do you take the complement of a space for example?
>
> I could be wrong however but this is the conclusion I got when I tried
> thinking about HoTT knots.
>
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  reply	other threads:[~2018-07-19 15:32 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 [this message]
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
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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