Discussion of Homotopy Type Theory and Univalent Foundations
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From: Dan Christensen <jdc@uwo.ca>
To: "homotopytypetheory@googlegroups.com"
	<homotopytypetheory@googlegroups.com>
Subject: Re: [HoTT] Syllepsis in HoTT
Date: Mon, 8 Mar 2021 15:10:45 +0000	[thread overview]
Message-ID: <87blbtk7ey.fsf@uwo.ca> (raw)
In-Reply-To: <7d4b6fd7-3035-e0b9-c966-97dd89d8b457@gmail.com> (Kristina Sojakova's message of "Mon, 08 Mar 2021 15:31:13 +0100")

It's great to see this proved!

As a tangential remark, I mentioned after Jamie's talk that I had a
very short proof of Eckmann-Hilton, so I thought I should share it.
Kristina's proof is slightly different and is probably designed to
make the proof of syllepsis go through more easily, but here is mine.

Dan


Definition horizontal_vertical {A : Type} {x : A} {p q : x = x} (h : p = 1) (k : 1 = q)
  : h @ k = (concat_p1 p)^ @ (h @@ k) @ (concat_1p q).
Proof.
  by induction k; revert p h; rapply paths_ind_r.
Defined.

Definition horizontal_vertical' {A : Type} {x : A} {p q : x = x} (h : p = 1) (k : 1 = q)
  : h @ k = (concat_1p p)^ @ (k @@ h) @ (concat_p1 q).
Proof.
  by induction k; revert p h; rapply paths_ind_r.
Defined.

Definition eckmann_hilton' {A : Type} {x : A} (h k : 1 = 1 :> (x = x)) : h @ k = k @ h
  := (horizontal_vertical h k) @ (horizontal_vertical' k h)^.



On Mar  8, 2021, Kristina Sojakova <sojakova.kristina@gmail.com> wrote:

> Dear all,
>
> I formalized my proof of syllepsis in Coq:
> https://github.com/kristinas/HoTT/blob/kristina-pushoutalg/theories/Colimits/Syllepsis.v
>
>
> I am looking forward to see how it compares to the argument Egbert has
> been working on.
>
> Best,
>
> Kristina
>
> On 3/8/2021 2:38 PM, Noah Snyder wrote:
>
>     The generator of \pi_4(S^3) is the image of the generator of \pi_3
>     (S^2) under stabilization.  This is just the surjective the part
>     of Freudenthal.  So to see that this generator has order dividing
>     2 one needs exactly two things: the syllepsis, and my follow-up
>     question about EH giving the generator of \pi_3(S^2).  This is why
>     I asked the follow-up question.
>
>     Note that putting formalization aside, that EH gives the generator
>     of \pi_4(S^3) and the syllepsis the proof that it has order 2, are
>     well-known among mathematicians via framed bordism theory (already
>     Pontryagin knew these two facts almost a hundred years ago).  So
>     informally it’s clear to mathematicians that the syllepsis shows
>     this number is 1 or 2.  Formalizing this well-known result remains
>     an interesting question of course.
>
>     Best,
>
>     Noah 
>
>     On Mon, Mar 8, 2021 at 3:53 AM Egbert Rijke <e.m.rijke@gmail.com>
>     wrote:
>
>         Dear Noah, 
>
>         I don't think that your claim that syllepsis gives a proof
>         that Brunerie's number is 1 or 2 is accurate. Syllepsis gives
>         you that a certain element of pi_4(S^3) has order 1 or 2, but
>         it is an entirely different matter to show that this element
>         generates the group. There could be many elements of order 2.
>
>         Best wishes,
>         Egbert
>
>         On Mon, Mar 8, 2021 at 9:44 AM Egbert Rijke
>         <e.m.rijke@gmail.com> wrote:
>
>             Hi Kristina, 
>
>             I've been on it already, because I was in that talk, and
>             while my formalization isn't yet finished, I do have all
>             the pseudocode already.
>
>             Best wishes,
>             Egbert
>
>             On Sun, Mar 7, 2021 at 7:00 PM Noah Snyder
>             <nsnyder@gmail.com> wrote:
>
>                 On the subject of formalization and the syllepsis, has
>                 it ever been formalized that Eckman-Hilton gives the
>                 generator of \pi_3(S^2)?  That is, we can build a
>                 3-loop for S^2 by refl_refl_base --> surf \circ surf^
>                 {-1} --EH--> surf^{-1} \circ surf -->  refl_refl_base,
>                 and we want to show that under the equivalence \pi_3
>                 (S^2) --> Z constructed in the book that this 3-loop
>                 maps to \pm 1 (which sign you end up getting will
>                 depend on conventions). 
>
>                 There's another explicit way to construct a generating
>                 a 3-loop on S^2, namely refl_refl_base --> surf \circ
>                 surf \circ \surf^-1 \circ surf^-1 --EH whiskered refl
>                 refl--> surf \circ surf \circ surf^-1 \circ surf^-1 -
>                 -> refl_refl_base, where I've suppressed a lot of
>                 associators and other details.  One could also ask
>                 whether this generator is the same as the one in my
>                 first paragraph.  This should be of comparable
>                 difficulty to the syllepsis (perhaps easier), but is
>                 another good example of something that's "easy" with
>                 string diagrams but a lot of work to translate into
>                 formalized HoTT.
>
>                 Best,
>
>                 Noah
>
>                 On Fri, Mar 5, 2021 at 1:27 PM Kristina Sojakova
>                 <sojakova.kristina@gmail.com> wrote:
>
>                     Dear all,
>
>                     Ali told me that apparently the following problem
>                     could be of interest 
>                     to some people
>                     (https://www.youtube.com/watch?v=TSCggv_YE7M&t=4350s):
>                     
>
>                     Given two higher paths p, q : 1_x = 1_x,
>                     Eckmann-Hilton gives us a path 
>                     EH(p,q) : p @ = q @ p. Show that EH(p,q) @ EH(q,p)
>                     = 1_{p@q=q_p}.
>
>                     I just established the above in HoTT and am
>                     thinking of formalizing it, 
>                     unless someone already did it.
>
>                     Thanks,
>
>                     Kristina
>
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  reply	other threads:[~2021-03-08 15:10 UTC|newest]

Thread overview: 19+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2021-03-05 18:27 Kristina Sojakova
2021-03-05 18:40 ` Jamie Vicary
2021-03-05 19:18   ` Noah Snyder
2021-03-07 18:00 ` Noah Snyder
2021-03-08  8:44   ` Egbert Rijke
2021-03-08  8:53     ` Egbert Rijke
2021-03-08 13:38       ` Noah Snyder
2021-03-08 14:31         ` Kristina Sojakova
2021-03-08 15:10           ` Dan Christensen [this message]
2021-03-08 15:15             ` Kristina Sojakova
2021-03-08 15:23               ` Noah Snyder
2021-03-08 15:35                 ` Noah Snyder
2021-03-08 15:46                   ` Egbert Rijke
2021-03-08 15:49                     ` Kristina Sojakova
2021-03-08 16:25                     ` Dan Christensen
2021-03-08 16:27                       ` Kristina Sojakova
2021-03-08 16:38             ` Kristina Sojakova
2021-03-08 16:54               ` Egbert Rijke
2021-03-08 19:55                 ` 'Favonia' via Homotopy Type Theory

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