Re: [HoTT] Syllepsis in HoTT

[Kristina Sojakova <sojakova.kristina@gmail.com>]

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Thanks Egbert, I think it will be useful to have both proofs, as they 
offer different insights.

On 3/8/21 4:46 PM, Egbert Rijke wrote:
> Congratulations, Kristina, on doing it so fast.
>
> I had a different route in mind, much less efficient. There are three 
> kinds of concatenations in the third identity type, all three pairs of 
> them satisfy interchange laws, and there is a coherence law between 
> the three interchange laws. This is what I had already formalized, and 
> this coherence law induces the syllepsis. But it takes me a lot more 
> coding to do it in the way I had in mind, which is why it takes me 
> forever.
>
> Best,
> Egbert
>
> On Mon, Mar 8, 2021 at 4:36 PM Noah Snyder <nsnyder@gmail.com 
> <mailto:nsnyder@gmail.com>> wrote:
>
>     My funny remark is slightly inaccurate.  \pi_3(S^2) just
>     classifies proofs of EH where both 2-loops are the same as each
>     other.  It is true that there's also a Z-worth of proofs of EH in
>     the general case, but this is a subtler fact about \pi_3(S^2
>     \wedge S^2).  Nonetheless  the point remains that any two
>     reasonable proofs of EH will be equal or inverse to each other. 
>     Best,
>
>     Noah
>
>     On Mon, Mar 8, 2021 at 10:23 AM Noah Snyder <nsnyder@gmail.com
>     <mailto:nsnyder@gmail.com>> wrote:
>
>         One funny remark, that \pi_3(S^2) = Z exactly tells you that
>         any proof of Eckman-Hilton is given by repeatedly applying
>         either the standard proof or its inverse.
>
>         In a sense there are exactly two “good” proofs of EH (the
>         standard one and it’s inverse).  In principle it’s not so
>         automatic to see that a given proof is one of the good two,
>         but in practice it’d be hard to give a bad one accidentally. 
>         By contrast, put two people in two separate rooms and there’s
>         a good chance they’ll produce the two different good proofs
>         (ie the clockwise proof and the counterclockwise proof). Best,
>
>         Noah
>
>         On Mon, Mar 8, 2021 at 10:15 AM Kristina Sojakova
>         <sojakova.kristina@gmail.com
>         <mailto:sojakova.kristina@gmail.com>> wrote:
>
>             Thanks Dan! I think we should have no trouble showing that
>             what I used
>             is equal to your proof but packaged a bit differently.
>
>             On 3/8/21 4:10 PM, Dan Christensen wrote:
>             > 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
>             <mailto: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 <mailto: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
>             <mailto: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
>             <mailto: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
>             <mailto: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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