Syllepsis in HoTT

[Kristina Sojakova]

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

[Jamie Vicary]

Hi Kristina, that's great. I don't know that anyone's done this before.

> Given two higher paths p, q : 1_x = 1_x

I guess you mean p,q:1_(1_x) = 1_(1_x) ?

Best wishes,
Jamie

[Noah Snyder]

It'd be great to see this done!  I've been wanting to see this for a while, but haven't gotten anyone to do it.

One remark on that part of the video: the syllepsis gives the proof that the "Brunerie number" is 1 or 2, but it doesn't immediately let you exclude the possibility that it's 1.  I think my student Nachiket Karnick and I do understand how to show that the number is 2 (with a much more direct calculation than what's in the second half of Brunerie's thesis, but still using the James construction).  I have an outline of an even more direct proof, but the syllepsis is one of the calculations required to make this more direct approach work.  Which is all to say that I'm very interested in seeing this result, especially if it meant that related calculations of similar difficulty could be done thereby giving much more direct calculations of the small homotopy groups of spheres.

Best,

Noah

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[Noah Snyder]

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

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[Egbert Rijke]

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

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[Egbert Rijke]

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

[Noah Snyder]

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 

[Kristina Sojakova]

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

[Dan Christensen]

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...@gmail.com> wrote:

> Dear all,
>

[Kristina Sojakova]

Thanks Dan! I think we should have no trouble showing that what I used
is equal to your proof but packaged a bit differently.

[Noah Snyder]

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

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[Noah Snyder]

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

[Egbert Rijke]

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

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[Kristina Sojakova]

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

[Dan Christensen]

On Mar 8, 2021, Egbert Rijke <e.m....@gmail.com> wrote:

> 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.

In case anyone wants to play with this in Coq, in this branch

https://github.com/jdchristensen/HoTT/tree/Hurewicz

the file Smashing.v contains similar facts, e.g. pmagma_loops_shuffle.
(But no coherence law is proved.)

Dan

[Kristina Sojakova]

Is there a geometric interpretation for the proof I gave?

[Kristina Sojakova]

If I'm not mistaken, Favonia also found a very short proof of EH some
years ago.

On 3/8/21 4:10 PM, Dan Christensen wrote:

[Egbert Rijke]

My agda file with the the interchange laws and EH is here

https://github.com/HoTT-Intro/Agda/blob/master/extra/interchange.agda

And the coherence law is here

https://github.com/HoTT-Intro/Agda/blob/master/extra/syllepsis.agda

For anyone who is interested.

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[Favonia]

I remember multiple people (including me) discovered relatively short proofs. Some history on GitHub: https://github.com/HoTT/book/issues/27

Best,
Favonia

they/them/theirs

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