Re: [HoTT] Syllepsis for syllepsis

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

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Thanks for all this Noah! Not being an expert, I will need some time to 
digest all the details, but my first question is this: I proved the 
equality of (4) and (5) for 4-loops p, q. Do we expect this should hold 
at a lower dimension too?

On 7/10/2023 7:43 PM, Noah Snyder wrote:
> Very nice! I really like this line of research!
>
> Let me try my hand at sketching what consequences I think this has for 
> homotopy groups of spheres. This isn't exactly my area of expertise so 
> I may have messed up something here. There's a TL;DR below.
>
> As a warmup let's talk about Eckmann-Hilton itself. EH says that if x 
> and y are 2-loops, then xy = yx. Since this involves two variables 
> this is a statement about a homotopy group of a wedge of spheres S^2 
> wedge S^2, namely it says that the commutator xyx^-1y^-1 which is 
> non-trivial in \pi_1(S^1 wedge S^1) becomes trivial when you suspend 
> it to get an element of \pi_2(S^2 wedge S^2). In other words, it gives 
> a new relation (commutativity) in \pi_2(S^2 wedge S^2). From this 
> viewpoint we can also easily get a statements about homotopy groups of 
> wedges of spheres for more elaborate constructions. In particular, the 
> syllepsis says that a certain element of \pi_3(S^2 wedge S^2) vanishes 
> when you suspend it to \pi_4(S^3 wedge S^3). Finally, the "syllepsis 
> of the syllepsis" (henceforth SoS, though see the postscript) says 
> that a certain element of pi_5(S^3 wedge S^3) vanishes when you 
> suspend it to pi_6(S^4 wedge S^4).
>
> Ok, but people are usually more interested in homotopy groups of 
> spheres, rather than of wedges of two spheres. So let's go back to 
> Eckmann-Hilton and think some more. We can consider EH where both 
> variables are the same loop x (or, if you prefer, one is x and the 
> other is x^-1) so that now we're talking about homotopy groups of a 
> single sphere. Here something interesting happens, note that EH now 
> gives an equality xx = xx, but we already knew xx = xx! Indeed if you 
> look at the suspension map \pi_1(S^1) --> \pi_2 (S^2) it's an 
> isomorphism, so we're not adding a new relation. Instead we're saying 
> that xx = xx in two different ways! First xx = xx via refl but second 
> xx = xx via EH. If we compose one of these trivializations with the 
> inverse of the other, what we end up with is a new element of 
> pi_3(S^2). This is how EH is related to pi_3(S^2).
>
> Now let's think about what the syllepsis says about homotopy groups of 
> spheres. So now we again want to look at the syllepsis of x with 
> itself. This tells us that the element of pi_3(S^2) that we 
> constructed from EH composed with itself will become trivial when 
> suspended into pi_4(S^3). In this case this is killing 2 in Z, and so 
> it really does add a new relation.
>
> Ok, now let's turn to SoS, and again restrict our attention to SoS of 
> x with itself. This says that a certain element of pi_5(S^3) vanishes 
> when suspended to pi_6(S^4). But if you look at the homotopy groups 
> the map pi_5(S^3) --> pi_6(S^4) is already an isomorphism (this is 
> analogous to what happened for EH!), in particular the paths (4) and 
> (5) from Kristina's paper are already equal when p = q without 
> assuming they're 4-loops! (I haven't thought at all how one would go 
> about proving this though!) So instead we do what we did for EH, for a 
> 4-loop x we have two different ways of showing (4) = (5) and this 
> gives us an interesting element of pi_7(S^4). And looking in the table 
> there is an interesting new element of pi_7(S^4) that doesn't come 
> from pi_6(S^3), and I'd guess this construction gives this new 
> generator of pi_7(S^4).
>
> Remark: Note that in general there's not a 1-to-1 relationship between 
> interesting generators and relations in the homotopy groups of spheres 
> (which are operations of one variable!) and interesting operations of 
> two variables. You might need to write down a pretty elaborate 
> composition of operators in two variables to write down a generator or 
> relation in homotopy groups of spheres. In particular, the generator 
> of pi_4(S^2) is a more elaborate composition (it's essentially EH 
> applied to EH), the relation 2=0 in pi_4(S^2) is also more elaborate, 
> and the generator of pi_6(S^3) is much much more elaborate! (In 
> particular, the generator of pi_6(S^3) was essentially constructed by 
> Andre Henriques, but in globular instead of HoTT so it's missing all 
> the unitors and associators. Even without all the associators and 
> unitors it's already extremely complicated! See 
> http://globular.science/1702.001v2)
>
> TL;DR: First show that if you assume p = q then (4) = (5) is already 
> true for 3-loops. Then taking p to be a 4-loop compose the proof of 
> (4) = (5) using that p=q with the inverse of the syllepsis of the 
> syllepsis and you'll get an element of pi_7(S^4) which hopefully is 
> the generator of the copy of Z in Z x Z/12 = pi_7(S^4).
>
> Best,
>
> Noah
>
> p.s. I wanted to push back a little on this "syllepsis of the 
> syllepsis" name. The "syllepsis" gets its name because it's what you 
> need to turn a "braided monoidal 2-category" into a "sylleptic 
> monoidal 2-category." (Sylleptic in turn is just "symmetric" but 
> changing "m" to "l" to make it a little bit less symmetric.) The 
> "syllepsis of the syllepsis" by contrast is what's needed to turn a 
> "sylleptic monoidal 2-category" into a "symmetric monoidal 
> 2-category." That is, the parallel name would be the "symmetsis" or 
> something similar. Perhaps a better nomenclature would be to use the 
> E1 = monoidal, E2 = braided monoidal, E3, etc. phrasing which isn't 
> specific to 2-categories. So you might call Eckman-Hilton the 
> E2-axiom, the syllepsis the E3-axiom, and the SoS the E4-axiom. There 
> will also be an E5-axiom, though because of stability you won't see 
> that when studying 2-categories, it'll come up when you look at 
> 3-categories. Another way you might talk about it is the syllepsis is 
> the "coherence of EH" while the syllepsis of the syllepsis is "the 
> coherence of the coherence of EH" which I think maybe matches how 
> you're thinking about the word sylleptsis?
>
> On Sat, Jul 8, 2023 at 4:14 PM Kristina Sojakova 
> <sojakova.kristina@gmail.com> wrote:
>
>     Hello David,
>
>     On 7/8/2023 4:00 PM, David Roberts wrote:
>>     Dear Kristina,
>>
>>     Am I correct in assuming that the "syllepsis for syllepsis" is
>>     the equality of (4) and (5) in your paper?
>     Indeed, we show the equality between (4) and (5).
>>
>>     Is this related to the fact stable pi_2 is Z/2?
>
>     We do not yet understand the implications of this result, that's
>     another interesting question I guess. Do you have a conjecture here?
>
>     Kristina
>
>>
>>     Best,
>>
>>     On Sat, 8 July 2023, 10:46 pm Kristina Sojakova,
>>     <sojakova.kristina@gmail.com> wrote:
>>
>>         Dear Andrej,
>>
>>         Indeed, my message could have been more user-friendly. The
>>         file contains
>>         alternative proofs of Eckmann-Hilton and syllepsis, together
>>         with the
>>         proofs of the coherences described in Section 8 of
>>
>>         https://inria.hal.science/hal-03917004v1/file/syllepsis.pdf
>>
>>         The last coherence outlined in the paper is what I referred
>>         to as
>>         "syllepsis for syllepsis".
>>
>>         Best,
>>
>>         Kristina
>>
>>         On 7/8/2023 2:22 PM, andrej.bauer@andrej.com wrote:
>>         > Dear Kristina,
>>         >
>>         > any chance you could spare a few words in English on the
>>         content of your formalization? Not everyone reads Coq code
>>         for breakfast.
>>         >
>>         > Many thanks in advance!
>>         >
>>         > Andrej
>>         >
>>
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