Syllepsis for syllepsis

[Kristina Sojakova]

Dear all,

I proved the higher-dimensional version of syllepsis at

https://github.com/kristinas/HoTT/blob/syllepsis2/theories/Homotopy/syllepsis

if it is of interest.

I'll be happy to answer any questions.

Best,

kristina

[andrej...@andrej.com]

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

[Kristina Sojakova]

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

[David Roberts]

Dear Kristina,

Am I correct in assuming that the "syllepsis for syllepsis" is the equality of (4) and (5) in your paper? 

Is this related to the fact stable pi_2 is Z/2?

Best,

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[Kristina Sojakova]

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

[Noah Snyder]

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?

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[Kristina Sojakova]

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?

[Noah Snyder]

For general p and q you need to assume that they're 4-loops, but if p=q then it should hold for 3-loops! Analogously xx = xx for 1-loops, but xy = yx only for 2-loops. Best,

Noah

[Raymond Baker]

I stumbled upon this thread a (quite bit) late, so sorry for chiming in out of the blue. But I have a way of showing that EH(x,x) gets you a generator of pi3(S2) in HoTT. I a gave a short talk about it at CMU's HoTT2023, and should soon be submitting a revised write up to the special issue in MSCS.

Its quite cool to see the construction of the expected generator of pi7(S4). It makes me wonder how much one could adapt what goes on in the EH and Hopf proof for these higher dimensions.

Related, I think all thats missing from Noah's message that is required to compute pi4(S3) is a non-triviality proof. Is there a good reference for this in book HoTT?