Re: [HoTT] Syllepsis for syllepsis

[Noah Snyder <nsnyder@gmail.com>]

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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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