Re: [HoTT] Syllepsis in HoTT

[Egbert Rijke <e.m.rijke@gmail.com>]

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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> 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> 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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