Re: [HoTT] Different definitions of Sn

[Guillaume Brunerie <guillaume.brunerie@gmail.com>]

Hi,

We give a definition of S^n with one point and one n-loop by
introduction/elimination/reduction rules in our paper with Dan Licata
(https://guillaumebrunerie.github.io/pdf/lb13cpp.pdf), which can be
implemented in Agda (so Kristina’s question can be formulated and can
presumably be formalized in Agda) but I don’t think we actually proved
it.

Best,
Guillaume

Den ons 18 sep. 2019 kl 10:32 skrev Anders Mortberg <andersmortberg@gmail.com>:
>
> Hi Kristina,
>
> We have proofs for S^0, S^1, S^2 and S^3 in Cubical Agda:
> https://github.com/agda/cubical/blob/master/Cubical/HITs/Susp/Base.agda
>
> However, I don't think we can even write down the general version of
> S^n as a type with a point and an n-loop with the schema implemented
> in Cubical Agda. As far as I know no other schema for HITs support
> this kind of types either.
>
> --
> Anders
>
> On Tue, Sep 17, 2019 at 9:56 PM Kristina Sojakova
> <sojakova.kristina@gmail.com> wrote:
> >
> > Hello everybody,
> >
> > Is it worked out somewhere that the two definitions of Sn as a
> > suspension on one hand and a HIT with a point and an n-loop on the other
> > hand are equivalent? This is also an exercise in the book. I know
> > Favonia has some Agda code on spheres but I couldn't find this result in
> > there.
> >
> > Thanks,
> >
> > Kristina
> >
> >
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