```Discussion of Homotopy Type Theory and Univalent Foundations
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```From: "Anders Mörtberg" <andersmortberg@gmail.com>
Subject: Re: [HoTT] Different definitions of Sn
Date: Wed, 18 Sep 2019 05:00:29 -0700 (PDT)

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Let me elaborate a bit on my answer. One might naively try to copy Dan and
Guillaume's definition and write the following in Cubical Agda:

Omega : (A : Type₀) → A → Type₀
Omega A a = (a ≡ a)

itOmega : ℕ → (A : Type₀) → A → Type₀
itOmega zero A a = Omega A a
itOmega (suc n) A a = itOmega n (Omega A a) refl

data Sn (n : ℕ) : Type₀ where
base : Sn n
surf : itOmega n (Sn n) base

However Agda will complain as surf is not constructing an element of Sn.
This might seem a bit funny as Cubical Agda is perfectly happy with

data S³ : Type₀ where
base : S³
surf : Path (Path (base ≡ base) refl refl) refl refl

But what is happening under the hood is that surf is a constructor taking
i, j, and k in the interval and returning an element of S^3 with boundary
"base" whenever i, j and k are 0 or 1. In cubicaltt we can write this HIT
in the following way:

data S3 = base
| surf <i j k> [ (i=0) -> base
, (i=1) -> base
, (j=0) -> base
, (j=1) -> base
, (k=0) -> base
, (k=1) -> base ]

The problem is then clearer: in order to write the surf constructor of Sn
we would have to quantify over n interval variables and specify the
boundary of the n-cell. This is what that is not supported by any of the
cubical schemas for HITs.

--
Anders

On Wednesday, September 18, 2019 at 11:00:22 AM UTC+2, Guillaume Brunerie
wrote:
>
> 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 <andersm...@gmail.com
> <javascript:>>:
> >
> > 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...@gmail.com <javascript:>> 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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```next prev parent reply index

Thread overview: 6+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-09-17 19:56 Kristina Sojakova
2019-09-18  8:32 ` Anders Mortberg
2019-09-18  9:00   ` Guillaume Brunerie
2019-09-18 12:00     ` Anders Mörtberg [this message]
2019-09-18 16:27       ` Licata, Dan
2019-09-18 19:19         ` Michael Shulman
```

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```Discussion of Homotopy Type Theory and Univalent Foundations