Discussion of Homotopy Type Theory and Univalent Foundations
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From: "Anders Mörtberg" <andersmortberg@gmail.com>
To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Different definitions of Sn
Date: Wed, 18 Sep 2019 05:00:29 -0700 (PDT)
Message-ID: <db712ad2-396f-4328-bb73-898dcb956834@googlegroups.com> (raw)
In-Reply-To: <CAFJ3QW+znvLhFgBODptrjnKcY8QaEH8BxD3adrgRUNvo5fU+EQ@mail.gmail.com>

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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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  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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