Different definitions of Sn

[Kristina Sojakova]

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

[Anders Mortberg]

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

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[Guillaume Brunerie]

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

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[Anders Mörtberg]

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

[Licata, Dan]

Thanks Anders. To be clear, I don’t think we have any schemas or semantics of HITs that accept this definition of S^n (with an n-loop for an internally-specified n) in any non-cubical settings either; at the time, I was thinking of it more as exploring what you can do with the definition. Proving the two definitions equivalent should be the same (modulo which definitional equalities you get) as implementing the one-n-loop rules in terms of suspensions, so that would be one way to justify these rules.

Also, the proof of pi_n(S^n) = Z that Guillaume mentioned, which predated Freudenthal, is a lot more work than the one that you get from Freudenthal, so I’m not sure we have any motivating examples for why the one-n-loop definition is better than the suspension definition for arbitrary n.

-Dan

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[Michael Shulman]

What I find more interesting is that it's even *possible* to
parametrize a definition of S^n with one n-loop by an internal n. It
would be intriguing to try to find a schema or semantics for HITs that
allow paths of arbitrary internally-specified dimensions, and see
whether there are other examples that are more useful. The only other
vaguely similar example that I know of is mentioned at the end of
https://homotopytypetheory.org/2014/11/02/universal-properties-without-function-extensionality/
(and implemented in the HoTT/Coq library), a definition of
localization that works in the absence of funext.

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