module Homotopy.Space.Sinfty where

# The Infinite Sphere🔗

The $(n+1)$-sphere can be constructed from the $n$-sphere via suspension. By writing a recursive HIT, we can define a type which is its own suspension. It stands to reason that this definition is a good candidate for being conisdered the infinitary limit of the process of iterated suspension and is thus referred to as the $\infty$-sphere.

data S∞ : Type where N S : S∞ merid : S∞ → N ≡ S

In classical topology, there are several definitions of the
$\infty$-sphere.
However, independently of the approach taken, the resulting space is
always contractible. In Homotopy Type Theory, then, the only meaningful
statement that can be made of `S∞`

is that it is contractible.
We prove this in two different ways.

## The Book HoTT Approach🔗

open is-contr private pathsS∞′ : (x : S∞) → N ≡ x pathsS∞′ N = refl pathsS∞′ S = merid N pathsS∞′ (merid x i) =

First we reduce the problem from constructing a dependent path over
`(λ i → N ≡ merid x i)`

from `refl`

to `merid N`

to the problem of constructing a
path in `N ≡ S`

from `transport (λ j → N ≡ merid x j) refl`

to `merid N`

.

to-pathp {A = λ j → N ≡ merid x j}

The proof goes as follows: by the characterisation of transport in
path types the LHS is identified with `refl ∙ merid x`

. We get rid of the `refl`

and then a a path between
`merid x`

and `merid N`

can be obtained from applying
`merid`

to the
recursive call `pathsS∞′ x`

.

(transport (λ j → N ≡ merid x j) refl ≡⟨ subst-path-right refl (merid x) ⟩≡ refl ∙ merid x ≡⟨ ∙-idl (merid x) ⟩≡ merid x ≡⟨ ap merid (sym (pathsS∞′ x)) ⟩≡ merid N ∎) i is-contrS∞′ : is-contr S∞ is-contrS∞′ .centre = N is-contrS∞′ .paths = pathsS∞′

## The Cubical Approach🔗

The cubical approach essentially acomplishes the same thing as the
previous proof, without using any helper lemmas, by way of drawing a
slightly clever cube. The case of the definition for the higher
constructor requires a square in which two of the sides are
`merid N`

and `merid x`

. We start with a square in
which both of these sides are `merid N`

(specifically
`merid N (i ∧ j)`

), and then construct a cube in which one of
the faces morphs `merid N`

into `merid x`

. This is
something that we can easily do since we have a path `N ≡ x`

via the recursive call `pathsS∞ x`

.

private pathsS∞ : (x : S∞) → N ≡ x pathsS∞ N = refl pathsS∞ S = merid N pathsS∞ (merid x i) j = hcomp (∂ i ∨ ∂ j) λ where k (k = i0) → merid N (i ∧ j) k (i = i0) → N k (j = i0) → N k (i = i1) → merid N j k (j = i1) → merid (pathsS∞ x k) i is-contrS∞ : is-contr S∞ is-contrS∞ .centre = N is-contrS∞ .paths = pathsS∞