module Homotopy.Space.Sinfty where

The infinite sphere🔗

The can be constructed from the 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 considered the infinitary limit of the process of iterated suspension and is thus referred to as the

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

In classical topology, there are several definitions of the 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

  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 i N ≡ merid x i) from refl to merid N to the problem of constructing a path in N ≡ S from transport 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  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 accomplishes 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.

  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∞