TheHoTTGame/1FundamentalGroup/Quest2Part0.md

Comparison maps between Ω S¹ base and ℤ

In Quest1 we have defined the map loop_times : ℤ → Ω S¹ base. Creating the inverse map is difficult without access to the entire circle. Similarly to how we used doubleCover to distinguish refl and base, the idea is to replace Bool with ℤ, allowing us to distinguish between all loops on S¹.

The plan is :

• Define a function sucℤ : ℤ → ℤ that increases every integer by one
• Prove that sucℤ is an isomorphism by constructing an inverse map predℤ : ℤ → ℤ.
• Turn sucℤ into a path sucPath : ℤ ≡ ℤ using isoToPath
• Define helix : S¹ → Type by mapping base to ℤ and a generic point loop i to sucPath i.
• Use helix and endPt to define the map base ≡ x → helix x on all x : S¹, in particular giving us Ω S¹ base → ℤ when applied to base.