TheHoTTGame/1FundamentalGroup/Quest1Part0.md

Loop Space

In this quest, we continue to formalise the problem statement.

The fundamental group of S¹ is ℤ.

Intuitively, the fundamental group of S¹ at base is consists of loops based as base up to homotopy of paths. In homotopy type theory, we have a native description of loops based at base : it is the space base ≡ base.

In general the loop space of a space A at a point a is defined as follows :

Ω : (A : Type) (a : A) → Type
Ω A a = a ≡ a 

Warning : the loop space can contain higher homotopical information that the fundamental group does not capture. For example, consider S².

data S² : Type where
  base : S²
  loop : base ≡ base
  northHemisphere : loop ≡ refl
  southHemisphere : refl ≡ loop

What is `refl`?

For any space A and point a : A, refl is the constant path at a. Technically speaking, we should write refl a to indicate the point we are at, however agda is often smart enough to figure that out.

Intuitively, all loops in S² based at base is homotopic to the constant path refl. In other words, the fundamental group at base of S² is trivial. However, the 'composition' of the path southHemisphere with northHemisphere in base ≡ base gives the surface of S², which intuitively is not homotopic to the constant point base. So base ≡ base has non-trivial path structure.