Added Quest1Part0.md

1FundamentalGroup/Quest0Part3.md

@@ -24,7 +24,7 @@ see `1FundamentalGroup/Quest0SideQuests/SideQuest1`.
   it should be asking for a path from `true` to `false`.
 
 To give this path we need to visualise 'lifting' `Refl`, `loop`
-and the homotopy `h : refl ≡ loop`
+and the homotopy `h : Refl ≡ loop`
 along the Boolean-bundle `doubleCover`.
 When we 'lift' `Refl` - starting at the point `true : doubleCover base` -
 it will still be a constant path at `true`,
@@ -34,8 +34,8 @@ it will look like
 
  <!-- [insert picture] -->
 
-The homotopy `h : refl ≡ loop` is 'lifted'
-(starting at 'lifted `refl`')
+The homotopy `h : Refl ≡ loop` is 'lifted'
+(starting at 'lifted `Refl`')
 to some kind of surface 
 
 <!-- [insert picture] -->
@@ -83,7 +83,7 @@ You can verify our expectation that `endPtOfTrue Refl` is `true`
 and `endPtOfTrue loop` is `false` using `C-c C-n`.
 
 Lastly we need to make the function `endPtOfTrue`
-take the path `h : refl ≡ loop` to a path from `true` to `false`.
+take the path `h : Refl ≡ loop` to a path from `true` to `false`.
 In general if `f : A → B` is a function and `p` is a path
 between points `x y : A` then we get a map `cong f p`
 from `f x` to `f y`.

1FundamentalGroup/Quest1Part0.md

@@ -0,0 +1,51 @@
+# 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 :
+
+```agda
+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`.
+```agda
+data S² : Type where
+  base : S²
+  loop : base ≡ base
+  northHemisphere : loop ≡ refl
+  southHemisphere : refl ≡ loop
+```
+
+<p>
+<details>
+<summary>What is `refl`?</summary>
+
+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.
+</details>
+</p>
+
+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.