52 lines
1.4 KiB
Markdown
52 lines
1.4 KiB
Markdown
# Loop Space
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In this quest,
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we continue to formalise the problem statement.
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> The fundamental group of `S¹` is `ℤ`.
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Intuitively,
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the fundamental group of `S¹` at `base` is
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consists of loops based as `base` up to homotopy of paths.
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In homotopy type theory,
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we have a native description of loops based at `base` :
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it is the space `base ≡ base`.
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In general the _loop space_ of a space `A` at a point `a` is defined as follows :
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```agda
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Ω : (A : Type) (a : A) → Type
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Ω A a = a ≡ a
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```
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Warning :
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the loop space can contain higher homotopical information that
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the fundamental group does not capture.
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For example, consider `S²`.
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```agda
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data S² : Type where
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base : S²
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loop : base ≡ base
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northHemisphere : loop ≡ refl
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southHemisphere : refl ≡ loop
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```
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<p>
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<details>
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<summary>What is `refl`?</summary>
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For any space `A` and point `a : A`,
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`refl` is the constant path at `a`.
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Technically speaking, we should write `refl a` to indicate the point we are at,
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however `agda` is often smart enough to figure that out.
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</details>
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</p>
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Intuitively, all loops in `S²` based at `base` is homotopic to
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the constant path `refl`.
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In other words, the fundamental group at `base` of `S²` is trivial.
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However, the 'composition' of the path `southHemisphere` with `northHemisphere`
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in `base ≡ base` gives the surface of `S²`,
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which intuitively is not homotopic to the constant point `base`.
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So `base ≡ base` has non-trivial path structure.
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