75 lines
1.7 KiB
Agda
75 lines
1.7 KiB
Agda
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{-# OPTIONS --without-K #-}
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module CircleFundamentalGroup where
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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renaming (Set to Type) public
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open import Agda.Builtin.Int
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-- Path
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data _≡_ {l : Level} {A : Type l} : A → A → Type l where
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refl : (x : A) → x ≡ x
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path-ind : ∀ {ℓ : Level} {A : Type}
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-- Motive
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→ (C : (x y : A) → x ≡ y → Type ℓ)
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-- What happens in the case of refl
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→ (c : (x : A) → C x x (refl x))
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-- Actual path to eliminate
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→ {x y : A} → (p : x ≡ y)
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-- Result
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→ C x y p
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path-ind C c {x} (refl x) = c x
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-- Id
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id : {ℓ : Level} {A : Set ℓ} → A → A
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id x = x
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-- Transport
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transport : {X : Set} (P : X → Set) {x y : X}
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→ x ≡ y
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→ P x → P y
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transport P (refl x) = id
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-- apd
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-- Lemma 2.3.4 of HoTT book
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apd : {A : Type} {B : A → Type}
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-- The function that we want to apply
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→ (f : (a : A) → B a)
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-- The path to apply it over
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→ {x y : A} → (p : x ≡ y)
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-- Result
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→ (transport B p) (f x) ≡ f y
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apd {A} {B} f p = path-ind D d p
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where
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D : (x y : A) → (p : x ≡ y) → Type
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D x y p = (transport B p) (f x) ≡ f y
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d : (x : A) → D x x (refl x)
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d x = refl (f x)
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-- Circle (S¹)
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postulate
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S¹ : Set
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base : S¹
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loop : base ≡ base
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S¹-ind : (C : S¹ → Type)
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→ (c-base : C base)
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→ (c-loop : c-base ≡ c-base)
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→ (s : S¹) → C s
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-- Fundamental group of a circle
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loop-space : {A : Type} → (a : A) → Set
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loop-space a = a ≡ a
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π₁ : {A : Type} → (a : A) → Type
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π₁ a = ?
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π₁[S¹]≡ℤ : π₁ base ≡ Int
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π₁[S¹]≡ℤ = ?
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-- References:
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-- - https://homotopytypetheory.org/2011/04/29/a-formal-proof-that-pi1s1-is-z/
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-- - HoTT book ch. 6.11
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-- - https://en.wikipedia.org/wiki/Fundamental_group
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