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Cleanup in nattrans

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Frederik Hanghøj Iversen 2018-03-05 15:04:16 +01:00
parent ddd5f17c05
commit b26ea18257
1 changed files with 7 additions and 10 deletions
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17
src/Cat/Category/NaturalTransformation.agda
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@ -34,7 +34,11 @@ open import Cat.Wishlist
module NaturalTransformation {ℓc ℓc' ℓd ℓd' : Level} module NaturalTransformation {ℓc ℓc' ℓd ℓd' : Level}
(ℂ : Category ℓc ℓc') (𝔻 : Category ℓd ℓd') where (ℂ : Category ℓc ℓc') (𝔻 : Category ℓd ℓd') where
open Category using (Object ; 𝟙) open Category using (Object ; 𝟙)
private
module ℂ = Category ℂ
module 𝔻 = Category 𝔻
module _ (F G : Functor ℂ 𝔻) where module _ (F G : Functor ℂ 𝔻) where
private private
@ -74,7 +78,6 @@ module NaturalTransformation {ℓc ℓc' ℓd ℓd' : Level}
where where
module F = Functor F module F = Functor F
F→ = F.func→ F→ = F.func→
module 𝔻 = Category 𝔻
identity : (F : Functor ℂ 𝔻) → NaturalTransformation F F identity : (F : Functor ℂ 𝔻) → NaturalTransformation F F
identity F = identityTrans F , identityNatural F identity F = identityTrans F , identityNatural F
@ -91,21 +94,15 @@ module NaturalTransformation {ℓc ℓc' ℓd ℓd' : Level}
proj₁ NT[ (θ , _) ∘ (η , _) ] = T[ θ ∘ η ] proj₁ NT[ (θ , _) ∘ (η , _) ] = T[ θ ∘ η ]
proj₂ NT[ (θ , θNat) ∘ (η , ηNat) ] {A} {B} f = begin proj₂ NT[ (θ , θNat) ∘ (η , ηNat) ] {A} {B} f = begin
𝔻 [ T[ θ ∘ η ] B ∘ F.func→ f ] ≡⟨⟩ 𝔻 [ T[ θ ∘ η ] B ∘ F.func→ f ] ≡⟨⟩
𝔻 [ 𝔻 [ θ B ∘ η B ] ∘ F.func→ f ] ≡⟨ sym isAssociative ⟩ 𝔻 [ 𝔻 [ θ B ∘ η B ] ∘ F.func→ f ] ≡⟨ sym 𝔻.isAssociative ⟩
𝔻 [ θ B ∘ 𝔻 [ η B ∘ F.func→ f ] ] ≡⟨ cong (λ φ → 𝔻 [ θ B ∘ φ ]) (ηNat f) ⟩ 𝔻 [ θ B ∘ 𝔻 [ η B ∘ F.func→ f ] ] ≡⟨ cong (λ φ → 𝔻 [ θ B ∘ φ ]) (ηNat f) ⟩
𝔻 [ θ B ∘ 𝔻 [ G.func→ f ∘ η A ] ] ≡⟨ isAssociative ⟩ 𝔻 [ θ B ∘ 𝔻 [ G.func→ f ∘ η A ] ] ≡⟨ 𝔻.isAssociative ⟩
𝔻 [ 𝔻 [ θ B ∘ G.func→ f ] ∘ η A ] ≡⟨ cong (λ φ → 𝔻 [ φ ∘ η A ]) (θNat f) ⟩ 𝔻 [ 𝔻 [ θ B ∘ G.func→ f ] ∘ η A ] ≡⟨ cong (λ φ → 𝔻 [ φ ∘ η A ]) (θNat f) ⟩
𝔻 [ 𝔻 [ H.func→ f ∘ θ A ] ∘ η A ] ≡⟨ sym isAssociative ⟩ 𝔻 [ 𝔻 [ H.func→ f ∘ θ A ] ∘ η A ] ≡⟨ sym 𝔻.isAssociative ⟩
𝔻 [ H.func→ f ∘ 𝔻 [ θ A ∘ η A ] ] ≡⟨⟩ 𝔻 [ H.func→ f ∘ 𝔻 [ θ A ∘ η A ] ] ≡⟨⟩
𝔻 [ H.func→ f ∘ T[ θ ∘ η ] A ] ∎ 𝔻 [ H.func→ f ∘ T[ θ ∘ η ] A ] ∎
where
open Category 𝔻
module _ {F G : Functor ℂ 𝔻} where module _ {F G : Functor ℂ 𝔻} where
private
open Category using (Object ; 𝟙)
module 𝔻 = Category 𝔻
transformationIsSet : isSet (Transformation F G) transformationIsSet : isSet (Transformation F G)
transformationIsSet _ _ p q i j C = 𝔻.arrowsAreSets _ _ (λ l → p l C) (λ l → q l C) i j transformationIsSet _ _ p q i j C = 𝔻.arrowsAreSets _ _ (λ l → p l C) (λ l → q l C) i j