Cleanup in nattrans
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@ -34,7 +34,11 @@ open import Cat.Wishlist
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module NaturalTransformation {ℓc ℓc' ℓd ℓd' : Level}
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(ℂ : Category ℓc ℓc') (𝔻 : Category ℓd ℓd') where
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open Category using (Object ; 𝟙)
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private
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module ℂ = Category ℂ
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module 𝔻 = Category 𝔻
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module _ (F G : Functor ℂ 𝔻) where
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private
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@ -74,7 +78,6 @@ module NaturalTransformation {ℓc ℓc' ℓd ℓd' : Level}
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where
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module F = Functor F
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F→ = F.func→
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module 𝔻 = Category 𝔻
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identity : (F : Functor ℂ 𝔻) → NaturalTransformation F F
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identity F = identityTrans F , identityNatural F
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@ -91,21 +94,15 @@ module NaturalTransformation {ℓc ℓc' ℓd ℓd' : Level}
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proj₁ NT[ (θ , _) ∘ (η , _) ] = T[ θ ∘ η ]
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proj₂ NT[ (θ , θNat) ∘ (η , ηNat) ] {A} {B} f = begin
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𝔻 [ T[ θ ∘ η ] B ∘ F.func→ f ] ≡⟨⟩
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𝔻 [ 𝔻 [ θ B ∘ η B ] ∘ F.func→ f ] ≡⟨ sym isAssociative ⟩
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𝔻 [ 𝔻 [ θ B ∘ η B ] ∘ F.func→ f ] ≡⟨ sym 𝔻.isAssociative ⟩
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𝔻 [ θ B ∘ 𝔻 [ η B ∘ F.func→ f ] ] ≡⟨ cong (λ φ → 𝔻 [ θ B ∘ φ ]) (ηNat f) ⟩
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𝔻 [ θ B ∘ 𝔻 [ G.func→ f ∘ η A ] ] ≡⟨ isAssociative ⟩
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𝔻 [ θ B ∘ 𝔻 [ G.func→ f ∘ η A ] ] ≡⟨ 𝔻.isAssociative ⟩
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𝔻 [ 𝔻 [ θ B ∘ G.func→ f ] ∘ η A ] ≡⟨ cong (λ φ → 𝔻 [ φ ∘ η A ]) (θNat f) ⟩
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𝔻 [ 𝔻 [ H.func→ f ∘ θ A ] ∘ η A ] ≡⟨ sym isAssociative ⟩
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𝔻 [ 𝔻 [ H.func→ f ∘ θ A ] ∘ η A ] ≡⟨ sym 𝔻.isAssociative ⟩
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𝔻 [ H.func→ f ∘ 𝔻 [ θ A ∘ η A ] ] ≡⟨⟩
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𝔻 [ H.func→ f ∘ T[ θ ∘ η ] A ] ∎
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where
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open Category 𝔻
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module _ {F G : Functor ℂ 𝔻} where
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private
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open Category using (Object ; 𝟙)
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module 𝔻 = Category 𝔻
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transformationIsSet : isSet (Transformation F G)
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transformationIsSet _ _ p q i j C = 𝔻.arrowsAreSets _ _ (λ l → p l C) (λ l → q l C) i j
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