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Composition of Natural Transformation
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import algebra.category.functor | |
open category eq eq.ops functor | |
inductive natural_transformation {C D : Category} (F G : C ⇒ D) : Type := | |
mk : Π (η : Π(a : C), hom (F a) (G a)), (Π{a b : C} (f : hom a b), G f ∘ η a = η b ∘ F f) | |
→ natural_transformation F G | |
infixl `⟹`:25 := natural_transformation -- \==> | |
namespace natural_transformation | |
variables {C D : Category} {F G H I : functor C D} | |
definition natural_map [coercion] (η : F ⟹ G) : Π(a : C), F a ⟶ G a := | |
rec (λ x y, x) η | |
theorem naturality (η : F ⟹ G) : Π⦃a b : C⦄ (f : a ⟶ b), G f ∘ η a = η b ∘ F f := | |
rec (λ x y, y) η | |
protected definition compose (η : G ⟹ H) (θ : F ⟹ G) : F ⟹ H := | |
natural_transformation.mk | |
(λ a, η a ∘ θ a) | |
(λ a b f, | |
calc | |
H f ∘ (η a ∘ θ a) = (H f ∘ η a) ∘ θ a : assoc | |
... = (η b ∘ G f) ∘ θ a : naturality η f | |
... = η b ∘ (G f ∘ θ a) : assoc | |
... = η b ∘ (θ b ∘ F f) : naturality θ f | |
... = (η b ∘ θ b) ∘ F f : assoc) | |
infixr `∘n`:60 := compose | |
protected theorem assoc (η₃ : H ⟹ I) (η₂ : G ⟹ H) (η₁ : F ⟹ G) : | |
η₃ ∘n (η₂ ∘n η₁) = (η₃ ∘n η₂) ∘n η₁ := | |
dcongr_arg2 mk (funext (take x, !assoc)) !proof_irrel | |
protected definition id {C D : Category} {F : functor C D} : natural_transformation F F := | |
mk (λa, id) (λa b f, !id_right ⬝ symm !id_left) | |
protected definition ID {C D : Category} (F : functor C D) : natural_transformation F F := id | |
protected theorem id_left (η : F ⟹ G) : natural_transformation.compose id η = η := | |
rec (λf H, dcongr_arg2 mk (funext (take x, !id_left)) !proof_irrel) η | |
protected theorem id_right (η : F ⟹ G) : natural_transformation.compose η id = η := | |
rec (λf H, dcongr_arg2 mk (funext (take x, !id_right)) !proof_irrel) η | |
end natural_transformation |
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