53 lines
2.4 KiB
Text
53 lines
2.4 KiB
Text
/-
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Copyright (c) 2014 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: algebra.category.natural_transformation
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Author: Floris van Doorn
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-/
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import .functor
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open category eq eq.ops functor
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inductive natural_transformation {C D : Category} (F G : C ⇒ D) : Type :=
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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)
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→ natural_transformation F G
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infixl `⟹`:25 := natural_transformation -- \==>
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namespace natural_transformation
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variables {C D : Category} {F G H I : functor C D}
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definition natural_map [coercion] (η : F ⟹ G) : Π(a : C), F a ⟶ G a :=
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natural_transformation.rec (λ x y, x) η
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theorem naturality (η : F ⟹ G) : Π⦃a b : C⦄ (f : a ⟶ b), G f ∘ η a = η b ∘ F f :=
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natural_transformation.rec (λ x y, y) η
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protected definition compose (η : G ⟹ H) (θ : F ⟹ G) : F ⟹ H :=
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natural_transformation.mk
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(λ a, η a ∘ θ a)
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(λ a b f,
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calc
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H f ∘ (η a ∘ θ a) = (H f ∘ η a) ∘ θ a : assoc
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... = (η b ∘ G f) ∘ θ a : naturality η f
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... = η b ∘ (G f ∘ θ a) : assoc
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... = η b ∘ (θ b ∘ F f) : naturality θ f
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... = (η b ∘ θ b) ∘ F f : assoc)
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--congr_arg (λx, η b ∘ x) (naturality θ f) -- this needed to be explicit for some reason (on Oct 24)
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infixr `∘n`:60 := compose
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protected theorem assoc (η₃ : H ⟹ I) (η₂ : G ⟹ H) (η₁ : F ⟹ G) :
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η₃ ∘n (η₂ ∘n η₁) = (η₃ ∘n η₂) ∘n η₁ :=
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dcongr_arg2 mk (funext (take x, !assoc)) !proof_irrel
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protected definition id {C D : Category} {F : functor C D} : natural_transformation F F :=
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mk (λa, id) (λa b f, !id_right ⬝ symm !id_left)
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protected definition ID {C D : Category} (F : functor C D) : natural_transformation F F := natural_transformation.id
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protected theorem id_left (η : F ⟹ G) : natural_transformation.compose natural_transformation.id η = η :=
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natural_transformation.rec (λf H, dcongr_arg2 mk (funext (take x, !id_left)) !proof_irrel) η
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protected theorem id_right (η : F ⟹ G) : natural_transformation.compose η natural_transformation.id = η :=
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natural_transformation.rec (λf H, dcongr_arg2 mk (funext (take x, !id_right)) !proof_irrel) η
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end natural_transformation
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