d8a616fa70
I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
49 lines
2.1 KiB
Text
49 lines
2.1 KiB
Text
-- 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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-- Author: Floris van Doorn
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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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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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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 := id
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protected theorem id_left (η : F ⟹ G) : natural_transformation.compose id η = η :=
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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 id = η :=
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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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