124 lines
4.6 KiB
Text
124 lines
4.6 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, Jakob von Raumer
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import .functor types.pi
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open eq precategory functor truncation equiv sigma.ops sigma is_equiv function pi
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inductive natural_transformation {C D : Precategory} (F G : C ⇒ D) : Type :=
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mk : Π (η : Π (a : C), hom (F a) (G a))
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(nat : 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 : Precategory} {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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infixr `∘n`:60 := compose
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protected theorem congr
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{C : Precategory} {D : Precategory}
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(F G : C ⇒ D)
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(η₁ η₂ : Π (a : C), hom (F a) (G a))
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(nat₁ : Π (a b : C) (f : hom a b), G f ∘ η₁ a = η₁ b ∘ F f)
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(nat₂ : Π (a b : C) (f : hom a b), G f ∘ η₂ a = η₂ b ∘ F f)
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(p₁ : η₁ = η₂) (p₂ : p₁ ▹ nat₁ = nat₂)
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: @natural_transformation.mk C D F G η₁ nat₁ = @natural_transformation.mk C D F G η₂ nat₂
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:=
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begin
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apply (dcongr_arg2 (@natural_transformation.mk C D F G) p₁ p₂),
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end
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protected definition 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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begin
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apply (rec_on η₃), intros (η₃1, η₃2),
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apply (rec_on η₂), intros (η₂1, η₂2),
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apply (rec_on η₁), intros (η₁1, η₁2),
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fapply natural_transformation.congr,
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apply funext.path_pi, intro a,
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apply assoc,
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apply funext.path_pi, intro a,
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apply funext.path_pi, intro b,
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apply funext.path_pi, intro f,
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apply (@is_hset.elim), apply !homH,
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end
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protected definition id {C D : Precategory} {F : functor C D} : natural_transformation F F :=
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mk (λa, id) (λa b f, !id_right ⬝ (!id_left⁻¹))
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protected definition ID {C D : Precategory} (F : functor C D) : natural_transformation F F :=
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id
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protected definition id_left (η : F ⟹ G) : id ∘n η = η :=
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begin
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apply (rec_on η), intros (η₁, nat₁),
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fapply (natural_transformation.congr F G),
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apply funext.path_pi, intro a,
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apply id_left,
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apply funext.path_pi, intro a,
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apply funext.path_pi, intro b,
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apply funext.path_pi, intro f,
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apply (@is_hset.elim), apply !homH,
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end
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protected definition id_right (η : F ⟹ G) : η ∘n id = η :=
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begin
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apply (rec_on η), intros (η₁, nat₁),
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fapply (natural_transformation.congr F G),
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apply funext.path_pi, intro a,
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apply id_right,
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apply funext.path_pi, intro a,
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apply funext.path_pi, intro b,
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apply funext.path_pi, intro f,
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apply (@is_hset.elim), apply !homH,
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end
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protected definition sigma_char :
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(Σ (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) ≃ (F ⟹ G) :=
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/-begin
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intro what,
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fapply equiv.mk,
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intro S, apply natural_transformation.mk, exact (S.2),
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fapply adjointify,
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intro H, apply (natural_transformation.rec_on H), intros (η, natu),
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exact (sigma.mk η @natu),
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intro H, apply (natural_transformation.rec_on _ _ _),
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intro S2,
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end-/
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/-(λ x, equiv.mk (λ S, natural_transformation.mk S.1 S.2)
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(adjointify (λ S, natural_transformation.mk S.1 S.2)
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(λ H, natural_transformation.rec_on H (λ η nat, sigma.mk η nat))
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(λ H, natural_transformation.rec_on H (λ η nat, refl (natural_transformation.mk η nat)))
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(λ S, sigma.rec_on S (λ η nat, refl (sigma.mk η nat)))))-/
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sorry
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protected definition to_hset : is_hset (F ⟹ G) :=
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begin
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apply trunc_equiv, apply (equiv.to_is_equiv sigma_char),
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apply trunc_sigma,
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apply trunc_pi, intro a, exact (@homH (objects D) _ (F a) (G a)),
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intro η, apply trunc_pi, intro a,
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apply trunc_pi, intro b, apply trunc_pi, intro f,
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apply succ_is_trunc, apply trunc_succ, exact (@homH (objects D) _ (F a) (G b)),
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end
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end natural_transformation
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