132 lines
5 KiB
Text
132 lines
5 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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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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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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set_option apply.class_instance false -- disable class instance resolution in the apply tactic
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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 (natural_transformation.rec_on η₃), intros (η₃1, η₃2),
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apply (natural_transformation.rec_on η₂), intros (η₂1, η₂2),
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apply (natural_transformation.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 (natural_transformation.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 (natural_transformation.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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--set_option pp.implicit true
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protected definition sigma_char (F G : C ⇒ D) :
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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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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,
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fapply sigma.mk,
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intro a, exact (H a),
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intros (a, b, f), exact (naturality H f),
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intro H, apply (natural_transformation.rec_on H),
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intros (eta, nat), unfold function.id,
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fapply natural_transformation.congr,
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apply idp,
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repeat ( apply funext.path_pi ; intro a ),
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apply (@is_hset.elim), apply !homH,
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intro S,
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fapply sigma.path,
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apply funext.path_pi, intro a,
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apply idp,
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repeat ( apply funext.path_pi ; intro a ),
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apply (@is_hset.elim), apply !homH,
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end
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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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