92 lines
3.5 KiB
Text
92 lines
3.5 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.precategory.nat_trans
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Author: Floris van Doorn, Jakob von Raumer
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-/
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import .functor .morphism
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open eq category functor is_trunc equiv sigma.ops sigma is_equiv function pi funext
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structure nat_trans {C D : Precategory} (F G : C ⇒ D) :=
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(natural_map : Π (a : C), hom (F a) (G a))
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(naturality : Π {a b : C} (f : hom a b), G f ∘ natural_map a = natural_map b ∘ F f)
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namespace nat_trans
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infixl `⟹`:25 := nat_trans -- \==>
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variables {C D : Precategory} {F G H I : C ⇒ D}
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attribute natural_map [coercion]
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protected definition compose [reducible] (η : G ⟹ H) (θ : F ⟹ G) : F ⟹ H :=
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nat_trans.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 definition id [reducible] {C D : Precategory} {F : functor C D} : nat_trans F F :=
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mk (λa, id) (λa b f, !id_right ⬝ !id_left⁻¹)
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protected definition ID [reducible] {C D : Precategory} (F : functor C D) : nat_trans F F :=
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id
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local attribute is_hprop_eq_hom [instance]
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definition nat_trans_eq_mk' {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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(p : η₁ ∼ η₂)
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: nat_trans.mk η₁ nat₁ = nat_trans.mk η₂ nat₂ :=
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apD011 nat_trans.mk (eq_of_homotopy p) !is_hprop.elim
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definition nat_trans_eq_mk {η₁ η₂ : F ⟹ G} : natural_map η₁ ∼ natural_map η₂ → η₁ = η₂ :=
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nat_trans.rec_on η₁ (λf₁ nat₁, nat_trans.rec_on η₂ (λf₂ nat₂ p, !nat_trans_eq_mk' p))
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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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nat_trans_eq_mk (λa, !assoc)
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protected definition id_left (η : F ⟹ G) : id ∘n η = η :=
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nat_trans_eq_mk (λa, !id_left)
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protected definition id_right (η : F ⟹ G) : η ∘n id = η :=
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nat_trans_eq_mk (λa, !id_right)
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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 nat_trans.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 η, apply nat_trans_eq_mk, intro a, apply idp,
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intro S,
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fapply sigma_eq,
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apply eq_of_homotopy, intro a,
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apply idp,
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apply is_hprop.elim,
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end
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set_option apply.class_instance false
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definition is_hset_nat_trans : is_hset (F ⟹ G) :=
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begin
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apply is_trunc_is_equiv_closed, apply (equiv.to_is_equiv !sigma_char),
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apply is_trunc_sigma,
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apply is_trunc_pi, intro a, exact (@homH (Precategory.carrier D) _ (F a) (G a)),
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intro η, apply is_trunc_pi, intro a,
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apply is_trunc_pi, intro b, apply is_trunc_pi, intro f,
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apply is_trunc_eq, apply is_trunc_succ, exact (@homH (Precategory.carrier D) _ (F a) (G b)),
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end
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end nat_trans
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