158 lines
5.1 KiB
Text
158 lines
5.1 KiB
Text
/-
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Copyright (c) 2015 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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Authors: Floris van Doorn, Jakob von Raumer
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Opposite precategory and (TODO) category
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-/
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import ..nat_trans ..category
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open eq functor iso equiv is_equiv nat_trans
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namespace category
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definition opposite [reducible] [constructor] {ob : Type} (C : precategory ob) : precategory ob :=
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precategory.mk' (λ a b, hom b a)
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(λ a b c f g, g ∘ f)
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(λ a, id)
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(λ a b c d f g h, !assoc')
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(λ a b c d f g h, !assoc)
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(λ a b f, !id_right)
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(λ a b f, !id_left)
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(λ a, !id_id)
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(λ a b, !is_hset_hom)
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definition Opposite [reducible] [constructor] (C : Precategory) : Precategory :=
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precategory.Mk (opposite C)
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infixr `∘op`:60 := @comp _ (opposite _) _ _ _
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postfix `ᵒᵖ`:(max+2) := Opposite
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variables {C D E : Precategory} {a b c : C}
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definition compose_op {f : hom a b} {g : hom b c} : f ∘op g = g ∘ f :=
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by reflexivity
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definition opposite_opposite' {ob : Type} (C : precategory ob) : opposite (opposite C) = C :=
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by cases C; apply idp
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definition opposite_opposite : (Cᵒᵖ)ᵒᵖ = C :=
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(ap (Precategory.mk C) (opposite_opposite' C)) ⬝ !Precategory.eta
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theorem opposite_hom_of_eq {ob : Type} [C : precategory ob] {c c' : ob} (p : c = c')
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: @hom_of_eq ob (opposite C) c c' p = inv_of_eq p :=
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by induction p; reflexivity
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theorem opposite_inv_of_eq {ob : Type} [C : precategory ob] {c c' : ob} (p : c = c')
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: @inv_of_eq ob (opposite C) c c' p = hom_of_eq p :=
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by induction p; reflexivity
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definition opposite_functor [constructor] (F : C ⇒ D) : Cᵒᵖ ⇒ Dᵒᵖ :=
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begin
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apply functor.mk,
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intros, apply respect_id F,
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intros, apply @respect_comp C D
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end
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definition opposite_functor_rev [constructor] (F : Cᵒᵖ ⇒ Dᵒᵖ) : C ⇒ D :=
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begin
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apply functor.mk,
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intros, apply respect_id F,
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intros, apply @respect_comp Cᵒᵖ Dᵒᵖ
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end
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postfix `ᵒᵖᶠ`:(max+2) := opposite_functor
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postfix `ᵒᵖ'`:(max+2) := opposite_functor_rev
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definition functor_id_op (C : Precategory) : (1 : C ⇒ C)ᵒᵖᶠ = 1 :=
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idp
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definition opposite_rev_opposite_functor (F : Cᵒᵖ ⇒ Dᵒᵖ) : Fᵒᵖ' ᵒᵖᶠ = F :=
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begin
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fapply functor_eq: esimp,
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{ intro c c' f, esimp, exact !id_right ⬝ !id_left}
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end
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definition opposite_opposite_rev_functor (F : C ⇒ D) : Fᵒᵖᶠᵒᵖ' = F :=
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begin
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fapply functor_eq: esimp,
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{ intro c c' f, esimp, exact !id_leftright}
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end
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definition opposite_compose (G : D ⇒ E) (F : C ⇒ D) : (G ∘f F)ᵒᵖᶠ = Gᵒᵖᶠ ∘f Fᵒᵖᶠ :=
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idp
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definition opposite_nat_trans [constructor] {F G : C ⇒ D} (η : F ⟹ G) : Gᵒᵖᶠ ⟹ Fᵒᵖᶠ :=
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begin
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fapply nat_trans.mk: esimp,
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{ intro c, exact η c},
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{ intro c c' f, exact !naturality⁻¹},
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end
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definition opposite_rev_nat_trans [constructor] {F G : Cᵒᵖ ⇒ Dᵒᵖ} (η : F ⟹ G) : Gᵒᵖ' ⟹ Fᵒᵖ' :=
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begin
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fapply nat_trans.mk: esimp,
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{ intro c, exact η c},
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{ intro c c' f, exact !(@naturality Cᵒᵖ Dᵒᵖ)⁻¹},
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end
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definition opposite_nat_trans_rev [constructor] {F G : C ⇒ D} (η : Fᵒᵖᶠ ⟹ Gᵒᵖᶠ) : G ⟹ F :=
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begin
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fapply nat_trans.mk: esimp,
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{ intro c, exact η c},
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{ intro c c' f, exact !(@naturality Cᵒᵖ Dᵒᵖ _ _ η)⁻¹},
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end
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definition opposite_rev_nat_trans_rev [constructor] {F G : Cᵒᵖ ⇒ Dᵒᵖ} (η : Fᵒᵖ' ⟹ Gᵒᵖ') : G ⟹ F :=
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begin
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fapply nat_trans.mk: esimp,
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{ intro c, exact η c},
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{ intro c c' f, exact (naturality η f)⁻¹},
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end
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definition opposite_iso [constructor] {ob : Type} [C : precategory ob] {a b : ob}
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(H : @iso _ C a b) : @iso _ (opposite C) a b :=
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begin
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fapply @iso.MK _ (opposite C),
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{ exact to_inv H},
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{ exact to_hom H},
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{ exact to_left_inverse H},
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{ exact to_right_inverse H},
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end
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definition iso_of_opposite_iso [constructor] {ob : Type} [C : precategory ob] {a b : ob}
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(H : @iso _ (opposite C) a b) : @iso _ C a b :=
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begin
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fapply iso.MK,
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{ exact to_inv H},
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{ exact to_hom H},
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{ exact to_left_inverse H},
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{ exact to_right_inverse H},
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end
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definition opposite_iso_equiv [constructor] {ob : Type} [C : precategory ob] (a b : ob)
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: @iso _ (opposite C) a b ≃ @iso _ C a b :=
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begin
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fapply equiv.MK,
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{ exact iso_of_opposite_iso},
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{ exact opposite_iso},
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{ intro H, apply iso_eq, reflexivity},
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{ intro H, apply iso_eq, reflexivity},
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end
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definition is_univalent_opposite (C : Category) : is_univalent (Opposite C) :=
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begin
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intro x y,
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fapply is_equiv_of_equiv_of_homotopy,
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{ refine @eq_equiv_iso C C x y ⬝e _, symmetry, esimp at *, apply opposite_iso_equiv},
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{ intro p, induction p, reflexivity}
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end
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definition category_opposite [constructor] (C : Category) : category (Opposite C) :=
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category.mk _ (is_univalent_opposite C)
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definition Category_opposite [constructor] (C : Category) : Category :=
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Category.mk _ (category_opposite C)
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end category
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