176 lines
6 KiB
Text
176 lines
6 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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Functor precategory and category
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-/
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import ..nat_trans ..category
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open eq functor is_trunc nat_trans iso is_equiv
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namespace category
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definition precategory_functor [instance] [reducible] [constructor] (D C : Precategory)
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: precategory (functor C D) :=
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precategory.mk (λa b, nat_trans a b)
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(λ a b c g f, nat_trans.compose g f)
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(λ a, nat_trans.id)
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(λ a b c d h g f, !nat_trans.assoc)
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(λ a b f, !nat_trans.id_left)
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(λ a b f, !nat_trans.id_right)
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definition Precategory_functor [reducible] (D C : Precategory) : Precategory :=
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precategory.Mk (precategory_functor D C)
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infixr `^c`:35 := Precategory_functor
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section
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/- we prove that if a natural transformation is pointwise an iso, then it is an iso -/
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variables {C D : Precategory} {F G : C ⇒ D} (η : F ⟹ G) [iso : Π(a : C), is_iso (η a)]
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include iso
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definition nat_trans_inverse : G ⟹ F :=
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nat_trans.mk
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(λc, (η c)⁻¹)
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(λc d f,
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begin
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apply comp_inverse_eq_of_eq_comp,
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transitivity (natural_map η d)⁻¹ ∘ to_fun_hom G f ∘ natural_map η c,
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{apply eq_inverse_comp_of_comp_eq, symmetry, apply naturality},
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{apply assoc}
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end)
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definition nat_trans_left_inverse : nat_trans_inverse η ∘n η = nat_trans.id :=
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begin
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fapply (apd011 nat_trans.mk),
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apply eq_of_homotopy, intro c, apply left_inverse,
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apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros,
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apply is_hset.elim
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end
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definition nat_trans_right_inverse : η ∘n nat_trans_inverse η = nat_trans.id :=
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begin
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fapply (apd011 nat_trans.mk),
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apply eq_of_homotopy, intro c, apply right_inverse,
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apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros,
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apply is_hset.elim
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end
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definition is_iso_nat_trans : is_iso η :=
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is_iso.mk (nat_trans_left_inverse η) (nat_trans_right_inverse η)
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end
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section
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/- and conversely, if a natural transformation is an iso, it is componentwise an iso -/
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variables {C D : Precategory} {F G : D ^c C} (η : hom F G) [isoη : is_iso η] (c : C)
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include isoη
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definition componentwise_is_iso : is_iso (η c) :=
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@is_iso.mk _ _ _ _ _ (natural_map η⁻¹ c) (ap010 natural_map ( left_inverse η) c)
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(ap010 natural_map (right_inverse η) c)
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local attribute componentwise_is_iso [instance]
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definition natural_map_inverse : natural_map η⁻¹ c = (η c)⁻¹ := idp
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definition naturality_iso {c c' : C} (f : c ⟶ c') : G f = η c' ∘ F f ∘ (η c)⁻¹ :=
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calc
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G f = (G f ∘ η c) ∘ (η c)⁻¹ : by rewrite comp_inverse_cancel_right
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... = (η c' ∘ F f) ∘ (η c)⁻¹ : by rewrite naturality
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... = η c' ∘ F f ∘ (η c)⁻¹ : by rewrite assoc
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definition naturality_iso' {c c' : C} (f : c ⟶ c') : (η c')⁻¹ ∘ G f ∘ η c = F f :=
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calc
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(η c')⁻¹ ∘ G f ∘ η c = (η c')⁻¹ ∘ η c' ∘ F f : by rewrite naturality
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... = F f : by rewrite inverse_comp_cancel_left
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omit isoη
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definition componentwise_iso (η : F ≅ G) (c : C) : F c ≅ G c :=
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@iso.mk _ _ _ _ (natural_map (to_hom η) c)
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(@componentwise_is_iso _ _ _ _ (to_hom η) (struct η) c)
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definition componentwise_iso_id (c : C) : componentwise_iso (iso.refl F) c = iso.refl (F c) :=
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iso_eq (idpath (ID (F c)))
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definition componentwise_iso_iso_of_eq (p : F = G) (c : C)
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: componentwise_iso (iso_of_eq p) c = iso_of_eq (ap010 to_fun_ob p c) :=
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eq.rec_on p !componentwise_iso_id
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definition natural_map_hom_of_eq (p : F = G) (c : C)
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: natural_map (hom_of_eq p) c = hom_of_eq (ap010 to_fun_ob p c) :=
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eq.rec_on p idp
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definition natural_map_inv_of_eq (p : F = G) (c : C)
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: natural_map (inv_of_eq p) c = hom_of_eq (ap010 to_fun_ob p c)⁻¹ :=
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eq.rec_on p idp
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end
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namespace functor
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variables {C : Precategory} {D : Category} {F G : D ^c C}
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definition eq_of_iso_ob (η : F ≅ G) (c : C) : F c = G c :=
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by apply eq_of_iso; apply componentwise_iso; exact η
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local attribute functor.to_fun_hom [quasireducible]
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definition eq_of_iso (η : F ≅ G) : F = G :=
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begin
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fapply functor_eq,
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{exact (eq_of_iso_ob η)},
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{intro c c' f,
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esimp [eq_of_iso_ob, inv_of_eq, hom_of_eq, eq_of_iso],
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rewrite [*right_inv iso_of_eq],
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esimp [function.id],
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symmetry, apply @naturality_iso _ _ _ _ _ (iso.struct _)
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}
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end
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definition iso_of_eq_eq_of_iso (η : F ≅ G) : iso_of_eq (eq_of_iso η) = η :=
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begin
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apply iso_eq,
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apply nat_trans_eq,
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intro c,
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rewrite natural_map_hom_of_eq, esimp [eq_of_iso],
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rewrite ap010_functor_eq, esimp [hom_of_eq,eq_of_iso_ob],
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rewrite (right_inv iso_of_eq),
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end
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definition eq_of_iso_iso_of_eq (p : F = G) : eq_of_iso (iso_of_eq p) = p :=
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begin
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apply functor_eq2,
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intro c,
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esimp [eq_of_iso],
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rewrite ap010_functor_eq,
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esimp [eq_of_iso_ob],
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rewrite componentwise_iso_iso_of_eq,
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rewrite (left_inv iso_of_eq)
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end
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definition is_univalent (D : Category) (C : Precategory) : is_univalent (D ^c C) :=
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λF G, adjointify _ eq_of_iso
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iso_of_eq_eq_of_iso
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eq_of_iso_iso_of_eq
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end functor
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definition category_functor [instance] (D : Category) (C : Precategory)
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: category (D ^c C) :=
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category.mk (D ^c C) (functor.is_univalent D C)
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definition Category_functor (D : Category) (C : Precategory) : Category :=
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category.Mk (D ^c C) !category_functor
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--this definition is only useful if the exponent is a category,
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-- and the elaborator has trouble with inserting the coercion
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definition Category_functor' (D C : Category) : Category :=
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Category_functor D C
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namespace ops
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infixr `^c2`:35 := Category_functor
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end ops
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end category
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