426 lines
16 KiB
Text
426 lines
16 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 .opposite
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open eq category is_trunc nat_trans iso is_equiv category.hom
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namespace functor
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definition precategory_functor [instance] [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] [constructor] (D C : Precategory) : Precategory :=
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precategory.Mk (precategory_functor D C)
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infixr ` ^c `:80 := 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 [constructor] : 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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abstract 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 end)
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definition nat_trans_left_inverse : nat_trans_inverse η ∘n η = 1 :=
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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_set.elim
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end
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definition nat_trans_right_inverse : η ∘n nat_trans_inverse η = 1 :=
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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_set.elim
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end
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definition is_natural_iso [constructor] : is_iso η :=
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is_iso.mk _ (nat_trans_left_inverse η) (nat_trans_right_inverse η)
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variable (iso)
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definition natural_iso.mk [constructor] : F ≅ G :=
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iso.mk _ (is_natural_iso η)
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omit iso
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variables (F G)
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definition is_natural_inverse (η : Πc, F c ≅ G c)
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(nat : Π⦃a b : C⦄ (f : hom a b), G f ∘ to_hom (η a) = to_hom (η b) ∘ F f)
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{a b : C} (f : hom a b) : F f ∘ to_inv (η a) = to_inv (η b) ∘ G f :=
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let η' : F ⟹ G := nat_trans.mk (λc, to_hom (η c)) @nat in
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naturality (nat_trans_inverse η') f
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definition is_natural_inverse' (η₁ : Πc, F c ≅ G c) (η₂ : F ⟹ G) (p : η₁ ~ η₂)
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{a b : C} (f : hom a b) : F f ∘ to_inv (η₁ a) = to_inv (η₁ b) ∘ G f :=
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is_natural_inverse F G η₁ abstract λa b g, (p a)⁻¹ ▸ (p b)⁻¹ ▸ naturality η₂ g end f
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variables {F G}
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definition natural_iso.MK [constructor]
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(η : Πc, F c ⟶ G c) (p : Π(c c' : C) (f : c ⟶ c'), G f ∘ η c = η c' ∘ F f)
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(θ : Πc, G c ⟶ F c) (r : Πc, θ c ∘ η c = id) (q : Πc, η c ∘ θ c = id) : F ≅ G :=
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iso.mk (nat_trans.mk η p) (@(is_natural_iso _) (λc, is_iso.mk (θ c) (r c) (q c)))
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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 {A B C D : Precategory} {F G : C ⇒ D} (η : hom F G) [isoη : is_iso η] (c : C)
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include isoη
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definition componentwise_is_iso [constructor] : 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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variable {isoη}
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definition natural_map_inverse : natural_map η⁻¹ c = (η c)⁻¹ := idp
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variable [isoη]
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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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theorem naturality_iso_id {F : C ⇒ C} (η : F ≅ 1) (c : C)
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: componentwise_iso η (F c) = F (componentwise_iso η c) :=
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comp.cancel_left (to_hom (componentwise_iso η c))
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((naturality (to_hom η)) (to_hom (componentwise_iso η c)))
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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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definition hom_of_eq_compose_right {H : B ⇒ C} (p : F = G)
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: hom_of_eq (ap (λx, x ∘f H) p) = hom_of_eq p ∘nf H :=
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eq.rec_on p idp
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definition inv_of_eq_compose_right {H : B ⇒ C} (p : F = G)
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: inv_of_eq (ap (λx, x ∘f H) p) = inv_of_eq p ∘nf H :=
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eq.rec_on p idp
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definition hom_of_eq_compose_left {H : D ⇒ C} (p : F = G)
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: hom_of_eq (ap (λx, H ∘f x) p) = H ∘fn hom_of_eq p :=
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by induction p; exact !fn_id⁻¹
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definition inv_of_eq_compose_left {H : D ⇒ C} (p : F = G)
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: inv_of_eq (ap (λx, H ∘f x) p) = H ∘fn inv_of_eq p :=
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by induction p; exact !fn_id⁻¹
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definition assoc_natural [constructor] (H : C ⇒ D) (G : B ⇒ C) (F : A ⇒ B)
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: H ∘f (G ∘f F) ⟹ (H ∘f G) ∘f F :=
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change_natural_map (hom_of_eq !functor.assoc)
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(λa, id)
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(λa, !natural_map_hom_of_eq ⬝ ap hom_of_eq !ap010_assoc)
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definition assoc_natural_rev [constructor] (H : C ⇒ D) (G : B ⇒ C) (F : A ⇒ B)
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: (H ∘f G) ∘f F ⟹ H ∘f (G ∘f F) :=
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change_natural_map (inv_of_eq !functor.assoc)
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(λa, id)
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(λa, !natural_map_inv_of_eq ⬝ ap (λx, hom_of_eq x⁻¹) !ap010_assoc)
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definition id_left_natural [constructor] (F : C ⇒ D) : functor.id ∘f F ⟹ F :=
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change_natural_map
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(hom_of_eq !functor.id_left)
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(λc, id)
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(λc, by induction F; exact !natural_map_hom_of_eq ⬝ ap hom_of_eq !ap010_functor_mk_eq_constant)
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definition id_left_natural_rev [constructor] (F : C ⇒ D) : F ⟹ functor.id ∘f F :=
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change_natural_map
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(inv_of_eq !functor.id_left)
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(λc, id)
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(λc, by induction F; exact !natural_map_inv_of_eq ⬝
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ap (λx, hom_of_eq x⁻¹) !ap010_functor_mk_eq_constant)
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definition id_right_natural [constructor] (F : C ⇒ D) : F ∘f functor.id ⟹ F :=
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change_natural_map
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(hom_of_eq !functor.id_right)
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(λc, id)
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(λc, by induction F; exact !natural_map_hom_of_eq ⬝ ap hom_of_eq !ap010_functor_mk_eq_constant)
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definition id_right_natural_rev [constructor] (F : C ⇒ D) : F ⟹ F ∘f functor.id :=
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change_natural_map
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(inv_of_eq !functor.id_right)
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(λc, id)
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(λc, by induction F; exact !natural_map_inv_of_eq ⬝
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ap (λx, hom_of_eq x⁻¹) !ap010_functor_mk_eq_constant)
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end
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section
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variables {C D E : Precategory} {G G' : D ⇒ E} {F F' : C ⇒ D} {J : D ⇒ D}
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definition is_iso_nf_compose [constructor] (G : D ⇒ E) (η : F ⟹ F') [H : is_iso η]
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: is_iso (G ∘fn η) :=
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is_iso.mk
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(G ∘fn @inverse (C ⇒ D) _ _ _ η _)
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abstract !fn_n_distrib⁻¹ ⬝ ap (λx, G ∘fn x) (@left_inverse (C ⇒ D) _ _ _ η _) ⬝ !fn_id end
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abstract !fn_n_distrib⁻¹ ⬝ ap (λx, G ∘fn x) (@right_inverse (C ⇒ D) _ _ _ η _) ⬝ !fn_id end
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definition is_iso_fn_compose [constructor] (η : G ⟹ G') (F : C ⇒ D) [H : is_iso η]
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: is_iso (η ∘nf F) :=
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is_iso.mk
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(@inverse (D ⇒ E) _ _ _ η _ ∘nf F)
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abstract !n_nf_distrib⁻¹ ⬝ ap (λx, x ∘nf F) (@left_inverse (D ⇒ E) _ _ _ η _) ⬝ !id_nf end
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abstract !n_nf_distrib⁻¹ ⬝ ap (λx, x ∘nf F) (@right_inverse (D ⇒ E) _ _ _ η _) ⬝ !id_nf end
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definition functor_iso_compose [constructor] (G : D ⇒ E) (η : F ≅ F') : G ∘f F ≅ G ∘f F' :=
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iso.mk _ (is_iso_nf_compose G (to_hom η))
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definition iso_functor_compose [constructor] (η : G ≅ G') (F : C ⇒ D) : G ∘f F ≅ G' ∘f F :=
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iso.mk _ (is_iso_fn_compose (to_hom η) F)
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infixr ` ∘fi ` :62 := functor_iso_compose
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infixr ` ∘if ` :62 := iso_functor_compose
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/- TODO: also needs n_nf_distrib and id_nf for these compositions
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definition nidf_compose [constructor] (η : J ⟹ 1) (F : C ⇒ D) [H : is_iso η]
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: is_iso (η ∘n1f F) :=
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is_iso.mk
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(@inverse (D ⇒ D) _ _ _ η _ ∘1nf F)
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abstract _ end
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_
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definition idnf_compose [constructor] (η : 1 ⟹ J) (F : C ⇒ D) [H : is_iso η]
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: is_iso (η ∘1nf F) :=
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is_iso.mk _
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_
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_
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definition fnid_compose [constructor] (F : D ⇒ E) (η : J ⟹ 1) [H : is_iso η]
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: is_iso (F ∘fn1 η) :=
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is_iso.mk _
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_
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_
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definition fidn_compose [constructor] (F : D ⇒ E) (η : 1 ⟹ J) [H : is_iso η]
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: is_iso (F ∘f1n η) :=
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is_iso.mk _
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_
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_
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-/
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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 [reducible]
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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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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] [constructor] (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 [constructor] (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' [constructor] (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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namespace functor
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variables {C : Precategory} {D : Category} {F G : D ^c C}
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definition eq_of_pointwise_iso (η : F ⟹ G) (iso : Π(a : C), is_iso (η a)) : F = G :=
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eq_of_iso (natural_iso.mk η iso)
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definition iso_of_eq_eq_of_pointwise_iso (η : F ⟹ G) (iso : Π(c : C), is_iso (η c))
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: iso_of_eq (eq_of_pointwise_iso η iso) = natural_iso.mk η iso :=
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!iso_of_eq_eq_of_iso
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definition hom_of_eq_eq_of_pointwise_iso (η : F ⟹ G) (iso : Π(c : C), is_iso (η c))
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: hom_of_eq (eq_of_pointwise_iso η iso) = η :=
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!hom_of_eq_eq_of_iso
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definition inv_of_eq_eq_of_pointwise_iso (η : F ⟹ G) (iso : Π(c : C), is_iso (η c))
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: inv_of_eq (eq_of_pointwise_iso η iso) = nat_trans_inverse η :=
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!inv_of_eq_eq_of_iso
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end functor
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/-
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functors involving only the functor category
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(see ..functor.curry for some other functors involving also products)
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-/
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variables {C D I : Precategory}
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definition constant2_functor [constructor] (F : I ⇒ D ^c C) (c : C) : I ⇒ D :=
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functor.mk (λi, to_fun_ob (F i) c)
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(λi j f, natural_map (F f) c)
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abstract (λi, ap010 natural_map !respect_id c ⬝ proof idp qed) end
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abstract (λi j k g f, ap010 natural_map !respect_comp c) end
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definition constant2_functor_natural [constructor] (F : I ⇒ D ^c C) {c d : C} (f : c ⟶ d)
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: constant2_functor F c ⟹ constant2_functor F d :=
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nat_trans.mk (λi, to_fun_hom (F i) f)
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(λi j k, (naturality (F k) f)⁻¹)
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definition functor_flip [constructor] (F : I ⇒ D ^c C) : C ⇒ D ^c I :=
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functor.mk (constant2_functor F)
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@(constant2_functor_natural F)
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abstract begin intros, apply nat_trans_eq, intro i, esimp, apply respect_id end end
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abstract begin intros, apply nat_trans_eq, intro i, esimp, apply respect_comp end end
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definition eval_functor [constructor] (C D : Precategory) (d : D) : C ^c D ⇒ C :=
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begin
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fapply functor.mk: esimp,
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{ intro F, exact F d},
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{ intro G F η, exact η d},
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{ intro F, reflexivity},
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{ intro H G F η θ, reflexivity},
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end
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definition precomposition_functor [constructor] {C D} (E) (F : C ⇒ D)
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: E ^c D ⇒ E ^c C :=
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begin
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fapply functor.mk: esimp,
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{ intro G, exact G ∘f F},
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{ intro G H η, exact η ∘nf F},
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{ intro G, reflexivity},
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{ intro G H I η θ, reflexivity},
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end
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definition postcomposition_functor [constructor] {C D} (E) (F : C ⇒ D)
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: C ^c E ⇒ D ^c E :=
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begin
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fapply functor.mk: esimp,
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{ intro G, exact F ∘f G},
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{ intro G H η, exact F ∘fn η},
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{ intro G, apply fn_id},
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{ intro G H I η θ, apply fn_n_distrib},
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end
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definition constant_diagram [constructor] (C D) : C ⇒ C ^c D :=
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begin
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fapply functor.mk: esimp,
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{ intro c, exact constant_functor D c},
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{ intro c d f, exact constant_nat_trans D f},
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{ intro c, fapply nat_trans_eq, reflexivity},
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{ intro c d e g f, fapply nat_trans_eq, reflexivity},
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end
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definition opposite_functor_opposite_left [constructor] (C D : Precategory)
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: (C ^c D)ᵒᵖ ⇒ Cᵒᵖ ^c Dᵒᵖ :=
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begin
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fapply functor.mk: esimp,
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{ exact opposite_functor},
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{ intro F G, exact opposite_nat_trans},
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{ intro F, apply nat_trans_eq, reflexivity},
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{ intro u v w g f, apply nat_trans_eq, reflexivity}
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end
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definition opposite_functor_opposite_right [constructor] (C D : Precategory)
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: Cᵒᵖ ^c Dᵒᵖ ⇒ (C ^c D)ᵒᵖ :=
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begin
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fapply functor.mk: esimp,
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{ exact opposite_functor_rev},
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{ apply @opposite_rev_nat_trans},
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{ intro F, apply nat_trans_eq, intro d, reflexivity},
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{ intro F G H η θ, apply nat_trans_eq, intro d, reflexivity}
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end
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definition constant_diagram_opposite [constructor] (C D)
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: (constant_diagram C D)ᵒᵖᶠ = opposite_functor_opposite_right C D ∘f constant_diagram Cᵒᵖ Dᵒᵖ :=
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begin
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fapply functor_eq,
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{ reflexivity},
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{ intro c c' f, esimp at *, refine !nat_trans.id_right ⬝ !nat_trans.id_left ⬝ _,
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apply nat_trans_eq, intro d, reflexivity}
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end
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end functor
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