416 lines
16 KiB
Text
416 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
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Functors which are equivalences or isomorphisms
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-/
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import .adjoint
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open eq functor iso prod nat_trans is_equiv equiv is_trunc
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namespace category
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variables {C D : Precategory} {F : C ⇒ D} {G : D ⇒ C}
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structure is_equivalence [class] (F : C ⇒ D) extends is_left_adjoint F :=
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mk' ::
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(is_iso_unit : is_iso η)
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(is_iso_counit : is_iso ε)
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abbreviation inverse := @is_equivalence.G
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postfix ⁻¹ := inverse
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--a second notation for the inverse, which is not overloaded (there is no unicode superscript F)
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postfix [parsing_only] `⁻¹ᴱ`:std.prec.max_plus := inverse
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definition is_isomorphism [class] (F : C ⇒ D) := fully_faithful F × is_equiv (to_fun_ob F)
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structure equivalence (C D : Precategory) :=
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(to_functor : C ⇒ D)
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(struct : is_equivalence to_functor)
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structure isomorphism (C D : Precategory) :=
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(to_functor : C ⇒ D)
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(struct : is_isomorphism to_functor)
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infix ` ≃c `:25 := equivalence
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infix ` ≅c `:25 := isomorphism
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attribute equivalence.struct isomorphism.struct [instance] [priority 1500]
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attribute equivalence.to_functor isomorphism.to_functor [coercion]
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definition is_iso_unit [instance] (F : C ⇒ D) [H : is_equivalence F] : is_iso (unit F) :=
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!is_equivalence.is_iso_unit
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definition is_iso_counit [instance] (F : C ⇒ D) [H : is_equivalence F] : is_iso (counit F) :=
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!is_equivalence.is_iso_counit
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definition iso_unit (F : C ⇒ D) [H : is_equivalence F] : F⁻¹ᴱ ∘f F ≅ 1 :=
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(@(iso.mk _) !is_iso_unit)⁻¹ⁱ
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definition iso_counit (F : C ⇒ D) [H : is_equivalence F] : F ∘f F⁻¹ᴱ ≅ 1 :=
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@(iso.mk _) !is_iso_counit
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definition split_essentially_surjective_of_is_equivalence (F : C ⇒ D)
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[H : is_equivalence F] : split_essentially_surjective F :=
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begin
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intro d, fconstructor,
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{ exact F⁻¹ d},
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{ exact componentwise_iso (@(iso.mk (counit F)) !is_iso_counit) d}
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end
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end category
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namespace category
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section
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parameters {C D : Precategory} {F : C ⇒ D} {G : D ⇒ C} (η : G ∘f F ≅ 1) (ε : F ∘f G ≅ 1)
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private definition ηn : 1 ⟹ G ∘f F := to_inv η
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private definition εn : F ∘f G ⟹ 1 := to_hom ε
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private definition ηi (c : C) : G (F c) ≅ c := componentwise_iso η c
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private definition εi (d : D) : F (G d) ≅ d := componentwise_iso ε d
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private definition ηi' (c : C) : G (F c) ≅ c :=
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to_fun_iso G (to_fun_iso F (ηi c)⁻¹ⁱ) ⬝i to_fun_iso G (εi (F c)) ⬝i ηi c
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local attribute ηn εn ηi εi ηi' [reducible]
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private theorem adj_η_natural {c c' : C} (f : hom c c')
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: G (F f) ∘ to_inv (ηi' c) = to_inv (ηi' c') ∘ f :=
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let ηi'_nat : G ∘f F ⟹ 1 :=
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calc
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G ∘f F ⟹ (G ∘f F) ∘f 1 : id_right_natural_rev (G ∘f F)
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... ⟹ (G ∘f F) ∘f (G ∘f F) : (G ∘f F) ∘fn ηn
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... ⟹ ((G ∘f F) ∘f G) ∘f F : assoc_natural (G ∘f F) G F
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... ⟹ (G ∘f (F ∘f G)) ∘f F : assoc_natural_rev G F G ∘nf F
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... ⟹ (G ∘f 1) ∘f F : (G ∘fn εn) ∘nf F
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... ⟹ G ∘f F : id_right_natural G ∘nf F
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... ⟹ 1 : to_hom η
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in
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begin
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refine is_natural_inverse' (G ∘f F) functor.id ηi' ηi'_nat _ f,
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intro c, esimp, rewrite [+id_left,id_right]
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end
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private theorem adjointify_adjH (c : C) :
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to_hom (εi (F c)) ∘ F (to_hom (ηi' c))⁻¹ = id :=
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begin
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rewrite [respect_inv], apply comp_inverse_eq_of_eq_comp,
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rewrite [id_left,↑ηi',+respect_comp,+respect_inv',assoc], apply eq_comp_inverse_of_comp_eq,
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rewrite [↑εi,-naturality_iso_id ε (F c)],
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symmetry, exact naturality εn (F (to_hom (ηi c)))
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end
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private theorem adjointify_adjK (d : D) :
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G (to_hom (εi d)) ∘ to_hom (ηi' (G d))⁻¹ⁱ = id :=
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begin
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apply comp_inverse_eq_of_eq_comp,
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rewrite [id_left,↑ηi',+respect_inv',assoc], apply eq_comp_inverse_of_comp_eq,
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rewrite [↑ηi,-naturality_iso_id η (G d),↑εi,naturality_iso_id ε d],
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exact naturality (to_hom η) (G (to_hom (εi d))),
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end
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parameter (G)
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include η ε
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definition is_equivalence.mk : is_equivalence F :=
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begin
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fapply is_equivalence.mk',
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{ exact G},
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{ fapply nat_trans.mk,
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{ intro c, exact to_inv (ηi' c)},
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{ intro c c' f, exact adj_η_natural f}},
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{ exact εn},
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{ exact adjointify_adjH},
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{ exact adjointify_adjK},
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{ exact @(is_natural_iso _) (λc, !is_iso_inverse)},
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{ unfold εn, apply iso.struct, },
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end
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definition equivalence.MK : C ≃c D :=
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equivalence.mk F is_equivalence.mk
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end
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variables {C D E : Precategory} {F : C ⇒ D}
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--TODO: add variants
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definition unit_eq_counit_inv (F : C ⇒ D) [H : is_equivalence F] (c : C) :
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to_fun_hom F (natural_map (unit F) c) =
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@(is_iso.inverse (counit F (F c))) (@(componentwise_is_iso (counit F)) !is_iso_counit (F c)) :=
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begin
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apply eq_inverse_of_comp_eq_id, apply counit_unit_eq
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end
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definition fully_faithful_of_is_equivalence (F : C ⇒ D) [H : is_equivalence F]
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: fully_faithful F :=
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begin
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intro c c',
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fapply adjointify,
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{ intro g, exact natural_map (@(iso.inverse (unit F)) !is_iso_unit) c' ∘ F⁻¹ g ∘ unit F c},
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{ intro g, rewrite [+respect_comp,▸*],
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xrewrite [natural_map_inverse (unit F) c', respect_inv'],
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apply inverse_comp_eq_of_eq_comp,
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rewrite [+unit_eq_counit_inv],
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esimp, exact naturality (counit F)⁻¹ _},
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{ intro f, xrewrite [▸*,natural_map_inverse (unit F) c'], apply inverse_comp_eq_of_eq_comp,
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apply naturality (unit F)},
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end
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definition is_isomorphism.mk [constructor] {F : C ⇒ D} (G : D ⇒ C)
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(p : G ∘f F = 1) (q : F ∘f G = 1) : is_isomorphism F :=
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begin
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constructor,
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{ apply fully_faithful_of_is_equivalence, fapply is_equivalence.mk,
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{ exact G},
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{ apply iso_of_eq p},
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{ apply iso_of_eq q}},
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{ fapply adjointify,
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{ exact G},
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{ exact ap010 to_fun_ob q},
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{ exact ap010 to_fun_ob p}}
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end
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definition isomorphism.MK [constructor] (F : C ⇒ D) (G : D ⇒ C)
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(p : G ∘f F = 1) (q : F ∘f G = 1) : C ≅c D :=
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isomorphism.mk F (is_isomorphism.mk G p q)
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definition is_equiv_ob_of_is_isomorphism [instance] [unfold 4] (F : C ⇒ D)
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[H : is_isomorphism F] : is_equiv (to_fun_ob F) :=
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pr2 H
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definition is_fully_faithful_of_is_isomorphism [instance] [unfold 4] (F : C ⇒ D)
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[H : is_isomorphism F] : fully_faithful F :=
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pr1 H
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definition strict_inverse [constructor] (F : C ⇒ D) [H : is_isomorphism F] : D ⇒ C :=
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begin
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fapply functor.mk,
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{ intro d, exact (to_fun_ob F)⁻¹ᶠ d},
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{ intro d d' g, exact (to_fun_hom F)⁻¹ᶠ (inv_of_eq !right_inv ∘ g ∘ hom_of_eq !right_inv)},
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{ intro d, apply inv_eq_of_eq, rewrite [respect_id,id_left], apply left_inverse},
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{ intro d₁ d₂ d₃ g₂ g₁, apply inv_eq_of_eq, rewrite [respect_comp F,+right_inv (to_fun_hom F)],
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rewrite [+assoc], esimp, /-apply ap (λx, (x ∘ _) ∘ _), FAILS-/ refine ap (λx, (x ∘ _) ∘ _) _,
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refine !id_right⁻¹ ⬝ _, rewrite [▸*,-+assoc], refine ap (λx, _ ∘ _ ∘ x) _,
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exact !right_inverse⁻¹},
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end
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postfix /-[parsing-only]-/ `⁻¹ˢ`:std.prec.max_plus := strict_inverse
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definition strict_right_inverse (F : C ⇒ D) [H : is_isomorphism F] : F ∘f F⁻¹ˢ = 1 :=
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begin
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fapply functor_eq,
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{ intro d, esimp, apply right_inv},
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{ intro d d' g,
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rewrite [▸*, right_inv (to_fun_hom F), +assoc],
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rewrite [↑[hom_of_eq,inv_of_eq,iso.to_inv], right_inverse],
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rewrite [id_left], apply comp_inverse_cancel_right},
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end
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definition strict_left_inverse (F : C ⇒ D) [H : is_isomorphism F] : F⁻¹ˢ ∘f F = 1 :=
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begin
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fapply functor_eq,
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{ intro d, esimp, apply left_inv},
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{ intro d d' g, esimp, apply comp_eq_of_eq_inverse_comp, apply comp_inverse_eq_of_eq_comp,
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apply inv_eq_of_eq, rewrite [+respect_comp,-assoc], apply ap011 (λx y, x ∘ F g ∘ y),
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{ rewrite [adj], rewrite [▸*,respect_inv_of_eq F]},
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{ rewrite [adj,▸*,respect_hom_of_eq F]}},
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end
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definition is_equivalence_of_is_isomorphism [instance] [constructor] (F : C ⇒ D) [H : is_isomorphism F]
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: is_equivalence F :=
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begin
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fapply is_equivalence.mk,
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{ apply F⁻¹ˢ},
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{ apply iso_of_eq !strict_left_inverse},
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{ apply iso_of_eq !strict_right_inverse},
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end
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definition equivalence_of_isomorphism [constructor] (F : C ≅c D) : C ≃c D :=
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equivalence.mk F _
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theorem is_prop_is_equivalence [instance] {C : Category} {D : Precategory} (F : C ⇒ D)
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: is_prop (is_equivalence F) :=
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begin
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assert f : is_equivalence F ≃ Σ(H : is_left_adjoint F), is_iso (unit F) × is_iso (counit F),
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{ fapply equiv.MK,
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{ intro H, induction H, fconstructor: constructor, repeat (esimp;assumption) },
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{ intro H, induction H with H1 H2, induction H1, induction H2, constructor,
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repeat (esimp at *;assumption)},
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{ intro H, induction H with H1 H2, induction H1, induction H2, reflexivity},
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{ intro H, induction H, reflexivity}},
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apply is_trunc_equiv_closed_rev, exact f,
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end
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theorem is_prop_is_isomorphism [instance] (F : C ⇒ D) : is_prop (is_isomorphism F) :=
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by unfold is_isomorphism; exact _
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/- closure properties -/
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definition is_isomorphism_id [instance] [constructor] (C : Precategory)
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: is_isomorphism (1 : C ⇒ C) :=
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is_isomorphism.mk 1 !functor.id_right !functor.id_right
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definition is_isomorphism_strict_inverse [constructor] (F : C ⇒ D) [K : is_isomorphism F]
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: is_isomorphism F⁻¹ˢ :=
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is_isomorphism.mk F !strict_right_inverse !strict_left_inverse
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definition is_isomorphism_compose [constructor] (G : D ⇒ E) (F : C ⇒ D)
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[H : is_isomorphism G] [K : is_isomorphism F] : is_isomorphism (G ∘f F) :=
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is_isomorphism.mk
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(F⁻¹ˢ ∘f G⁻¹ˢ)
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abstract begin
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rewrite [functor.assoc,-functor.assoc F⁻¹ˢ,strict_left_inverse,functor.id_right,
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strict_left_inverse]
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end end
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abstract begin
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rewrite [functor.assoc,-functor.assoc G,strict_right_inverse,functor.id_right,
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strict_right_inverse]
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end end
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definition is_equivalence_id [constructor] (C : Precategory) : is_equivalence (1 : C ⇒ C) := _
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definition is_equivalence_inverse [constructor] (F : C ⇒ D) [K : is_equivalence F]
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: is_equivalence F⁻¹ᴱ :=
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is_equivalence.mk F (iso_counit F) (iso_unit F)
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definition is_equivalence_compose [constructor] (G : D ⇒ E) (F : C ⇒ D)
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[H : is_equivalence G] [K : is_equivalence F] : is_equivalence (G ∘f F) :=
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is_equivalence.mk
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(F⁻¹ᴱ ∘f G⁻¹ᴱ)
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abstract begin
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rewrite [functor.assoc,-functor.assoc F⁻¹ᴱ],
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refine ((_ ∘fi !iso_unit) ∘if _) ⬝i _,
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refine (iso_of_eq !functor.id_right ∘if _) ⬝i _,
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apply iso_unit
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end end
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abstract begin
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rewrite [functor.assoc,-functor.assoc G],
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refine ((_ ∘fi !iso_counit) ∘if _) ⬝i _,
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refine (iso_of_eq !functor.id_right ∘if _) ⬝i _,
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apply iso_counit
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end end
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variable (C)
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definition equivalence.refl [refl] [constructor] : C ≃c C :=
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equivalence.mk _ !is_equivalence_id
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definition isomorphism.refl [refl] [constructor] : C ≅c C :=
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isomorphism.mk _ !is_isomorphism_id
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variable {C}
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definition equivalence.symm [symm] [constructor] (H : C ≃c D) : D ≃c C :=
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equivalence.mk _ (is_equivalence_inverse H)
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definition isomorphism.symm [symm] [constructor] (H : C ≅c D) : D ≅c C :=
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isomorphism.mk _ (is_isomorphism_strict_inverse H)
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definition equivalence.trans [trans] [constructor] (H : C ≃c D) (K : D ≃c E) : C ≃c E :=
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equivalence.mk _ (is_equivalence_compose K H)
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definition isomorphism.trans [trans] [constructor] (H : C ≅c D) (K : D ≅c E) : C ≅c E :=
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isomorphism.mk _ (is_isomorphism_compose K H)
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definition equivalence.to_strict_inverse [unfold 3] (H : C ≃c D) : D ⇒ C :=
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H⁻¹ᴱ
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definition isomorphism.to_strict_inverse [unfold 3] (H : C ≅c D) : D ⇒ C :=
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H⁻¹ˢ
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definition is_isomorphism_of_is_equivalence [constructor] {C D : Category} (F : C ⇒ D)
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[H : is_equivalence F] : is_isomorphism F :=
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begin
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fapply is_isomorphism.mk,
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{ exact F⁻¹ᴱ},
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{ apply eq_of_iso, apply iso_unit},
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{ apply eq_of_iso, apply iso_counit},
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end
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definition isomorphism_of_equivalence [constructor] {C D : Category} (F : C ≃c D) : C ≅c D :=
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isomorphism.mk F !is_isomorphism_of_is_equivalence
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definition equivalence_eq {C : Category} {D : Precategory} {F F' : C ≃c D}
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(p : equivalence.to_functor F = equivalence.to_functor F') : F = F' :=
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begin
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induction F, induction F', exact apd011 equivalence.mk p !is_prop.elim
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end
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definition isomorphism_eq {F F' : C ≅c D}
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(p : isomorphism.to_functor F = isomorphism.to_functor F') : F = F' :=
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begin
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induction F, induction F', exact apd011 isomorphism.mk p !is_prop.elim
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end
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definition is_equiv_isomorphism_of_equivalence [constructor] (C D : Category)
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: is_equiv (@equivalence_of_isomorphism C D) :=
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begin
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fapply adjointify,
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{ exact isomorphism_of_equivalence},
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{ intro F, apply equivalence_eq, reflexivity},
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{ intro F, apply isomorphism_eq, reflexivity},
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end
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definition isomorphism_equiv_equivalence [constructor] (C D : Category)
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: (C ≅c D) ≃ (C ≃c D) :=
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equiv.mk _ !is_equiv_isomorphism_of_equivalence
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definition isomorphism_of_eq [constructor] {C D : Precategory} (p : C = D) : C ≅c D :=
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isomorphism.MK (functor_of_eq p)
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(functor_of_eq p⁻¹)
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(by induction p; reflexivity)
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(by induction p; reflexivity)
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definition equiv_ob_of_isomorphism [constructor] {C D : Precategory} (H : C ≅c D) : C ≃ D :=
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equiv.mk H _
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definition equiv_hom_of_isomorphism [constructor] {C D : Precategory} (H : C ≅c D) (c c' : C)
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: c ⟶ c' ≃ H c ⟶ H c' :=
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equiv.mk (to_fun_hom (isomorphism.to_functor H)) _
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/- TODO
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definition is_equiv_isomorphism_of_eq [constructor] (C D : Precategory)
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: is_equiv (@isomorphism_of_eq C D) :=
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begin
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fapply adjointify,
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{ intro H, fapply Precategory_eq_of_equiv,
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{ apply equiv_ob_of_isomorphism H},
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{ exact equiv_hom_of_isomorphism H},
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{ /-exact sorry FAILS-/ intros, esimp, apply respect_comp}},
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{ intro H, apply isomorphism_eq, esimp, fapply functor_eq: esimp,
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{ intro c, exact sorry},
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{ exact sorry}},
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{ intro p, induction p, esimp, exact sorry},
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end
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definition eq_equiv_isomorphism [constructor] (C D : Precategory)
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: (C = D) ≃ (C ≅c D) :=
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equiv.mk _ !is_equiv_isomorphism_of_eq
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definition equivalence_of_eq [unfold 3] [reducible] {C D : Precategory} (p : C = D) : C ≃c D :=
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equivalence_of_isomorphism (isomorphism_of_eq p)
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definition eq_equiv_equivalence [constructor] (C D : Category) : (C = D) ≃ (C ≃c D) :=
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!eq_equiv_isomorphism ⬝e !isomorphism_equiv_equivalence
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definition is_equivalence_equiv [constructor] (F : C ⇒ D)
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: is_equivalence F ≃ (fully_faithful F × split_essentially_surjective F) :=
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sorry
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definition is_equivalence_equiv_is_weak_equivalence [constructor] {C D : Category}
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(F : C ⇒ D) : is_equivalence F ≃ is_weak_equivalence F :=
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sorry
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-/
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/- TODO?
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definition is_isomorphism_equiv1 (F : C ⇒ D) : is_equivalence F
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≃ Σ(G : D ⇒ C) (η : 1 = G ∘f F) (ε : F ∘f G = 1),
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sorry ⬝ ap (λ(H : C ⇒ C), F ∘f H) η ⬝ sorry = ap (λ(H : D ⇒ D), H ∘f F) ε⁻¹ :=
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sorry
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definition is_isomorphism_equiv2 (F : C ⇒ D) : is_equivalence F
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≃ ∃(G : D ⇒ C), 1 = G ∘f F × F ∘f G = 1 :=
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sorry
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-/
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end category
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