162 lines
5.7 KiB
Text
162 lines
5.7 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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Module: algebra.precategory.adjoint
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Authors: Floris van Doorn
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-/
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import algebra.category.constructions .constructions types.function arity
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open category functor nat_trans eq is_trunc iso equiv prod trunc function
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namespace category
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variables {C D : Precategory} {F : C ⇒ D}
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-- do we want to have a structure "is_adjoint" and define
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-- structure is_left_adjoint (F : C ⇒ D) :=
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-- (right_adjoint : D ⇒ C) -- G
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-- (is_adjoint : adjoint F right_adjoint)
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structure is_left_adjoint [class] (F : C ⇒ D) :=
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(G : D ⇒ C)
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(η : functor.id ⟹ G ∘f F)
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(ε : F ∘f G ⟹ functor.id)
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(H : Π(c : C), (ε (F c)) ∘ (F (η c)) = ID (F c))
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(K : Π(d : D), (G (ε d)) ∘ (η (G d)) = ID (G d))
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abbreviation right_adjoint := @is_left_adjoint.G
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abbreviation unit := @is_left_adjoint.η
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abbreviation counit := @is_left_adjoint.ε
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-- structure is_left_adjoint [class] (F : C ⇒ D) :=
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-- (right_adjoint : D ⇒ C) -- G
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-- (unit : functor.id ⟹ right_adjoint ∘f F) -- η
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-- (counit : F ∘f right_adjoint ⟹ functor.id) -- ε
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-- (H : Π(c : C), (counit (F c)) ∘ (F (unit c)) = ID (F c))
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-- (K : Π(d : D), (right_adjoint (counit d)) ∘ (unit (right_adjoint d)) = ID (right_adjoint d))
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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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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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--TODO: review and change
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--TODO: make some or all of these structures?
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definition faithful (F : C ⇒ D) :=
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Π⦃c c' : C⦄ (f f' : c ⟶ c'), F f = F f' → f = f'
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definition full (F : C ⇒ D) := Π⦃c c' : C⦄ (g : F c ⟶ F c'), ∃(f : c ⟶ c'), F f = g
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definition fully_faithful [reducible] (F : C ⇒ D) := Π⦃c c' : C⦄, is_equiv (@to_fun_hom _ _ F c c')
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definition split_essentially_surjective (F : C ⇒ D) :=
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Π⦃d : D⦄, Σ(c : C), F c ≅ d
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definition essentially_surjective (F : C ⇒ D) :=
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Π⦃d : D⦄, ∃(c : C), F c ≅ d
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definition is_weak_equivalence (F : C ⇒ D) :=
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fully_faithful F × essentially_surjective F
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definition is_isomorphism (F : C ⇒ D) :=
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fully_faithful F × is_equiv (to_fun_ob F)
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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 `⊣`:55 := adjoint
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infix `⋍`:25 := equivalence -- \backsimeq or \equiv
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infix `≌`:25 := isomorphism -- \backcong or \iso
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definition is_hprop_is_left_adjoint {C : Category} {D : Precategory} (F : C ⇒ D)
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: is_hprop (is_left_adjoint F) :=
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begin
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apply is_hprop.mk,
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intro G G', cases G with G η ε H K, cases G' with G' η' ε' H' K',
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fapply (apd011111 is_left_adjoint.mk),
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{ fapply functor_eq,
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{ intro d, apply eq_of_iso, fapply iso.MK,
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{ exact (G' (ε d) ∘ η' (G d))},
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{ exact (G (ε' d) ∘ η (G' d))},
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{ apply sorry /-rewrite [assoc, -{((G (ε' d)) ∘ (η (G' d))) ∘ (G' (ε d))}(assoc)],-/
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-- apply concat, apply (ap (λc, c ∘ η' _)), rewrite -assoc, apply idp
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},
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--/-rewrite [-nat_trans.assoc]-/apply sorry
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---assoc (G (ε' d)) (η (G' d)) (G' (ε d))
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{ apply sorry}},
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{ apply sorry},
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},
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{ apply sorry},
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{ apply sorry},
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{ apply is_hprop.elim},
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{ apply is_hprop.elim},
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end
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definition is_equivalence.mk (F : C ⇒ D) (G : D ⇒ C) (η : G ∘f F ≅ functor.id)
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(ε : F ∘f G ≅ functor.id) : is_equivalence F :=
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sorry
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definition full_of_fully_faithful (H : fully_faithful F) : full F :=
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sorry --λc c' g, trunc.elim _ _
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definition faithful_of_fully_faithful (H : fully_faithful F) : faithful F :=
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λc c' f f' p, is_injective_of_is_embedding p
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definition fully_faithful_of_full_of_faithful (H : faithful F) (K : full F) : fully_faithful F :=
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sorry
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definition fully_faithful_equiv (F : C ⇒ D) : fully_faithful F ≃ (faithful F × full F) :=
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sorry
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definition is_equivalence_equiv (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_hprop_is_weak_equivalence (F : C ⇒ D) : is_hprop (is_weak_equivalence F) :=
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sorry
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definition is_hprop_is_equivalence {C D : Category} (F : C ⇒ D) : is_hprop (is_equivalence F) :=
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sorry
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definition is_equivalence_equiv_is_weak_equivalence {C D : Category} (F : C ⇒ D)
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: is_equivalence F ≃ is_weak_equivalence F :=
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sorry
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definition is_hprop_is_isomorphism (F : C ⇒ D) : is_hprop (is_isomorphism F) :=
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sorry
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definition is_isomorphism_equiv1 (F : C ⇒ D) : is_equivalence F
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≃ Σ(G : D ⇒ C) (η : functor.id = G ∘f F) (ε : F ∘f G = functor.id),
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sorry ▹ ap (λ(H : C ⇒ C), F ∘f H) η = 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), functor.id = G ∘f F × F ∘f G = functor.id :=
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sorry
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definition is_equivalence_of_isomorphism (H : is_isomorphism F) : is_equivalence F :=
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sorry
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definition is_isomorphism_of_is_equivalence {C D : Category} {F : C ⇒ D} (H : is_equivalence F)
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: is_isomorphism F :=
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sorry
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definition isomorphism_of_eq {C D : Precategory} (p : C = D) : C ≌ D :=
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sorry
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definition is_equiv_isomorphism_of_eq (C D : Precategory) : is_equiv (@isomorphism_of_eq C D) :=
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sorry
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definition equivalence_of_eq {C D : Precategory} (p : C = D) : C ⋍ D :=
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sorry
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definition is_equiv_equivalence_of_eq (C D : Category) : is_equiv (@equivalence_of_eq C D) :=
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sorry
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end category
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