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Mathlib.CategoryTheory.Localization.Bousfield

Bousfield localization #

Given a predicate P : ObjectProperty C on the objects of a category C, we define Localization.LeftBousfield.W P : MorphismProperty C as the class of morphisms f : X ⟶ Y such that for any Z : C such that P Z, the precomposition with f induces a bijection (Y ⟶ Z) ≃ (X ⟶ Z).

(This construction is part of left Bousfield localization in the context of model categories.)

When G ⊣ F is an adjunction with F : C ⥤ D fully faithful, then G : D ⥤ C is a localization functor for the class W (· ∈ Set.range F.obj), which then identifies to the inverse image by G of the class of isomorphisms in C.

References #

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def CategoryTheory.Localization.LeftBousfield.W {C : Type u_1} [Category.{u_3, u_1} C] (P : ObjectProperty C) :
MorphismProperty C

Given P : ObjectProperty C, this is the class of morphisms f : X ⟶ Y such that for all Z : C such that P Z, the precomposition with f induces a bijection (Y ⟶ Z) ≃ (X ⟶ Z).

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    noncomputable def CategoryTheory.Localization.LeftBousfield.W.homEquiv {C : Type u_1} [Category.{u_3, u_1} C] {P : ObjectProperty C} {X Y : C} {f : X ⟶ Y} (hf : W P f) (Z : C) (hZ : P Z) :
    (Y ⟶ Z) ≃ (X ⟶ Z)

    The bijection (Y ⟶ Z) ≃ (X ⟶ Z) induced by f : X ⟶ Y when LeftBousfield.W P f and P Z.

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      @[simp]
      theorem CategoryTheory.Localization.LeftBousfield.W.homEquiv_apply {C : Type u_1} [Category.{u_3, u_1} C] {P : ObjectProperty C} {X Y : C} {f : X ⟶ Y} (hf : W P f) (Z : C) (hZ : P Z) (g : Y ⟶ Z) :
      (hf.homEquiv Z hZ) g = CategoryStruct.comp f g
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      theorem CategoryTheory.Localization.LeftBousfield.W_isoClosure {C : Type u_1} [Category.{u_3, u_1} C] (P : ObjectProperty C) :
      W P.isoClosure = W P
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      instance CategoryTheory.Localization.LeftBousfield.instIsMultiplicativeW {C : Type u_1} [Category.{u_3, u_1} C] (P : ObjectProperty C) :
      (W P).IsMultiplicative
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      instance CategoryTheory.Localization.LeftBousfield.instHasTwoOutOfThreePropertyW {C : Type u_1} [Category.{u_3, u_1} C] (P : ObjectProperty C) :
      (W P).HasTwoOutOfThreeProperty
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      theorem CategoryTheory.Localization.LeftBousfield.W_of_isIso {C : Type u_1} [Category.{u_3, u_1} C] (P : ObjectProperty C) {X Y : C} (f : X ⟶ Y) [IsIso f] :
      W P f
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      theorem CategoryTheory.Localization.LeftBousfield.W_iff_isIso {C : Type u_1} [Category.{u_3, u_1} C] (P : ObjectProperty C) {X Y : C} (f : X ⟶ Y) (hX : P X) (hY : P Y) :
      W P f ↔ IsIso f
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      instance CategoryTheory.Localization.LeftBousfield.instRespectsIsoW {C : Type u_1} [Category.{u_3, u_1} C] (P : ObjectProperty C) :
      (W P).RespectsIso
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      theorem CategoryTheory.Localization.LeftBousfield.W_adj_unit_app {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_3, u_2} D] {F : Functor C D} {G : Functor D C} (adj : G ⊣ F) [F.Full] [F.Faithful] (X : D) :
      W (fun (x : D) => x ∈ Set.range F.obj) (adj.unit.app X)
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      theorem CategoryTheory.Localization.LeftBousfield.W_iff_isIso_map {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_3, u_2} D] {F : Functor C D} {G : Functor D C} (adj : G ⊣ F) [F.Full] [F.Faithful] {X Y : D} (f : X ⟶ Y) :
      W (fun (x : D) => x ∈ Set.range F.obj) f ↔ IsIso (G.map f)
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      theorem CategoryTheory.Localization.LeftBousfield.W_eq_inverseImage_isomorphisms {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_3, u_2} D] {F : Functor C D} {G : Functor D C} (adj : G ⊣ F) [F.Full] [F.Faithful] :
      (W fun (x : D) => x ∈ Set.range F.obj) = (MorphismProperty.isomorphisms C).inverseImage G
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      theorem CategoryTheory.Localization.LeftBousfield.isLocalization {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_3, u_2} D] {F : Functor C D} {G : Functor D C} (adj : G ⊣ F) [F.Full] [F.Faithful] :
      G.IsLocalization (W fun (x : D) => x ∈ Set.range F.obj)