File: Mathlib/CategoryTheory/Localization/DerivabilityStructure/Constructor.lean

full_name: CategoryTheory.LocalizerMorphism.IsRightDerivabilityStructure.Constructor.fromRightResolution
code: /-- Given `Φ : LocalizerMorphism W₁ W₂`, `L : C₂ ⥤ D` a localization functor for `W₂` and
a morphism `y : L.obj X₂ ⟶ X₃`, this is the functor which sends `R : Φ.RightResolution d` to
`(isoOfHom L W₂ R.w R.hw).inv ≫ y` in the category `w.CostructuredArrowDownwards y`
where `w` is `TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 _) (Functor.rightUnitor _).inv`. -/
@[simps]
noncomputable def fromRightResolution :
    Φ.RightResolution X₂ ⥤ (TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 _)
      (Functor.rightUnitor _).inv).CostructuredArrowDownwards y 
goal: CategoryTheory.Functor (Φ.RightResolution X₂)
  ((CategoryTheory.TwoSquare.mk Φ.functor (Φ.functor.comp L) L (CategoryTheory.Functor.id D)
        (Φ.functor.comp L).rightUnitor.inv).CostructuredArrowDownwards
    y)

full_name: CategoryTheory.LocalizerMorphism.IsRightDerivabilityStructure.Constructor.fromRightResolution_map
code: /-- Given `Φ : LocalizerMorphism W₁ W₂`, `L : C₂ ⥤ D` a localization functor for `W₂` and
a morphism `y : L.obj X₂ ⟶ X₃`, this is the functor which sends `R : Φ.RightResolution d` to
`(isoOfHom L W₂ R.w R.hw).inv ≫ y` in the category `w.CostructuredArrowDownwards y`
where `w` is `TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 _) (Functor.rightUnitor _).inv`. -/
@[simps]
noncomputable def fromRightResolution :
    Φ.RightResolution X₂ ⥤ (TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 _)
      (Functor.rightUnitor _).inv).CostructuredArrowDownwards y 
goal: (CategoryTheory.LocalizerMorphism.IsRightDerivabilityStructure.Constructor.fromRightResolution Φ L y).map φ =
  CategoryTheory.CostructuredArrow.homMk (CategoryTheory.StructuredArrow.homMk φ.f ⋯) ⋯

full_name: CategoryTheory.LocalizerMorphism.IsRightDerivabilityStructure.Constructor.fromRightResolution_obj
code: /-- Given `Φ : LocalizerMorphism W₁ W₂`, `L : C₂ ⥤ D` a localization functor for `W₂` and
a morphism `y : L.obj X₂ ⟶ X₃`, this is the functor which sends `R : Φ.RightResolution d` to
`(isoOfHom L W₂ R.w R.hw).inv ≫ y` in the category `w.CostructuredArrowDownwards y`
where `w` is `TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 _) (Functor.rightUnitor _).inv`. -/
@[simps]
noncomputable def fromRightResolution :
    Φ.RightResolution X₂ ⥤ (TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 _)
      (Functor.rightUnitor _).inv).CostructuredArrowDownwards y 
goal: (CategoryTheory.LocalizerMorphism.IsRightDerivabilityStructure.Constructor.fromRightResolution Φ L y).obj R =
  CategoryTheory.CostructuredArrow.mk
    (CategoryTheory.StructuredArrow.homMk
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Localization.isoOfHom L W₂ R.w ⋯).inv y) ⋯)

full_name: CategoryTheory.LocalizerMorphism.IsRightDerivabilityStructure.Constructor.isConnected
code: lemma isConnected :
    IsConnected ((TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 _)
      (Functor.rightUnitor _).inv).CostructuredArrowDownwards y) 
goal: CategoryTheory.IsConnected
  ((CategoryTheory.TwoSquare.mk Φ.functor (Φ.functor.comp L) L (CategoryTheory.Functor.id D)
        (Φ.functor.comp L).rightUnitor.inv).CostructuredArrowDownwards
    y)

full_name: CategoryTheory.LocalizerMorphism.IsRightDerivabilityStructure.mk'
code: /-- If a localizer morphism `Φ` is a localized equivalence, then it is a right
derivability structure if the categories of right resolutions are connected and the
categories of right resolutions of arrows are nonempty. -/
lemma mk' : Φ.IsRightDerivabilityStructure 
goal: Φ.IsRightDerivabilityStructure