File: Mathlib/CategoryTheory/Localization/HomEquiv.lean

full_name: CategoryTheory.LocalizerMorphism.homMap
code: /-- If `Φ : LocalizerMorphism W₁ W₂` is a morphism of localizers, `L₁` and `L₂`
are localization functors for `W₁` and `W₂`, then this is the induced map
`(L₁.obj X ⟶ L₁.obj Y) ⟶ (L₂.obj (Φ.functor.obj X) ⟶ L₂.obj (Φ.functor.obj Y))`
for all objects `X` and `Y`. -/
noncomputable def homMap (f : L₁.obj X ⟶ L₁.obj Y) :
    L₂.obj (Φ.functor.obj X) ⟶ L₂.obj (Φ.functor.obj Y) 
goal: L₂.obj (Φ.functor.obj X) ⟶ L₂.obj (Φ.functor.obj Y)

full_name: CategoryTheory.LocalizerMorphism.homMap_map
code: @[simp]
lemma homMap_map (f : X ⟶ Y) :
    Φ.homMap L₁ L₂ (L₁.map f) = L₂.map (Φ.functor.map f) 
goal: Φ.homMap L₁ L₂ (L₁.map f) = L₂.map (Φ.functor.map f)

full_name: CategoryTheory.LocalizerMorphism.homMap_id
code: @[simp]
lemma homMap_id  :
    Φ.homMap L₁ L₂ (𝟙 (L₁.obj X)) = 𝟙 (L₂.obj (Φ.functor.obj X)) 
goal: Φ.homMap L₁ L₂ (CategoryTheory.CategoryStruct.id (L₁.obj X)) =
  CategoryTheory.CategoryStruct.id (L₂.obj (Φ.functor.obj X))

full_name: CategoryTheory.LocalizerMorphism.homMap_comp
code: @[reassoc]
lemma homMap_comp (f : L₁.obj X ⟶ L₁.obj Y) (g : L₁.obj Y ⟶ L₁.obj Z) :
    Φ.homMap L₁ L₂ (f ≫ g) = Φ.homMap L₁ L₂ f ≫ Φ.homMap L₁ L₂ g 
goal: Φ.homMap L₁ L₂ (CategoryTheory.CategoryStruct.comp f g) =
  CategoryTheory.CategoryStruct.comp (Φ.homMap L₁ L₂ f) (Φ.homMap L₁ L₂ g)

full_name: CategoryTheory.LocalizerMorphism.homMap_apply
code: @[reassoc]
lemma homMap_apply (G : D₁ ⥤ D₂) (e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G) (f : L₁.obj X ⟶ L₁.obj Y) :
    Φ.homMap L₁ L₂ f = e.hom.app X ≫ G.map f ≫ e.inv.app Y 
goal: Φ.homMap L₁ L₂ f =
  CategoryTheory.CategoryStruct.comp (e.hom.app X) (CategoryTheory.CategoryStruct.comp (G.map f) (e.inv.app Y))

full_name: CategoryTheory.LocalizerMorphism.id_homMap
code: @[simp]
lemma id_homMap (f : L₁.obj X ⟶ L₁.obj Y) :
    (id W₁).homMap L₁ L₁ f = f 
goal: (CategoryTheory.LocalizerMorphism.id W₁).homMap L₁ L₁ f = f

full_name: CategoryTheory.LocalizerMorphism.homMap_homMap
code: @[simp]
lemma homMap_homMap (f : L₁.obj X ⟶ L₁.obj Y) :
    Ψ.homMap L₂ L₃ (Φ.homMap L₁ L₂ f) = (Φ.comp Ψ).homMap L₁ L₃ f 
goal: Ψ.homMap L₂ L₃ (Φ.homMap L₁ L₂ f) = (Φ.comp Ψ).homMap L₁ L₃ f

full_name: CategoryTheory.Localization.homEquiv
code: /-- Bijection between types of morphisms in two localized categories
for the same class of morphisms `W`. -/
@[simps (config := .lemmasOnly) apply]
noncomputable def homEquiv :
    (L₁.obj X ⟶ L₁.obj Y) ≃ (L₂.obj X ⟶ L₂.obj Y) 
goal: (L₁.obj X ⟶ L₁.obj Y) ≃ (L₂.obj X ⟶ L₂.obj Y)

full_name: CategoryTheory.Localization.homEquiv_apply
code: /-- Bijection between types of morphisms in two localized categories
for the same class of morphisms `W`. -/
@[simps (config := .lemmasOnly) apply]
noncomputable def homEquiv :
    (L₁.obj X ⟶ L₁.obj Y) ≃ (L₂.obj X ⟶ L₂.obj Y) 
goal: (CategoryTheory.Localization.homEquiv W L₁ L₂) f = (CategoryTheory.LocalizerMorphism.id W).homMap L₁ L₂ f

full_name: CategoryTheory.Localization.homEquiv_symm_apply
code: @[simp]
lemma homEquiv_symm_apply (g : L₂.obj X ⟶ L₂.obj Y) :
    (homEquiv W L₁ L₂).symm g = homEquiv W L₂ L₁ g 
goal: (CategoryTheory.Localization.homEquiv W L₁ L₂).symm g = (CategoryTheory.Localization.homEquiv W L₂ L₁) g

full_name: CategoryTheory.Localization.homEquiv_eq
code: lemma homEquiv_eq (G : D₁ ⥤ D₂) (e : L₁ ⋙ G ≅ L₂) (f : L₁.obj X ⟶ L₁.obj Y) :
    homEquiv W L₁ L₂ f = e.inv.app X ≫ G.map f ≫ e.hom.app Y 
goal: (CategoryTheory.Localization.homEquiv W L₁ L₂) f =
  CategoryTheory.CategoryStruct.comp (e.inv.app X) (CategoryTheory.CategoryStruct.comp (G.map f) (e.hom.app Y))

full_name: CategoryTheory.Localization.homEquiv_refl
code: @[simp]
lemma homEquiv_refl (f : L₁.obj X ⟶ L₁.obj Y) :
    homEquiv W L₁ L₁ f = f 
goal: (CategoryTheory.Localization.homEquiv W L₁ L₁) f = f

full_name: CategoryTheory.Localization.homEquiv_trans
code: lemma homEquiv_trans (f : L₁.obj X ⟶ L₁.obj Y) :
    homEquiv W L₂ L₃ (homEquiv W L₁ L₂ f) = homEquiv W L₁ L₃ f 
goal: (CategoryTheory.Localization.homEquiv W L₂ L₃) ((CategoryTheory.Localization.homEquiv W L₁ L₂) f) =
  (CategoryTheory.Localization.homEquiv W L₁ L₃) f

full_name: CategoryTheory.Localization.homEquiv_comp
code: lemma homEquiv_comp (f : L₁.obj X ⟶ L₁.obj Y) (g : L₁.obj Y ⟶ L₁.obj Z) :
    homEquiv W L₁ L₂ (f ≫ g) = homEquiv W L₁ L₂ f ≫ homEquiv W L₁ L₂ g 
goal: (CategoryTheory.Localization.homEquiv W L₁ L₂) (CategoryTheory.CategoryStruct.comp f g) =
  CategoryTheory.CategoryStruct.comp ((CategoryTheory.Localization.homEquiv W L₁ L₂) f)
    ((CategoryTheory.Localization.homEquiv W L₁ L₂) g)

full_name: CategoryTheory.Localization.homEquiv_map
code: @[simp]
lemma homEquiv_map (f : X ⟶ Y) : homEquiv W L₁ L₂ (L₁.map f) = L₂.map f 
goal: (CategoryTheory.Localization.homEquiv W L₁ L₂) (L₁.map f) = L₂.map f

full_name: CategoryTheory.Localization.homEquiv_id
code: @[simp]
lemma homEquiv_id : homEquiv W L₁ L₂ (𝟙 (L₁.obj X)) = 𝟙 (L₂.obj X) 
goal: (CategoryTheory.Localization.homEquiv W L₁ L₂) (CategoryTheory.CategoryStruct.id (L₁.obj X)) =
  CategoryTheory.CategoryStruct.id (L₂.obj X)

full_name: CategoryTheory.Localization.homEquiv_isoOfHom_inv
code: lemma homEquiv_isoOfHom_inv (f : Y ⟶ X) (hf : W f) :
    homEquiv W L₁ L₂ (isoOfHom L₁ W f hf).inv = (isoOfHom L₂ W f hf).inv 
goal: (CategoryTheory.Localization.homEquiv W L₁ L₂) (CategoryTheory.Localization.isoOfHom L₁ W f hf).inv =
  (CategoryTheory.Localization.isoOfHom L₂ W f hf).inv