Theorems · Theorem · category theory

CategoryTheory.LocalizerMorphism.rightDerivedFunctorComparison_fac_app_assoc

∀ {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [inst : CategoryTheory.Category.{v₁, u₁} C₁]
  [inst_1 : CategoryTheory.Category.{v₂, u₂} C₂] [inst_2 : CategoryTheory.Category.{v₃, u₃} H] {D₁ : Type u₄}
  {D₂ : Type u₅} [inst_3 : CategoryTheory.Category.{v₄, u₄} D₁] [inst_4 : CategoryTheory.Category.{v₅, u₅} D₂]
  {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂}
  (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) (L₂ : CategoryTheory.Functor C₂ D₂)
  [inst_5 : L₁.IsLocalization W₁] [inst_6 : L₂.IsLocalization W₂] (F : CategoryTheory.Functor C₂ H)
  (F₁ : CategoryTheory.Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : CategoryTheory.Functor D₂ H)
  (α₂ : F ⟶ L₂.comp F₂) [inst_7 : F₁.IsRightDerivedFunctor α₁ W₁] (X : C₁) {Z : H}
  (h : F₂.obj ((Φ.localizedFunctor L₁ L₂).obj (L₁.obj X)) ⟶ Z),
  CategoryTheory.CategoryStruct.comp (α₁.app X)
      (CategoryTheory.CategoryStruct.comp ((Φ.rightDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂).app (L₁.obj X)) h) =
    CategoryTheory.CategoryStruct.comp (α₂.app (Φ.functor.obj X))
      (CategoryTheory.CategoryStruct.comp
        (F₂.map ((CategoryTheory.CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).hom.app X)) h)

Defined in

Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived

Cited by

0 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsLocalizationCategoryTheory.Functor.IsLocalizationCategoryTheory.Functor.IsRightDerivedFunctor