Existence of pointwise right derived functors via derivability structures

In this file, we show how a right derivability structure can be used in order to construct (pointwise) right derived functors. Let Φ be a right derivability structure from W₁ : MorphismProperty C₁ to W₂ : MorphismProperty C₂. Let F : C₂ ⥤ H be a functor. Then, the lemma hasPointwiseRightDerivedFunctor_iff_of_isRightDerivabilityStructure says that F has a pointwise right derived functor with respect to W₂ if and only if Φ.functor ⋙ F has a pointwise right derived functor with respect to W₁. This is essentially the Proposition 5.5 from the article Structures de dérivabilité by Bruno Kahn and Georges Maltsiniotis (there, it was stated in terms of absolute derived functors).

In particular, if Φ.functor ⋙ F inverts W₁, it follows that the right derived functor of F with respect to W₂ exists.

References

• Bruno Kahn and Georges Maltsiniotis, Structures de dérivabilité

noncomputable def CategoryTheory.LocalizerMorphism.rightDerivedFunctorComparison {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [Category.{v₁, u₁} C₁] [Category.{v₂, u₂} C₂] [Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [Category.{v₄, u₄} D₁] [Category.{v₅, u₅} D₂] {W₁ : MorphismProperty C₁} {W₂ : MorphismProperty C₂} (Φ : LocalizerMorphism W₁ W₂) (L₁ : Functor C₁ D₁) (L₂ : Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : Functor C₂ H) (F₁ : Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] : F₁ ⟶ (Φ.localizedFunctor L₁ L₂).comp F₂

If Φ is a localizer morphism from W₁ : MorphismProperty C₁ to W₂ : MorphismProperty C₂, if L₁ : C₁ ⥤ D₁ and L₂ : C₂ ⥤ D₂ are localization functors for W₁ and W₂, if F : C₂ ⥤ H is a functor, if F₁ : D₁ ⥤ H is a right derived functor of Φ.functor ⋙ F, and if F₂ : D₂ ⥤ H is a functor equipped with a natural transformation α₂ : F ⟶ L₂ ⋙ F₂, this is the canonical morphism F₁ ⟶ Φ.localizedFunctor L₁ L₂ ⋙ F₂.

Equations

• One or more equations did not get rendered due to their size.

theorem CategoryTheory.LocalizerMorphism.rightDerivedFunctorComparison_fac {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [Category.{v₁, u₁} C₁] [Category.{v₂, u₂} C₂] [Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [Category.{v₄, u₄} D₁] [Category.{v₅, u₅} D₂] {W₁ : MorphismProperty C₁} {W₂ : MorphismProperty C₂} (Φ : LocalizerMorphism W₁ W₂) (L₁ : Functor C₁ D₁) (L₂ : Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : Functor C₂ H) (F₁ : Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] : CategoryStruct.comp α₁ (L₁.whiskerLeft (Φ.rightDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂)) = CategoryStruct.comp (Φ.functor.whiskerLeft α₂) (CategoryStruct.comp (Φ.functor.associator L₂ F₂).inv (CategoryStruct.comp (Functor.whiskerRight (CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).hom F₂) (L₁.associator (Φ.localizedFunctor L₁ L₂) F₂).hom))

theorem CategoryTheory.LocalizerMorphism.rightDerivedFunctorComparison_fac_assoc {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [Category.{v₁, u₁} C₁] [Category.{v₂, u₂} C₂] [Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [Category.{v₄, u₄} D₁] [Category.{v₅, u₅} D₂] {W₁ : MorphismProperty C₁} {W₂ : MorphismProperty C₂} (Φ : LocalizerMorphism W₁ W₂) (L₁ : Functor C₁ D₁) (L₂ : Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : Functor C₂ H) (F₁ : Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] {Z : Functor C₁ H} (h : L₁.comp ((Φ.localizedFunctor L₁ L₂).comp F₂) ⟶ Z) : CategoryStruct.comp α₁ (CategoryStruct.comp (L₁.whiskerLeft (Φ.rightDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂)) h) = CategoryStruct.comp (Φ.functor.whiskerLeft α₂) (CategoryStruct.comp (Φ.functor.associator L₂ F₂).inv (CategoryStruct.comp (Functor.whiskerRight (CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).hom F₂) (CategoryStruct.comp (L₁.associator (Φ.localizedFunctor L₁ L₂) F₂).hom h)))

@[simp]

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

@[simp]

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

theorem CategoryTheory.LocalizerMorphism.hasPointwiseRightDerivedFunctorAt_iff_of_isRightDerivabilityStructure {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [Category.{v₁, u₁} C₁] [Category.{v₂, u₂} C₂] [Category.{v₃, u₃} H] {W₁ : MorphismProperty C₁} {W₂ : MorphismProperty C₂} (Φ : LocalizerMorphism W₁ W₂) (F : Functor C₂ H) [Φ.IsRightDerivabilityStructure] (X : C₁) : (Φ.functor.comp F).HasPointwiseRightDerivedFunctorAt W₁ X ↔ F.HasPointwiseRightDerivedFunctorAt W₂ (Φ.functor.obj X)

theorem CategoryTheory.LocalizerMorphism.hasPointwiseRightDerivedFunctor_iff_of_isRightDerivabilityStructure {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [Category.{v₁, u₁} C₁] [Category.{v₂, u₂} C₂] [Category.{v₃, u₃} H] {W₁ : MorphismProperty C₁} {W₂ : MorphismProperty C₂} (Φ : LocalizerMorphism W₁ W₂) (F : Functor C₂ H) [Φ.IsRightDerivabilityStructure] : F.HasPointwiseRightDerivedFunctor W₂ ↔ (Φ.functor.comp F).HasPointwiseRightDerivedFunctor W₁

instance CategoryTheory.LocalizerMorphism.instIsIsoFunctorRightDerivedFunctorComparison {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [Category.{v₁, u₁} C₁] [Category.{v₂, u₂} C₂] [Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [Category.{v₄, u₄} D₁] [Category.{v₅, u₅} D₂] {W₁ : MorphismProperty C₁} {W₂ : MorphismProperty C₂} (Φ : LocalizerMorphism W₁ W₂) (L₁ : Functor C₁ D₁) (L₂ : Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : Functor C₂ H) (F₁ : Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] [Φ.IsRightDerivabilityStructure] [(Φ.functor.comp F).HasPointwiseRightDerivedFunctor W₁] [F₂.IsRightDerivedFunctor α₂ W₂] : IsIso (Φ.rightDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂)

theorem CategoryTheory.LocalizerMorphism.isIso_iff_of_isRightDerivabilityStructure {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [Category.{v₁, u₁} C₁] [Category.{v₂, u₂} C₂] [Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [Category.{v₄, u₄} D₁] [Category.{v₅, u₅} D₂] {W₁ : MorphismProperty C₁} {W₂ : MorphismProperty C₂} (Φ : LocalizerMorphism W₁ W₂) (L₁ : Functor C₁ D₁) (L₂ : Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : Functor C₂ H) (F₁ : Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] [Φ.IsRightDerivabilityStructure] [(Φ.functor.comp F).HasPointwiseRightDerivedFunctor W₁] [F₂.IsRightDerivedFunctor α₂ W₂] (X : C₁) : IsIso (α₁.app X) ↔ IsIso (α₂.app (Φ.functor.obj X))

@[deprecated CategoryTheory.LocalizerMorphism.isIso_iff_of_isRightDerivabilityStructure (since := "2025-11-16")]

theorem CategoryTheory.LocalizerMorphism.isIso_α_iff_of_isRightDerivabilityStructure {C₁ : Type u₁} {C₂ : Type u₂} {H : Type u₃} [Category.{v₁, u₁} C₁] [Category.{v₂, u₂} C₂] [Category.{v₃, u₃} H] {D₁ : Type u₄} {D₂ : Type u₅} [Category.{v₄, u₄} D₁] [Category.{v₅, u₅} D₂] {W₁ : MorphismProperty C₁} {W₂ : MorphismProperty C₂} (Φ : LocalizerMorphism W₁ W₂) (L₁ : Functor C₁ D₁) (L₂ : Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : Functor C₂ H) (F₁ : Functor D₁ H) (α₁ : Φ.functor.comp F ⟶ L₁.comp F₁) (F₂ : Functor D₂ H) (α₂ : F ⟶ L₂.comp F₂) [F₁.IsRightDerivedFunctor α₁ W₁] [Φ.IsRightDerivabilityStructure] [(Φ.functor.comp F).HasPointwiseRightDerivedFunctor W₁] [F₂.IsRightDerivedFunctor α₂ W₂] (X : C₁) : IsIso (α₁.app X) ↔ IsIso (α₂.app (Φ.functor.obj X))

Alias of CategoryTheory.LocalizerMorphism.isIso_iff_of_isRightDerivabilityStructure.