Existence of pointwise left derived functors via derivability structures

In this file, we show how a left derivability structure can be used in order to construct (pointwise) left derived functors. Let Φ be a left derivability structure from W₁ : MorphismProperty C₁ to W₂ : MorphismProperty C₂. Let F : C₂ ⥤ H be a functor. Then, the lemma hasPointwiseLeftDerivedFunctor_iff_of_isLeftDerivabilityStructure says that F has a pointwise left derived functor with respect to W₂ if and only if Φ.functor ⋙ F has a pointwise left 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 left derived functor of F with respect to W₂ exists.

This file contains the dual results to those obtained in the file Mathlib/CategoryTheory/Localization/DerivabilityStructure/PointwiseRightDerived.lean.

References

• Bruno Kahn and Georges Maltsiniotis, Structures de dérivabilité

noncomputable def CategoryTheory.LocalizerMorphism.leftDerivedFunctorComparison {C₁ : Type u_1} {C₂ : Type u_2} {H : Type u_3} {D₁ : Type u_4} {D₂ : Type u_5} [Category.{v_1, u_1} C₁] [Category.{v_2, u_2} C₂] [Category.{v_3, u_3} H] [Category.{v_4, u_4} D₁] [Category.{v_5, u_5} 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 : L₁.comp F₁ ⟶ Φ.functor.comp F) (F₂ : Functor D₂ H) (α₂ : L₂.comp F₂ ⟶ F) [F₁.IsLeftDerivedFunctor α₁ W₁] : (Φ.localizedFunctor L₁ L₂).comp F₂ ⟶ F₁

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 left derived functor of Φ.functor ⋙ F, and if F₂ : D₂ ⥤ H is a functor equipped with a natural transformation α₂ : L₂ ⋙ F₂ ⟶ F, this is the canonical morphism Φ.localizedFunctor L₁ L₂ ⋙ F₂ ⟶ F₁.

Equations

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

theorem CategoryTheory.LocalizerMorphism.leftDerivedFunctorComparison_fac {C₁ : Type u_1} {C₂ : Type u_2} {H : Type u_3} {D₁ : Type u_4} {D₂ : Type u_5} [Category.{v_1, u_1} C₁] [Category.{v_2, u_2} C₂] [Category.{v_3, u_3} H] [Category.{v_4, u_4} D₁] [Category.{v_5, u_5} 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 : L₁.comp F₁ ⟶ Φ.functor.comp F) (F₂ : Functor D₂ H) (α₂ : L₂.comp F₂ ⟶ F) [F₁.IsLeftDerivedFunctor α₁ W₁] : CategoryStruct.comp (L₁.whiskerLeft (Φ.leftDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂)) α₁ = CategoryStruct.comp (L₁.associator (Φ.localizedFunctor L₁ L₂) F₂).inv (CategoryStruct.comp (Functor.whiskerRight (CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).inv F₂) (CategoryStruct.comp (Φ.functor.associator L₂ F₂).hom (Φ.functor.whiskerLeft α₂)))

theorem CategoryTheory.LocalizerMorphism.leftDerivedFunctorComparison_fac_assoc {C₁ : Type u_1} {C₂ : Type u_2} {H : Type u_3} {D₁ : Type u_4} {D₂ : Type u_5} [Category.{v_1, u_1} C₁] [Category.{v_2, u_2} C₂] [Category.{v_3, u_3} H] [Category.{v_4, u_4} D₁] [Category.{v_5, u_5} 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 : L₁.comp F₁ ⟶ Φ.functor.comp F) (F₂ : Functor D₂ H) (α₂ : L₂.comp F₂ ⟶ F) [F₁.IsLeftDerivedFunctor α₁ W₁] {Z : Functor C₁ H} (h : Φ.functor.comp F ⟶ Z) : CategoryStruct.comp (L₁.whiskerLeft (Φ.leftDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂)) (CategoryStruct.comp α₁ h) = CategoryStruct.comp (L₁.associator (Φ.localizedFunctor L₁ L₂) F₂).inv (CategoryStruct.comp (Functor.whiskerRight (CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).inv F₂) (CategoryStruct.comp (Φ.functor.associator L₂ F₂).hom (CategoryStruct.comp (Φ.functor.whiskerLeft α₂) h)))

@[simp]

theorem CategoryTheory.LocalizerMorphism.leftDerivedFunctorComparison_fac_app {C₁ : Type u_1} {C₂ : Type u_2} {H : Type u_3} {D₁ : Type u_4} {D₂ : Type u_5} [Category.{v_1, u_1} C₁] [Category.{v_2, u_2} C₂] [Category.{v_3, u_3} H] [Category.{v_4, u_4} D₁] [Category.{v_5, u_5} 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 : L₁.comp F₁ ⟶ Φ.functor.comp F) (F₂ : Functor D₂ H) (α₂ : L₂.comp F₂ ⟶ F) [F₁.IsLeftDerivedFunctor α₁ W₁] (X : C₁) : CategoryStruct.comp ((Φ.leftDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂).app (L₁.obj X)) (α₁.app X) = CategoryStruct.comp (F₂.map ((CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).inv.app X)) (α₂.app (Φ.functor.obj X))

@[simp]

theorem CategoryTheory.LocalizerMorphism.leftDerivedFunctorComparison_fac_app_assoc {C₁ : Type u_1} {C₂ : Type u_2} {H : Type u_3} {D₁ : Type u_4} {D₂ : Type u_5} [Category.{v_1, u_1} C₁] [Category.{v_2, u_2} C₂] [Category.{v_3, u_3} H] [Category.{v_4, u_4} D₁] [Category.{v_5, u_5} 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 : L₁.comp F₁ ⟶ Φ.functor.comp F) (F₂ : Functor D₂ H) (α₂ : L₂.comp F₂ ⟶ F) [F₁.IsLeftDerivedFunctor α₁ W₁] (X : C₁) {Z : H} (h : F.obj (Φ.functor.obj X) ⟶ Z) : CategoryStruct.comp ((Φ.leftDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂).app (L₁.obj X)) (CategoryStruct.comp (α₁.app X) h) = CategoryStruct.comp (F₂.map ((CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).inv.app X)) (CategoryStruct.comp (α₂.app (Φ.functor.obj X)) h)

theorem CategoryTheory.LocalizerMorphism.hasPointwiseLeftDerivedFunctorAt_iff_of_isLeftDerivabilityStructure {C₁ : Type u_1} {C₂ : Type u_2} {H : Type u_3} [Category.{v_1, u_1} C₁] [Category.{v_2, u_2} C₂] [Category.{v_3, u_3} H] {W₁ : MorphismProperty C₁} {W₂ : MorphismProperty C₂} (Φ : LocalizerMorphism W₁ W₂) (F : Functor C₂ H) [Φ.IsLeftDerivabilityStructure] (X : C₁) : (Φ.functor.comp F).HasPointwiseLeftDerivedFunctorAt W₁ X ↔ F.HasPointwiseLeftDerivedFunctorAt W₂ (Φ.functor.obj X)

theorem CategoryTheory.LocalizerMorphism.hasPointwiseLeftDerivedFunctor_iff_of_isLeftDerivabilityStructure {C₁ : Type u_1} {C₂ : Type u_2} {H : Type u_3} [Category.{v_1, u_1} C₁] [Category.{v_2, u_2} C₂] [Category.{v_3, u_3} H] {W₁ : MorphismProperty C₁} {W₂ : MorphismProperty C₂} (Φ : LocalizerMorphism W₁ W₂) (F : Functor C₂ H) [Φ.IsLeftDerivabilityStructure] : F.HasPointwiseLeftDerivedFunctor W₂ ↔ (Φ.functor.comp F).HasPointwiseLeftDerivedFunctor W₁

instance CategoryTheory.LocalizerMorphism.instIsIsoFunctorLeftDerivedFunctorComparison {C₁ : Type u_1} {C₂ : Type u_2} {H : Type u_3} {D₁ : Type u_4} {D₂ : Type u_5} [Category.{v_1, u_1} C₁] [Category.{v_2, u_2} C₂] [Category.{v_3, u_3} H] [Category.{v_4, u_4} D₁] [Category.{v_5, u_5} 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 : L₁.comp F₁ ⟶ Φ.functor.comp F) (F₂ : Functor D₂ H) (α₂ : L₂.comp F₂ ⟶ F) [F₁.IsLeftDerivedFunctor α₁ W₁] [Φ.IsLeftDerivabilityStructure] [(Φ.functor.comp F).HasPointwiseLeftDerivedFunctor W₁] [F₂.IsLeftDerivedFunctor α₂ W₂] : IsIso (Φ.leftDerivedFunctorComparison L₁ L₂ F F₁ α₁ F₂ α₂)

theorem CategoryTheory.LocalizerMorphism.isIso_iff_of_isLeftDerivabilityStructure {C₁ : Type u_1} {C₂ : Type u_2} {H : Type u_3} {D₁ : Type u_4} {D₂ : Type u_5} [Category.{v_1, u_1} C₁] [Category.{v_2, u_2} C₂] [Category.{v_3, u_3} H] [Category.{v_4, u_4} D₁] [Category.{v_5, u_5} 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 : L₁.comp F₁ ⟶ Φ.functor.comp F) (F₂ : Functor D₂ H) (α₂ : L₂.comp F₂ ⟶ F) [F₁.IsLeftDerivedFunctor α₁ W₁] [Φ.IsLeftDerivabilityStructure] [(Φ.functor.comp F).HasPointwiseLeftDerivedFunctor W₁] [F₂.IsLeftDerivedFunctor α₂ W₂] (X : C₁) : IsIso (α₁.app X) ↔ IsIso (α₂.app (Φ.functor.obj X))