File: Mathlib/Algebra/Homology/DerivedCategory/ExactFunctor.lean

full_name: CategoryTheory.Functor.mapDerivedCategory
code: /-- The functor `DerivedCategory C₁ ⥤ DerivedCategory C₂` induced
by an exact functor `F : C₁ ⥤ C₂` between abelian categories. -/
noncomputable def mapDerivedCategory : DerivedCategory C₁ ⥤ DerivedCategory C₂ 
goal: DerivedCategory C₁ ⥤ DerivedCategory C₂

full_name: CategoryTheory.Functor.mapDerivedCategoryFactors
code: /-- The functor `F.mapDerivedCategory` is induced
by `F.mapHomologicalComplex (ComplexShape.up ℤ)`. -/
noncomputable def mapDerivedCategoryFactors :
    DerivedCategory.Q ⋙ F.mapDerivedCategory ≅
      F.mapHomologicalComplex (ComplexShape.up ℤ) ⋙ DerivedCategory.Q 
goal: DerivedCategory.Q ⋙ F.mapDerivedCategory ≅ F.mapHomologicalComplex (ComplexShape.up ℤ) ⋙ DerivedCategory.Q

full_name: CategoryTheory.Functor.instLiftingCochainComplexIntDerivedCategoryQQuasiIsoUpCompHomologicalComplexMapHomologicalComplexMapDerivedCategory
code: noncomputable instance :
    Localization.Lifting DerivedCategory.Q
      (HomologicalComplex.quasiIso C₁ (ComplexShape.up ℤ))
      (F.mapHomologicalComplex _ ⋙ DerivedCategory.Q) F.mapDerivedCategory 
goal: CategoryTheory.Localization.Lifting DerivedCategory.Q (HomologicalComplex.quasiIso C₁ (ComplexShape.up ℤ))
  (F.mapHomologicalComplex (ComplexShape.up ℤ) ⋙ DerivedCategory.Q) F.mapDerivedCategory

full_name: CategoryTheory.Functor.mapDerivedCategoryFactorsh
code: /-- The functor `F.mapDerivedCategory` is induced
by `F.mapHomotopyCategory (ComplexShape.up ℤ)`. -/
noncomputable def mapDerivedCategoryFactorsh :
    DerivedCategory.Qh ⋙ F.mapDerivedCategory ≅
      F.mapHomotopyCategory (ComplexShape.up ℤ) ⋙ DerivedCategory.Qh 
goal: DerivedCategory.Qh ⋙ F.mapDerivedCategory ≅ F.mapHomotopyCategory (ComplexShape.up ℤ) ⋙ DerivedCategory.Qh

full_name: CategoryTheory.Functor.mapDerivedCategoryFactorsh_hom_app
code: lemma mapDerivedCategoryFactorsh_hom_app (K : CochainComplex C₁ ℤ) :
    F.mapDerivedCategoryFactorsh.hom.app ((HomotopyCategory.quotient _ _).obj K) =
      F.mapDerivedCategory.map ((DerivedCategory.quotientCompQhIso C₁).hom.app K) ≫
        F.mapDerivedCategoryFactors.hom.app K ≫
        (DerivedCategory.quotientCompQhIso C₂).inv.app _ ≫
        DerivedCategory.Qh.map ((F.mapHomotopyCategoryFactors (ComplexShape.up ℤ)).inv.app K) 
goal: F.mapDerivedCategoryFactorsh.hom.app ((HomotopyCategory.quotient C₁ (ComplexShape.up ℤ)).obj K) =
  F.mapDerivedCategory.map ((DerivedCategory.quotientCompQhIso C₁).hom.app K) ≫
    F.mapDerivedCategoryFactors.hom.app K ≫
      (DerivedCategory.quotientCompQhIso C₂).inv.app ((F.mapHomologicalComplex (ComplexShape.up ℤ)).obj K) ≫
        DerivedCategory.Qh.map ((F.mapHomotopyCategoryFactors (ComplexShape.up ℤ)).inv.app K)

full_name: CategoryTheory.Functor.instLiftingHomotopyCategoryIntUpDerivedCategoryQhQuasiIsoCompMapHomotopyCategoryMapDerivedCategory
code: noncomputable instance :
    Localization.Lifting DerivedCategory.Qh
      (HomotopyCategory.quasiIso C₁ (ComplexShape.up ℤ))
      (F.mapHomotopyCategory _ ⋙ DerivedCategory.Qh) F.mapDerivedCategory 
goal: CategoryTheory.Localization.Lifting DerivedCategory.Qh (HomotopyCategory.quasiIso C₁ (ComplexShape.up ℤ))
  (F.mapHomotopyCategory (ComplexShape.up ℤ) ⋙ DerivedCategory.Qh) F.mapDerivedCategory

full_name: CategoryTheory.Functor.instCommShiftDerivedCategoryMapDerivedCategoryInt
code: noncomputable instance : F.mapDerivedCategory.CommShift ℤ 
goal: F.mapDerivedCategory.CommShift ℤ

full_name: CategoryTheory.Functor.instCommShiftHomotopyCategoryIntUpDerivedCategoryHomMapDerivedCategoryFactorsh
code: instance : NatTrans.CommShift F.mapDerivedCategoryFactorsh.hom ℤ 
goal: CategoryTheory.NatTrans.CommShift F.mapDerivedCategoryFactorsh.hom ℤ

full_name: CategoryTheory.Functor.instCommShiftCochainComplexIntDerivedCategoryHomMapDerivedCategoryFactors
code: instance : NatTrans.CommShift F.mapDerivedCategoryFactors.hom ℤ 
goal: CategoryTheory.NatTrans.CommShift F.mapDerivedCategoryFactors.hom ℤ

full_name: CategoryTheory.Functor.instIsTriangulatedDerivedCategoryMapDerivedCategory
code: instance : F.mapDerivedCategory.IsTriangulated 
goal: F.mapDerivedCategory.IsTriangulated