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

full_name: DerivedCategory.homologyFunctor
code: /-- The homology functor `DerivedCategory C ⥤ C` in degree `n : ℤ`. -/
noncomputable def homologyFunctor (n : ℤ) : DerivedCategory C ⥤ C 
goal: DerivedCategory C ⥤ C

full_name: DerivedCategory.homologyFunctorFactors
code: /-- The homology functor on the derived category is induced by the homology
functor on the category of cochain complexes. -/
noncomputable def homologyFunctorFactors (n : ℤ) : Q ⋙ homologyFunctor C n ≅
    HomologicalComplex.homologyFunctor _ _ n 
goal: DerivedCategory.Q ⋙ DerivedCategory.homologyFunctor C n ≅ HomologicalComplex.homologyFunctor C (ComplexShape.up ℤ) n

full_name: DerivedCategory.homologyFunctorFactorsh
code: /-- The homology functor on the derived category is induced by the homology
functor on the homotopy category of cochain complexes. -/
noncomputable def homologyFunctorFactorsh (n : ℤ) : Qh ⋙ homologyFunctor C n ≅
    HomotopyCategory.homologyFunctor _ _ n 
goal: DerivedCategory.Qh ⋙ DerivedCategory.homologyFunctor C n ≅ HomotopyCategory.homologyFunctor C (ComplexShape.up ℤ) n

full_name: DerivedCategory.instIsHomologicalHomologyFunctor
code: instance (n : ℤ) : (homologyFunctor C n).IsHomological 
goal: (DerivedCategory.homologyFunctor C n).IsHomological

full_name: DerivedCategory.instShiftSequenceHomologyFunctorOfNatInt
code: /-- The functors `homologyFunctor C n : DerivedCategory C ⥤ C` for all `n : ℤ` are part
of a "shift sequence", i.e. they satisfy compatiblities with shifts. -/
noncomputable instance : (homologyFunctor C 0).ShiftSequence ℤ 
goal: (DerivedCategory.homologyFunctor C 0).ShiftSequence ℤ

full_name: DerivedCategory.HomologySequence.δ
code: /-- The connecting homomorphism on the homology sequence attached to a distinguished
triangle in the derived category. -/
noncomputable def δ : (homologyFunctor C n₀).obj T.obj₃ ⟶ (homologyFunctor C n₁).obj T.obj₁ 
goal: (DerivedCategory.homologyFunctor C n₀).obj T.obj₃ ⟶ (DerivedCategory.homologyFunctor C n₁).obj T.obj₁

full_name: DerivedCategory.HomologySequence.comp_δ
code: @[reassoc (attr := simp)]
lemma comp_δ : (homologyFunctor C n₀).map T.mor₂ ≫ δ T n₀ n₁ h = 0 
goal: (DerivedCategory.homologyFunctor C n₀).map T.mor₂ ≫ DerivedCategory.HomologySequence.δ T n₀ n₁ h = 0

full_name: DerivedCategory.HomologySequence.δ_comp
code: @[reassoc (attr := simp)]
lemma δ_comp : δ T n₀ n₁ h ≫ (homologyFunctor C n₁).map T.mor₁ = 0 
goal: DerivedCategory.HomologySequence.δ T n₀ n₁ h ≫ (DerivedCategory.homologyFunctor C n₁).map T.mor₁ = 0

full_name: DerivedCategory.HomologySequence.exact₂
code: lemma exact₂ :
    (ShortComplex.mk ((homologyFunctor C n₀).map T.mor₁) ((homologyFunctor C n₀).map T.mor₂)
      (by simp only [← Functor.map_comp, comp_distTriang_mor_zero₁₂ _ hT,
        Functor.map_zero])).Exact 
goal: (CategoryTheory.ShortComplex.mk ((DerivedCategory.homologyFunctor C n₀).map T.mor₁)
    ((DerivedCategory.homologyFunctor C n₀).map T.mor₂) ⋯).Exact

full_name: DerivedCategory.HomologySequence.exact₃
code: lemma exact₃ : (ShortComplex.mk _ _ (comp_δ T hT n₀ n₁ h)).Exact 
goal: (CategoryTheory.ShortComplex.mk ((DerivedCategory.homologyFunctor C n₀).map T.mor₂)
    (DerivedCategory.HomologySequence.δ T n₀ n₁ h) ⋯).Exact

full_name: DerivedCategory.HomologySequence.exact₁
code: lemma exact₁ : (ShortComplex.mk _ _ (δ_comp T hT n₀ n₁ h)).Exact 
goal: (CategoryTheory.ShortComplex.mk (DerivedCategory.HomologySequence.δ T n₀ n₁ h)
    ((DerivedCategory.homologyFunctor C n₁).map T.mor₁) ⋯).Exact

full_name: DerivedCategory.HomologySequence.epi_homologyMap_mor₁_iff
code: lemma epi_homologyMap_mor₁_iff :
    Epi ((homologyFunctor C n₀).map T.mor₁) ↔ (homologyFunctor C n₀).map T.mor₂ = 0 
goal: CategoryTheory.Epi ((DerivedCategory.homologyFunctor C n₀).map T.mor₁) ↔
  (DerivedCategory.homologyFunctor C n₀).map T.mor₂ = 0

full_name: DerivedCategory.HomologySequence.mono_homologyMap_mor₁_iff
code: lemma mono_homologyMap_mor₁_iff :
    Mono ((homologyFunctor C n₁).map T.mor₁) ↔ δ T n₀ n₁ h = 0 
goal: CategoryTheory.Mono ((DerivedCategory.homologyFunctor C n₁).map T.mor₁) ↔
  DerivedCategory.HomologySequence.δ T n₀ n₁ h = 0

full_name: DerivedCategory.HomologySequence.epi_homologyMap_mor₂_iff
code: lemma epi_homologyMap_mor₂_iff :
    Epi ((homologyFunctor C n₀).map T.mor₂) ↔ δ T n₀ n₁ h = 0 
goal: CategoryTheory.Epi ((DerivedCategory.homologyFunctor C n₀).map T.mor₂) ↔
  DerivedCategory.HomologySequence.δ T n₀ n₁ h = 0

full_name: DerivedCategory.HomologySequence.mono_homologyMap_mor₂_iff
code: lemma mono_homologyMap_mor₂_iff :
    Mono ((homologyFunctor C n₀).map T.mor₂) ↔ (homologyFunctor C n₀).map T.mor₁ = 0 
goal: CategoryTheory.Mono ((DerivedCategory.homologyFunctor C n₀).map T.mor₂) ↔
  (DerivedCategory.homologyFunctor C n₀).map T.mor₁ = 0