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

full_name: CategoryTheory.ShortComplex.ShortExact.singleδ
code: /-- The connecting homomorphism
`(singleFunctor C 0).obj S.X₃ ⟶ ((singleFunctor C 0).obj S.X₁)⟦(1 : ℤ)⟧` in the derived
category of `C` when `S` is a short exact short complex in `C`. -/
noncomputable def singleδ : (singleFunctor C 0).obj S.X₃ ⟶
    ((singleFunctor C 0).obj S.X₁)⟦(1 : ℤ)⟧ 
goal: (DerivedCategory.singleFunctor C 0).obj S.X₃ ⟶
  (CategoryTheory.shiftFunctor (DerivedCategory C) 1).obj ((DerivedCategory.singleFunctor C 0).obj S.X₁)

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangle_obj₃
code: /-- The (distinguished) triangle in the derived category of `C` given by a
short exact short complex in `C`. -/
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
goal: hS.singleTriangle.obj₃ = (DerivedCategory.singleFunctor C 0).obj S.X₃

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangle_mor₁
code: /-- The (distinguished) triangle in the derived category of `C` given by a
short exact short complex in `C`. -/
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
goal: hS.singleTriangle.mor₁ = (DerivedCategory.singleFunctor C 0).map S.f

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangle_obj₂
code: /-- The (distinguished) triangle in the derived category of `C` given by a
short exact short complex in `C`. -/
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
goal: hS.singleTriangle.obj₂ = (DerivedCategory.singleFunctor C 0).obj S.X₂

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangle
code: /-- The (distinguished) triangle in the derived category of `C` given by a
short exact short complex in `C`. -/
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
goal: CategoryTheory.Pretriangulated.Triangle (DerivedCategory C)

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangle_mor₂
code: /-- The (distinguished) triangle in the derived category of `C` given by a
short exact short complex in `C`. -/
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
goal: hS.singleTriangle.mor₂ = (DerivedCategory.singleFunctor C 0).map S.g

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangle_mor₃
code: /-- The (distinguished) triangle in the derived category of `C` given by a
short exact short complex in `C`. -/
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
goal: hS.singleTriangle.mor₃ = hS.singleδ

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangle_obj₁
code: /-- The (distinguished) triangle in the derived category of `C` given by a
short exact short complex in `C`. -/
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) 
goal: hS.singleTriangle.obj₁ = (DerivedCategory.singleFunctor C 0).obj S.X₁

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_hom_hom₁
code: /-- Given a short exact complex `S` in `C` that is short exact (`hS`), this is the
canonical isomorphism between the triangle `hS.singleTriangle` in the derived category
and the triangle attached to the corresponding short exact sequence of cochain complexes
after the application of the single functor. -/
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
goal: hS.singleTriangleIso.hom.hom₁ = ((DerivedCategory.singleFunctorsPostcompQIso C).hom.hom 0).app S.X₁

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_hom_hom₃
code: /-- Given a short exact complex `S` in `C` that is short exact (`hS`), this is the
canonical isomorphism between the triangle `hS.singleTriangle` in the derived category
and the triangle attached to the corresponding short exact sequence of cochain complexes
after the application of the single functor. -/
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
goal: hS.singleTriangleIso.hom.hom₃ = ((DerivedCategory.singleFunctorsPostcompQIso C).hom.hom 0).app S.X₃

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_inv_hom₃
code: /-- Given a short exact complex `S` in `C` that is short exact (`hS`), this is the
canonical isomorphism between the triangle `hS.singleTriangle` in the derived category
and the triangle attached to the corresponding short exact sequence of cochain complexes
after the application of the single functor. -/
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
goal: hS.singleTriangleIso.inv.hom₃ = ((DerivedCategory.singleFunctorsPostcompQIso C).inv.hom 0).app S.X₃

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangleIso
code: /-- Given a short exact complex `S` in `C` that is short exact (`hS`), this is the
canonical isomorphism between the triangle `hS.singleTriangle` in the derived category
and the triangle attached to the corresponding short exact sequence of cochain complexes
after the application of the single functor. -/
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
goal: hS.singleTriangle ≅ DerivedCategory.triangleOfSES ⋯

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_hom_hom₂
code: /-- Given a short exact complex `S` in `C` that is short exact (`hS`), this is the
canonical isomorphism between the triangle `hS.singleTriangle` in the derived category
and the triangle attached to the corresponding short exact sequence of cochain complexes
after the application of the single functor. -/
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
goal: hS.singleTriangleIso.hom.hom₂ = ((DerivedCategory.singleFunctorsPostcompQIso C).hom.hom 0).app S.X₂

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_inv_hom₂
code: /-- Given a short exact complex `S` in `C` that is short exact (`hS`), this is the
canonical isomorphism between the triangle `hS.singleTriangle` in the derived category
and the triangle attached to the corresponding short exact sequence of cochain complexes
after the application of the single functor. -/
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
goal: hS.singleTriangleIso.inv.hom₂ = ((DerivedCategory.singleFunctorsPostcompQIso C).inv.hom 0).app S.X₂

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_inv_hom₁
code: /-- Given a short exact complex `S` in `C` that is short exact (`hS`), this is the
canonical isomorphism between the triangle `hS.singleTriangle` in the derived category
and the triangle attached to the corresponding short exact sequence of cochain complexes
after the application of the single functor. -/
@[simps!]
noncomputable def singleTriangleIso :
    hS.singleTriangle ≅
      triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) 
goal: hS.singleTriangleIso.inv.hom₁ = ((DerivedCategory.singleFunctorsPostcompQIso C).inv.hom 0).app S.X₁

full_name: CategoryTheory.ShortComplex.ShortExact.singleTriangle_distinguished
code: /-- The distinguished triangle in the derived category of `C` given by a
short exact short complex in `C`. -/
lemma singleTriangle_distinguished :
    hS.singleTriangle ∈ distTriang (DerivedCategory C) 
goal: hS.singleTriangle ∈ CategoryTheory.Pretriangulated.distinguishedTriangles