File: Mathlib/Algebra/Homology/HomologySequenceLemmas.lean

full_name: HomologicalComplex.HomologySequence.mapSnakeInput_f₂
code: /-- The morphism `snakeInput hS₁ i j hij ⟶ snakeInput hS₂ i j hij` induced by
a morphism `φ : S₁ ⟶ S₂` of short complexes of homological complexes, that
are short exact (`hS₁ : S₁.ShortExact` and `hS₂ : S₁.ShortExact`). -/
@[simps]
noncomputable def mapSnakeInput (i j : ι) (hij : c.Rel i j) :
    snakeInput hS₁ i j hij ⟶ snakeInput hS₂ i j hij 
goal: (HomologicalComplex.HomologySequence.mapSnakeInput φ hS₁ hS₂ i j hij).f₂ =
  (HomologicalComplex.cyclesFunctor C c j).mapShortComplex.map φ

full_name: HomologicalComplex.HomologySequence.mapSnakeInput_f₃
code: /-- The morphism `snakeInput hS₁ i j hij ⟶ snakeInput hS₂ i j hij` induced by
a morphism `φ : S₁ ⟶ S₂` of short complexes of homological complexes, that
are short exact (`hS₁ : S₁.ShortExact` and `hS₂ : S₁.ShortExact`). -/
@[simps]
noncomputable def mapSnakeInput (i j : ι) (hij : c.Rel i j) :
    snakeInput hS₁ i j hij ⟶ snakeInput hS₂ i j hij 
goal: (HomologicalComplex.HomologySequence.mapSnakeInput φ hS₁ hS₂ i j hij).f₃ =
  (HomologicalComplex.homologyFunctor C c j).mapShortComplex.map φ

full_name: HomologicalComplex.HomologySequence.mapSnakeInput
code: /-- The morphism `snakeInput hS₁ i j hij ⟶ snakeInput hS₂ i j hij` induced by
a morphism `φ : S₁ ⟶ S₂` of short complexes of homological complexes, that
are short exact (`hS₁ : S₁.ShortExact` and `hS₂ : S₁.ShortExact`). -/
@[simps]
noncomputable def mapSnakeInput (i j : ι) (hij : c.Rel i j) :
    snakeInput hS₁ i j hij ⟶ snakeInput hS₂ i j hij 
goal: HomologicalComplex.HomologySequence.snakeInput hS₁ i j hij ⟶ HomologicalComplex.HomologySequence.snakeInput hS₂ i j hij

full_name: HomologicalComplex.HomologySequence.mapSnakeInput_f₁
code: /-- The morphism `snakeInput hS₁ i j hij ⟶ snakeInput hS₂ i j hij` induced by
a morphism `φ : S₁ ⟶ S₂` of short complexes of homological complexes, that
are short exact (`hS₁ : S₁.ShortExact` and `hS₂ : S₁.ShortExact`). -/
@[simps]
noncomputable def mapSnakeInput (i j : ι) (hij : c.Rel i j) :
    snakeInput hS₁ i j hij ⟶ snakeInput hS₂ i j hij 
goal: (HomologicalComplex.HomologySequence.mapSnakeInput φ hS₁ hS₂ i j hij).f₁ =
  (HomologicalComplex.opcyclesFunctor C c i).mapShortComplex.map φ

full_name: HomologicalComplex.HomologySequence.mapSnakeInput_f₀
code: /-- The morphism `snakeInput hS₁ i j hij ⟶ snakeInput hS₂ i j hij` induced by
a morphism `φ : S₁ ⟶ S₂` of short complexes of homological complexes, that
are short exact (`hS₁ : S₁.ShortExact` and `hS₂ : S₁.ShortExact`). -/
@[simps]
noncomputable def mapSnakeInput (i j : ι) (hij : c.Rel i j) :
    snakeInput hS₁ i j hij ⟶ snakeInput hS₂ i j hij 
goal: (HomologicalComplex.HomologySequence.mapSnakeInput φ hS₁ hS₂ i j hij).f₀ =
  (HomologicalComplex.homologyFunctor C c i).mapShortComplex.map φ

full_name: HomologicalComplex.HomologySequence.δ_naturality
code: @[reassoc]
lemma δ_naturality (i j : ι) (hij : c.Rel i j) :
    hS₁.δ i j hij ≫ HomologicalComplex.homologyMap φ.τ₁ _ =
      HomologicalComplex.homologyMap φ.τ₃ _ ≫ hS₂.δ i j hij 
goal: hS₁.δ i j hij ≫ HomologicalComplex.homologyMap φ.τ₁ j = HomologicalComplex.homologyMap φ.τ₃ i ≫ hS₂.δ i j hij

full_name: HomologicalComplex.HomologySequence.composableArrows₂
code: /-- The (exact) sequence `S.X₁.homology i ⟶ S.X₂.homology i ⟶ S.X₃.homology i` -/
@[simp]
noncomputable def composableArrows₂ (i : ι) : ComposableArrows C 2 
goal: CategoryTheory.ComposableArrows C 2

full_name: HomologicalComplex.HomologySequence.composableArrows₂_exact
code: lemma composableArrows₂_exact (i : ι) :
    (composableArrows₂ S₁ i).Exact 
goal: (HomologicalComplex.HomologySequence.composableArrows₂ S₁ i).Exact

full_name: HomologicalComplex.HomologySequence.composableArrows₅
code: /-- The (exact) sequence
`H_i(S.X₁) ⟶ H_i(S.X₂) ⟶ H_i(S.X₃) ⟶ H_j(S.X₁) ⟶ H_j(S.X₂) ⟶ H_j(S.X₃)` when `c.Rel i j`
and `S` is a short exact short complex of homological complexes in an abelian category. -/
@[simp]
noncomputable def composableArrows₅ (i j : ι) (hij : c.Rel i j) : ComposableArrows C 5 
goal: CategoryTheory.ComposableArrows C 5

full_name: HomologicalComplex.HomologySequence.composableArrows₅_exact
code: lemma composableArrows₅_exact (i j : ι) (hij : c.Rel i j) :
    (composableArrows₅ hS₁ i j hij).Exact 
goal: (HomologicalComplex.HomologySequence.composableArrows₅ hS₁ i j hij).Exact

full_name: HomologicalComplex.HomologySequence.mapComposableArrows₂
code: /-- The map between the exact sequences `S₁.X₁.homology i ⟶ S₁.X₂.homology i ⟶ S₁.X₃.homology i`
and `S₂.X₁.homology i ⟶ S₂.X₂.homology i ⟶ S₂.X₃.homology i` that is induced by `φ : S₁ ⟶ S₂`. -/
@[simp]
noncomputable def mapComposableArrows₂ (i : ι) : composableArrows₂ S₁ i ⟶ composableArrows₂ S₂ i 
goal: HomologicalComplex.HomologySequence.composableArrows₂ S₁ i ⟶ HomologicalComplex.HomologySequence.composableArrows₂ S₂ i

full_name: HomologicalComplex.HomologySequence.mapComposableArrows₅
code: /-- The map `composableArrows₅ hS₁ i j hij ⟶ composableArrows₅ hS₂ i j hij` of exact
sequences induced by a morphism `φ : S₁ ⟶ S₂` between short exact short complexes of
homological complexes. -/
@[simp]
noncomputable def mapComposableArrows₅ (i j : ι) (hij : c.Rel i j) :
    composableArrows₅ hS₁ i j hij ⟶ composableArrows₅ hS₂ i j hij 
goal: HomologicalComplex.HomologySequence.composableArrows₅ hS₁ i j hij ⟶
  HomologicalComplex.HomologySequence.composableArrows₅ hS₂ i j hij

full_name: HomologicalComplex.HomologySequence.mono_homologyMap_τ₃
code: lemma mono_homologyMap_τ₃ (i : ι)
    (h₁ : Epi (homologyMap φ.τ₁ i))
    (h₂ : Mono (homologyMap φ.τ₂ i))
    (h₃ : ∀ j, c.Rel i j → Mono (homologyMap φ.τ₁ j)) :
    Mono (homologyMap φ.τ₃ i) 
goal: CategoryTheory.Mono (HomologicalComplex.homologyMap φ.τ₃ i)

full_name: HomologicalComplex.HomologySequence.epi_homologyMap_τ₃
code: lemma epi_homologyMap_τ₃ (i : ι)
    (h₁ : Epi (homologyMap φ.τ₂ i))
    (h₂ : ∀ j, c.Rel i j → Epi (homologyMap φ.τ₁ j))
    (h₃ : ∀ j, c.Rel i j → Mono (homologyMap φ.τ₂ j)) :
    Epi (homologyMap φ.τ₃ i) 
goal: CategoryTheory.Epi (HomologicalComplex.homologyMap φ.τ₃ i)

full_name: HomologicalComplex.HomologySequence.isIso_homologyMap_τ₃
code: lemma isIso_homologyMap_τ₃ (i : ι)
    (h₁ : Epi (homologyMap φ.τ₁ i))
    (h₂ : IsIso (homologyMap φ.τ₂ i))
    (h₃ : ∀ j, c.Rel i j → IsIso (homologyMap φ.τ₁ j))
    (h₄ : ∀ j, c.Rel i j → Mono (homologyMap φ.τ₂ j)) :
    IsIso (homologyMap φ.τ₃ i) 
goal: CategoryTheory.IsIso (HomologicalComplex.homologyMap φ.τ₃ i)

full_name: HomologicalComplex.HomologySequence.quasiIso_τ₃
code: lemma quasiIso_τ₃ (h₁ : QuasiIso φ.τ₁) (h₂ : QuasiIso φ.τ₂) :
    QuasiIso φ.τ₃ 
goal: QuasiIso φ.τ₃