File: Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean

full_name: CategoryTheory.ShortComplex.QuasiIso.isIso'
code: /-- A morphism `φ : S₁ ⟶ S₂` of short complexes that have homology is a quasi-isomorphism if
the induced map `homologyMap φ : S₁.homology ⟶ S₂.homology` is an isomorphism. -/
class QuasiIso (φ : S₁ ⟶ S₂) : Prop where
  /-- the homology map is an isomorphism -/
  isIso' : IsIso (homologyMap φ)
goal: CategoryTheory.IsIso (CategoryTheory.ShortComplex.homologyMap φ)

full_name: CategoryTheory.ShortComplex.QuasiIso
code: /-- A morphism `φ : S₁ ⟶ S₂` of short complexes that have homology is a quasi-isomorphism if
the induced map `homologyMap φ : S₁.homology ⟶ S₂.homology` is an isomorphism. -/
class QuasiIso (φ : S₁ ⟶ S₂) : Prop where
  /-- the homology map is an isomorphism -/
  isIso' : IsIso (homologyMap φ)
goal: Prop

full_name: CategoryTheory.ShortComplex.QuasiIso.isIso
code: instance QuasiIso.isIso (φ : S₁ ⟶ S₂) [QuasiIso φ] : IsIso (homologyMap φ) 
goal: CategoryTheory.IsIso (CategoryTheory.ShortComplex.homologyMap φ)

full_name: CategoryTheory.ShortComplex.quasiIso_iff
code: lemma quasiIso_iff (φ : S₁ ⟶ S₂) :
    QuasiIso φ ↔ IsIso (homologyMap φ) 
goal: CategoryTheory.ShortComplex.QuasiIso φ ↔ CategoryTheory.IsIso (CategoryTheory.ShortComplex.homologyMap φ)

full_name: CategoryTheory.ShortComplex.quasiIso_of_isIso
code: instance quasiIso_of_isIso (φ : S₁ ⟶ S₂) [IsIso φ] : QuasiIso φ 
goal: CategoryTheory.ShortComplex.QuasiIso φ

full_name: CategoryTheory.ShortComplex.quasiIso_comp
code: instance quasiIso_comp (φ : S₁ ⟶ S₂) (φ' : S₂ ⟶ S₃) [hφ : QuasiIso φ] [hφ' : QuasiIso φ'] :
    QuasiIso (φ ≫ φ') 
goal: CategoryTheory.ShortComplex.QuasiIso (CategoryTheory.CategoryStruct.comp φ φ')

full_name: CategoryTheory.ShortComplex.quasiIso_of_comp_left
code: lemma quasiIso_of_comp_left (φ : S₁ ⟶ S₂) (φ' : S₂ ⟶ S₃)
    [hφ : QuasiIso φ] [hφφ' : QuasiIso (φ ≫ φ')] :
    QuasiIso φ' 
goal: CategoryTheory.ShortComplex.QuasiIso φ'

full_name: CategoryTheory.ShortComplex.quasiIso_iff_comp_left
code: lemma quasiIso_iff_comp_left (φ : S₁ ⟶ S₂) (φ' : S₂ ⟶ S₃) [hφ : QuasiIso φ] :
    QuasiIso (φ ≫ φ') ↔ QuasiIso φ' 
goal: CategoryTheory.ShortComplex.QuasiIso (CategoryTheory.CategoryStruct.comp φ φ') ↔ CategoryTheory.ShortComplex.QuasiIso φ'

full_name: CategoryTheory.ShortComplex.quasiIso_of_comp_right
code: lemma quasiIso_of_comp_right (φ : S₁ ⟶ S₂) (φ' : S₂ ⟶ S₃)
    [hφ' : QuasiIso φ'] [hφφ' : QuasiIso (φ ≫ φ')] :
    QuasiIso φ 
goal: CategoryTheory.ShortComplex.QuasiIso φ

full_name: CategoryTheory.ShortComplex.quasiIso_iff_comp_right
code: lemma quasiIso_iff_comp_right (φ : S₁ ⟶ S₂) (φ' : S₂ ⟶ S₃) [hφ' : QuasiIso φ'] :
    QuasiIso (φ ≫ φ') ↔ QuasiIso φ 
goal: CategoryTheory.ShortComplex.QuasiIso (CategoryTheory.CategoryStruct.comp φ φ') ↔ CategoryTheory.ShortComplex.QuasiIso φ

full_name: CategoryTheory.ShortComplex.quasiIso_of_arrow_mk_iso
code: lemma quasiIso_of_arrow_mk_iso (φ : S₁ ⟶ S₂) (φ' : S₃ ⟶ S₄) (e : Arrow.mk φ ≅ Arrow.mk φ')
    [hφ : QuasiIso φ] : QuasiIso φ' 
goal: CategoryTheory.ShortComplex.QuasiIso φ'

full_name: CategoryTheory.ShortComplex.quasiIso_iff_of_arrow_mk_iso
code: lemma quasiIso_iff_of_arrow_mk_iso (φ : S₁ ⟶ S₂) (φ' : S₃ ⟶ S₄) (e : Arrow.mk φ ≅ Arrow.mk φ') :
    QuasiIso φ ↔ QuasiIso φ' 
goal: CategoryTheory.ShortComplex.QuasiIso φ ↔ CategoryTheory.ShortComplex.QuasiIso φ'

full_name: CategoryTheory.ShortComplex.LeftHomologyMapData.quasiIso_iff
code: lemma LeftHomologyMapData.quasiIso_iff {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData}
    {h₂ : S₂.LeftHomologyData} (γ : LeftHomologyMapData φ h₁ h₂) :
    QuasiIso φ ↔ IsIso γ.φH 
goal: CategoryTheory.ShortComplex.QuasiIso φ ↔ CategoryTheory.IsIso γ.φH

full_name: CategoryTheory.ShortComplex.RightHomologyMapData.quasiIso_iff
code: lemma RightHomologyMapData.quasiIso_iff {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData}
    {h₂ : S₂.RightHomologyData} (γ : RightHomologyMapData φ h₁ h₂) :
    QuasiIso φ ↔ IsIso γ.φH 
goal: CategoryTheory.ShortComplex.QuasiIso φ ↔ CategoryTheory.IsIso γ.φH

full_name: CategoryTheory.ShortComplex.quasiIso_iff_isIso_leftHomologyMap'
code: lemma quasiIso_iff_isIso_leftHomologyMap' (φ : S₁ ⟶ S₂)
    (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) :
    QuasiIso φ ↔ IsIso (leftHomologyMap' φ h₁ h₂) 
goal: CategoryTheory.ShortComplex.QuasiIso φ ↔ CategoryTheory.IsIso (CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₂)

full_name: CategoryTheory.ShortComplex.quasiIso_iff_isIso_rightHomologyMap'
code: lemma quasiIso_iff_isIso_rightHomologyMap' (φ : S₁ ⟶ S₂)
    (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) :
    QuasiIso φ ↔ IsIso (rightHomologyMap' φ h₁ h₂) 
goal: CategoryTheory.ShortComplex.QuasiIso φ ↔ CategoryTheory.IsIso (CategoryTheory.ShortComplex.rightHomologyMap' φ h₁ h₂)

full_name: CategoryTheory.ShortComplex.quasiIso_iff_isIso_homologyMap'
code: lemma quasiIso_iff_isIso_homologyMap' (φ : S₁ ⟶ S₂)
    (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData) :
    QuasiIso φ ↔ IsIso (homologyMap' φ h₁ h₂) 
goal: CategoryTheory.ShortComplex.QuasiIso φ ↔ CategoryTheory.IsIso (CategoryTheory.ShortComplex.homologyMap' φ h₁ h₂)

full_name: CategoryTheory.ShortComplex.quasiIso_of_epi_of_isIso_of_mono
code: lemma quasiIso_of_epi_of_isIso_of_mono (φ : S₁ ⟶ S₂) [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] :
    QuasiIso φ 
goal: CategoryTheory.ShortComplex.QuasiIso φ

full_name: CategoryTheory.ShortComplex.quasiIso_opMap_iff
code: lemma quasiIso_opMap_iff (φ : S₁ ⟶ S₂) :
    QuasiIso (opMap φ) ↔ QuasiIso φ 
goal: CategoryTheory.ShortComplex.QuasiIso (CategoryTheory.ShortComplex.opMap φ) ↔ CategoryTheory.ShortComplex.QuasiIso φ

full_name: CategoryTheory.ShortComplex.quasiIso_opMap
code: lemma quasiIso_opMap (φ : S₁ ⟶ S₂) [QuasiIso φ] :
    QuasiIso (opMap φ) 
goal: CategoryTheory.ShortComplex.QuasiIso (CategoryTheory.ShortComplex.opMap φ)

full_name: CategoryTheory.ShortComplex.quasiIso_unopMap
code: lemma quasiIso_unopMap {S₁ S₂ : ShortComplex Cᵒᵖ} [S₁.HasHomology] [S₂.HasHomology]
    [S₁.unop.HasHomology] [S₂.unop.HasHomology]
    (φ : S₁ ⟶ S₂) [QuasiIso φ] : QuasiIso (unopMap φ) 
goal: CategoryTheory.ShortComplex.QuasiIso (CategoryTheory.ShortComplex.unopMap φ)

full_name: CategoryTheory.ShortComplex.quasiIso_iff_isIso_liftCycles
code: lemma quasiIso_iff_isIso_liftCycles (φ : S₁ ⟶ S₂)
    (hf₁ : S₁.f = 0) (hg₁ : S₁.g = 0) (hf₂ : S₂.f = 0) :
    QuasiIso φ ↔ IsIso (S₂.liftCycles φ.τ₂ (by rw [φ.comm₂₃, hg₁, zero_comp])) 
goal: CategoryTheory.ShortComplex.QuasiIso φ ↔ CategoryTheory.IsIso (S₂.liftCycles φ.τ₂ ⋯)

full_name: CategoryTheory.ShortComplex.quasiIso_iff_isIso_descOpcycles
code: lemma quasiIso_iff_isIso_descOpcycles (φ : S₁ ⟶ S₂)
    (hg₁ : S₁.g = 0) (hf₂ : S₂.f = 0) (hg₂ : S₂.g = 0) :
    QuasiIso φ ↔ IsIso (S₁.descOpcycles φ.τ₂ (by rw [← φ.comm₁₂, hf₂, comp_zero])) 
goal: CategoryTheory.ShortComplex.QuasiIso φ ↔ CategoryTheory.IsIso (S₁.descOpcycles φ.τ₂ ⋯)