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

full_name: CategoryTheory.ShortComplex.isLimitOfIsLimitπ
code: /-- If a cone with values in `ShortComplex C` is such that it becomes limit
when we apply the three projections `ShortComplex C ⥤ C`, then it is limit. -/
def isLimitOfIsLimitπ (c : Cone F)
    (h₁ : IsLimit (π₁.mapCone c)) (h₂ : IsLimit (π₂.mapCone c))
    (h₃ : IsLimit (π₃.mapCone c)) : IsLimit c 
goal: CategoryTheory.Limits.IsLimit c

full_name: CategoryTheory.ShortComplex.limitCone
code: /-- Construction of a limit cone for a functor `J ⥤ ShortComplex C` using the limits
of the three components `J ⥤ C`. -/
noncomputable def limitCone : Cone F 
goal: CategoryTheory.Limits.Cone F

full_name: CategoryTheory.ShortComplex.isLimitπ₁MapConeLimitCone
code: /-- `limitCone F` becomes limit after the application of `π₁ : ShortComplex C ⥤ C`. -/
noncomputable def isLimitπ₁MapConeLimitCone : IsLimit (π₁.mapCone (limitCone F)) 
goal: CategoryTheory.Limits.IsLimit (CategoryTheory.ShortComplex.π₁.mapCone (CategoryTheory.ShortComplex.limitCone F))

full_name: CategoryTheory.ShortComplex.isLimitπ₂MapConeLimitCone
code: /-- `limitCone F` becomes limit after the application of `π₂ : ShortComplex C ⥤ C`. -/
noncomputable def isLimitπ₂MapConeLimitCone : IsLimit (π₂.mapCone (limitCone F)) 
goal: CategoryTheory.Limits.IsLimit (CategoryTheory.ShortComplex.π₂.mapCone (CategoryTheory.ShortComplex.limitCone F))

full_name: CategoryTheory.ShortComplex.isLimitπ₃MapConeLimitCone
code: /-- `limitCone F` becomes limit after the application of `π₃ : ShortComplex C ⥤ C`. -/
noncomputable def isLimitπ₃MapConeLimitCone : IsLimit (π₃.mapCone (limitCone F)) 
goal: CategoryTheory.Limits.IsLimit (CategoryTheory.ShortComplex.π₃.mapCone (CategoryTheory.ShortComplex.limitCone F))

full_name: CategoryTheory.ShortComplex.isLimitLimitCone
code: /-- `limitCone F` is limit. -/
noncomputable def isLimitLimitCone : IsLimit (limitCone F) 
goal: CategoryTheory.Limits.IsLimit (CategoryTheory.ShortComplex.limitCone F)

full_name: CategoryTheory.ShortComplex.hasLimit_of_hasLimitπ
code: instance hasLimit_of_hasLimitπ : HasLimit F 
goal: CategoryTheory.Limits.HasLimit F

full_name: CategoryTheory.ShortComplex.instPreservesLimitπ₁
code: noncomputable instance : PreservesLimit F π₁ 
goal: CategoryTheory.Limits.PreservesLimit F CategoryTheory.ShortComplex.π₁

full_name: CategoryTheory.ShortComplex.instPreservesLimitπ₂
code: noncomputable instance : PreservesLimit F π₂ 
goal: CategoryTheory.Limits.PreservesLimit F CategoryTheory.ShortComplex.π₂

full_name: CategoryTheory.ShortComplex.instPreservesLimitπ₃
code: noncomputable instance : PreservesLimit F π₃ 
goal: CategoryTheory.Limits.PreservesLimit F CategoryTheory.ShortComplex.π₃

full_name: CategoryTheory.ShortComplex.hasLimitsOfShape
code: instance hasLimitsOfShape :
    HasLimitsOfShape J (ShortComplex C) 
goal: CategoryTheory.Limits.HasLimitsOfShape J (CategoryTheory.ShortComplex C)

full_name: CategoryTheory.ShortComplex.instPreservesLimitsOfShapeπ₁
code: noncomputable instance : PreservesLimitsOfShape J (π₁ : _ ⥤ C) 
goal: CategoryTheory.Limits.PreservesLimitsOfShape J CategoryTheory.ShortComplex.π₁

full_name: CategoryTheory.ShortComplex.instPreservesLimitsOfShapeπ₂
code: noncomputable instance : PreservesLimitsOfShape J (π₂ : _ ⥤ C) 
goal: CategoryTheory.Limits.PreservesLimitsOfShape J CategoryTheory.ShortComplex.π₂

full_name: CategoryTheory.ShortComplex.instPreservesLimitsOfShapeπ₃
code: noncomputable instance : PreservesLimitsOfShape J (π₃ : _ ⥤ C) 
goal: CategoryTheory.Limits.PreservesLimitsOfShape J CategoryTheory.ShortComplex.π₃

full_name: CategoryTheory.ShortComplex.hasFiniteLimits
code: instance hasFiniteLimits : HasFiniteLimits (ShortComplex C) 
goal: CategoryTheory.Limits.HasFiniteLimits (CategoryTheory.ShortComplex C)

full_name: CategoryTheory.ShortComplex.instPreservesFiniteLimitsπ₁
code: noncomputable instance : PreservesFiniteLimits (π₁ : _ ⥤ C) 
goal: CategoryTheory.Limits.PreservesFiniteLimits CategoryTheory.ShortComplex.π₁

full_name: CategoryTheory.ShortComplex.instPreservesFiniteLimitsπ₂
code: noncomputable instance : PreservesFiniteLimits (π₂ : _ ⥤ C) 
goal: CategoryTheory.Limits.PreservesFiniteLimits CategoryTheory.ShortComplex.π₂

full_name: CategoryTheory.ShortComplex.instPreservesFiniteLimitsπ₃
code: noncomputable instance : PreservesFiniteLimits (π₃ : _ ⥤ C) 
goal: CategoryTheory.Limits.PreservesFiniteLimits CategoryTheory.ShortComplex.π₃

full_name: CategoryTheory.ShortComplex.preservesMonomorphisms_π₁
code: instance preservesMonomorphisms_π₁ :
    Functor.PreservesMonomorphisms (π₁ : _ ⥤ C) 
goal: CategoryTheory.ShortComplex.π₁.PreservesMonomorphisms

full_name: CategoryTheory.ShortComplex.preservesMonomorphisms_π₂
code: instance preservesMonomorphisms_π₂ :
    Functor.PreservesMonomorphisms (π₂ : _ ⥤ C) 
goal: CategoryTheory.ShortComplex.π₂.PreservesMonomorphisms

full_name: CategoryTheory.ShortComplex.preservesMonomorphisms_π₃
code: instance preservesMonomorphisms_π₃ :
    Functor.PreservesMonomorphisms (π₃ : _ ⥤ C) 
goal: CategoryTheory.ShortComplex.π₃.PreservesMonomorphisms

full_name: CategoryTheory.ShortComplex.isColimitOfIsColimitπ
code: /-- If a cocone with values in `ShortComplex C` is such that it becomes colimit
when we apply the three projections `ShortComplex C ⥤ C`, then it is colimit. -/
def isColimitOfIsColimitπ (c : Cocone F)
    (h₁ : IsColimit (π₁.mapCocone c)) (h₂ : IsColimit (π₂.mapCocone c))
    (h₃ : IsColimit (π₃.mapCocone c)) : IsColimit c 
goal: CategoryTheory.Limits.IsColimit c

full_name: CategoryTheory.ShortComplex.colimitCocone
code: /-- Construction of a colimit cocone for a functor `J ⥤ ShortComplex C` using the colimits
of the three components `J ⥤ C`. -/
noncomputable def colimitCocone : Cocone F 
goal: CategoryTheory.Limits.Cocone F

full_name: CategoryTheory.ShortComplex.isColimitπ₁MapCoconeColimitCocone
code: /-- `colimitCocone F` becomes colimit after the application of `π₁ : ShortComplex C ⥤ C`. -/
noncomputable def isColimitπ₁MapCoconeColimitCocone :
    IsColimit (π₁.mapCocone (colimitCocone F)) 
goal: CategoryTheory.Limits.IsColimit (CategoryTheory.ShortComplex.π₁.mapCocone (CategoryTheory.ShortComplex.colimitCocone F))

full_name: CategoryTheory.ShortComplex.isColimitπ₂MapCoconeColimitCocone
code: /-- `colimitCocone F` becomes colimit after the application of `π₂ : ShortComplex C ⥤ C`. -/
noncomputable def isColimitπ₂MapCoconeColimitCocone :
    IsColimit (π₂.mapCocone (colimitCocone F)) 
goal: CategoryTheory.Limits.IsColimit (CategoryTheory.ShortComplex.π₂.mapCocone (CategoryTheory.ShortComplex.colimitCocone F))

full_name: CategoryTheory.ShortComplex.isColimitπ₃MapCoconeColimitCocone
code: /-- `colimitCocone F` becomes colimit after the application of `π₃ : ShortComplex C ⥤ C`. -/
noncomputable def isColimitπ₃MapCoconeColimitCocone :
    IsColimit (π₃.mapCocone (colimitCocone F)) 
goal: CategoryTheory.Limits.IsColimit (CategoryTheory.ShortComplex.π₃.mapCocone (CategoryTheory.ShortComplex.colimitCocone F))

full_name: CategoryTheory.ShortComplex.isColimitColimitCocone
code: /-- `colimitCocone F` is colimit. -/
noncomputable def isColimitColimitCocone : IsColimit (colimitCocone F) 
goal: CategoryTheory.Limits.IsColimit (CategoryTheory.ShortComplex.colimitCocone F)

full_name: CategoryTheory.ShortComplex.hasColimit_of_hasColimitπ
code: instance hasColimit_of_hasColimitπ : HasColimit F 
goal: CategoryTheory.Limits.HasColimit F

full_name: CategoryTheory.ShortComplex.instPreservesColimitπ₁
code: noncomputable instance : PreservesColimit F π₁ 
goal: CategoryTheory.Limits.PreservesColimit F CategoryTheory.ShortComplex.π₁

full_name: CategoryTheory.ShortComplex.instPreservesColimitπ₂
code: noncomputable instance : PreservesColimit F π₂ 
goal: CategoryTheory.Limits.PreservesColimit F CategoryTheory.ShortComplex.π₂

full_name: CategoryTheory.ShortComplex.instPreservesColimitπ₃
code: noncomputable instance : PreservesColimit F π₃ 
goal: CategoryTheory.Limits.PreservesColimit F CategoryTheory.ShortComplex.π₃

full_name: CategoryTheory.ShortComplex.hasColimitsOfShape
code: instance hasColimitsOfShape :
    HasColimitsOfShape J (ShortComplex C) 
goal: CategoryTheory.Limits.HasColimitsOfShape J (CategoryTheory.ShortComplex C)

full_name: CategoryTheory.ShortComplex.instPreservesColimitsOfShapeπ₁
code: noncomputable instance : PreservesColimitsOfShape J (π₁ : _ ⥤ C) 
goal: CategoryTheory.Limits.PreservesColimitsOfShape J CategoryTheory.ShortComplex.π₁

full_name: CategoryTheory.ShortComplex.instPreservesColimitsOfShapeπ₂
code: noncomputable instance : PreservesColimitsOfShape J (π₂ : _ ⥤ C) 
goal: CategoryTheory.Limits.PreservesColimitsOfShape J CategoryTheory.ShortComplex.π₂

full_name: CategoryTheory.ShortComplex.instPreservesColimitsOfShapeπ₃
code: noncomputable instance : PreservesColimitsOfShape J (π₃ : _ ⥤ C) 
goal: CategoryTheory.Limits.PreservesColimitsOfShape J CategoryTheory.ShortComplex.π₃

full_name: CategoryTheory.ShortComplex.hasFiniteColimits
code: instance hasFiniteColimits : HasFiniteColimits (ShortComplex C) 
goal: CategoryTheory.Limits.HasFiniteColimits (CategoryTheory.ShortComplex C)

full_name: CategoryTheory.ShortComplex.instPreservesFiniteColimitsπ₁
code: noncomputable instance : PreservesFiniteColimits (π₁ : _ ⥤ C) 
goal: CategoryTheory.Limits.PreservesFiniteColimits CategoryTheory.ShortComplex.π₁

full_name: CategoryTheory.ShortComplex.instPreservesFiniteColimitsπ₂
code: noncomputable instance : PreservesFiniteColimits (π₂ : _ ⥤ C) 
goal: CategoryTheory.Limits.PreservesFiniteColimits CategoryTheory.ShortComplex.π₂

full_name: CategoryTheory.ShortComplex.instPreservesFiniteColimitsπ₃
code: noncomputable instance : PreservesFiniteColimits (π₃ : _ ⥤ C) 
goal: CategoryTheory.Limits.PreservesFiniteColimits CategoryTheory.ShortComplex.π₃

full_name: CategoryTheory.ShortComplex.preservesEpimorphisms_π₁
code: instance preservesEpimorphisms_π₁ :
    Functor.PreservesEpimorphisms (π₁ : _ ⥤ C) 
goal: CategoryTheory.ShortComplex.π₁.PreservesEpimorphisms

full_name: CategoryTheory.ShortComplex.preservesEpimorphisms_π₂
code: instance preservesEpimorphisms_π₂ :
    Functor.PreservesEpimorphisms (π₂ : _ ⥤ C) 
goal: CategoryTheory.ShortComplex.π₂.PreservesEpimorphisms

full_name: CategoryTheory.ShortComplex.preservesEpimorphisms_π₃
code: instance preservesEpimorphisms_π₃ :
    Functor.PreservesEpimorphisms (π₃ : _ ⥤ C) 
goal: CategoryTheory.ShortComplex.π₃.PreservesEpimorphisms