File: Mathlib/Algebra/Homology/HomologicalComplexLimits.lean

full_name: HomologicalComplex.isLimitOfEval
code: /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
    (hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s 
goal: CategoryTheory.Limits.IsLimit s

full_name: HomologicalComplex.coneOfHasLimitEval_pt_d
code: /-- A cone for a functor `F : J ⥤ HomologicalComplex C c` which is given in degree `n` by
the limit `F ⋙ eval C c n`. -/
@[simps]
noncomputable def coneOfHasLimitEval : Cone F 
goal: (HomologicalComplex.coneOfHasLimitEval F).pt.d n m =
  CategoryTheory.Limits.limMap { app := fun j => (F.obj j).d n m, naturality := ⋯ }

full_name: HomologicalComplex.coneOfHasLimitEval
code: /-- A cone for a functor `F : J ⥤ HomologicalComplex C c` which is given in degree `n` by
the limit `F ⋙ eval C c n`. -/
@[simps]
noncomputable def coneOfHasLimitEval : Cone F 
goal: CategoryTheory.Limits.Cone F

full_name: HomologicalComplex.coneOfHasLimitEval_pt_X
code: /-- A cone for a functor `F : J ⥤ HomologicalComplex C c` which is given in degree `n` by
the limit `F ⋙ eval C c n`. -/
@[simps]
noncomputable def coneOfHasLimitEval : Cone F 
goal: (HomologicalComplex.coneOfHasLimitEval F).pt.X n = CategoryTheory.Limits.limit (F ⋙ HomologicalComplex.eval C c n)

full_name: HomologicalComplex.coneOfHasLimitEval_π_app_f
code: /-- A cone for a functor `F : J ⥤ HomologicalComplex C c` which is given in degree `n` by
the limit `F ⋙ eval C c n`. -/
@[simps]
noncomputable def coneOfHasLimitEval : Cone F 
goal: ((HomologicalComplex.coneOfHasLimitEval F).π.app j).f n =
  CategoryTheory.Limits.limit.π (F ⋙ HomologicalComplex.eval C c n) j

full_name: HomologicalComplex.isLimitConeOfHasLimitEval
code: /-- The cone `coneOfHasLimitEval F` is limit. -/
noncomputable def isLimitConeOfHasLimitEval : IsLimit (coneOfHasLimitEval F) 
goal: CategoryTheory.Limits.IsLimit (HomologicalComplex.coneOfHasLimitEval F)

full_name: HomologicalComplex.instHasLimit
code: instance : HasLimit F 
goal: CategoryTheory.Limits.HasLimit F

full_name: HomologicalComplex.instPreservesLimitEval
code: noncomputable instance (n : ι) : PreservesLimit F (eval C c n) 
goal: CategoryTheory.Limits.PreservesLimit F (HomologicalComplex.eval C c n)

full_name: HomologicalComplex.instHasLimitsOfShape
code: instance [HasLimitsOfShape J C] : HasLimitsOfShape J (HomologicalComplex C c) 
goal: CategoryTheory.Limits.HasLimitsOfShape J (HomologicalComplex C c)

full_name: HomologicalComplex.instPreservesLimitsOfShapeEvalOfHasLimitsOfShape
code: noncomputable instance [HasLimitsOfShape J C] (n : ι) :
  PreservesLimitsOfShape J (eval C c n) 
goal: CategoryTheory.Limits.PreservesLimitsOfShape J (HomologicalComplex.eval C c n)

full_name: HomologicalComplex.instHasFiniteLimits
code: instance [HasFiniteLimits C] : HasFiniteLimits (HomologicalComplex C c) 
goal: CategoryTheory.Limits.HasFiniteLimits (HomologicalComplex C c)

full_name: HomologicalComplex.instPreservesFiniteLimitsEvalOfHasFiniteLimits
code: noncomputable instance [HasFiniteLimits C] (n : ι) :
  PreservesFiniteLimits (eval C c n) 
goal: CategoryTheory.Limits.PreservesFiniteLimits (HomologicalComplex.eval C c n)

full_name: HomologicalComplex.instMonoFOfHasFiniteLimits
code: instance [HasFiniteLimits C] {K L : HomologicalComplex C c} (φ : K ⟶ L) [Mono φ] (n : ι) :
    Mono (φ.f n) 
goal: CategoryTheory.Mono (φ.f n)

full_name: HomologicalComplex.isColimitOfEval
code: /-- A cocone in `HomologicalComplex C c` is colimit if the induced cocones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are colimit. -/
def isColimitOfEval (s : Cocone F)
    (hs : ∀ (i : ι), IsColimit ((eval C c i).mapCocone s)) : IsColimit s 
goal: CategoryTheory.Limits.IsColimit s

full_name: HomologicalComplex.coconeOfHasColimitEval_ι_app_f
code: /-- A cocone for a functor `F : J ⥤ HomologicalComplex C c` which is given in degree `n` by
the colimit of `F ⋙ eval C c n`. -/
@[simps]
noncomputable def coconeOfHasColimitEval : Cocone F 
goal: ((HomologicalComplex.coconeOfHasColimitEval F).ι.app j).f n =
  CategoryTheory.Limits.colimit.ι (F ⋙ HomologicalComplex.eval C c n) j

full_name: HomologicalComplex.coconeOfHasColimitEval_pt_d
code: /-- A cocone for a functor `F : J ⥤ HomologicalComplex C c` which is given in degree `n` by
the colimit of `F ⋙ eval C c n`. -/
@[simps]
noncomputable def coconeOfHasColimitEval : Cocone F 
goal: (HomologicalComplex.coconeOfHasColimitEval F).pt.d n m =
  CategoryTheory.Limits.colimMap { app := fun j => (F.obj j).d n m, naturality := ⋯ }

full_name: HomologicalComplex.coconeOfHasColimitEval
code: /-- A cocone for a functor `F : J ⥤ HomologicalComplex C c` which is given in degree `n` by
the colimit of `F ⋙ eval C c n`. -/
@[simps]
noncomputable def coconeOfHasColimitEval : Cocone F 
goal: CategoryTheory.Limits.Cocone F

full_name: HomologicalComplex.coconeOfHasColimitEval_pt_X
code: /-- A cocone for a functor `F : J ⥤ HomologicalComplex C c` which is given in degree `n` by
the colimit of `F ⋙ eval C c n`. -/
@[simps]
noncomputable def coconeOfHasColimitEval : Cocone F 
goal: (HomologicalComplex.coconeOfHasColimitEval F).pt.X n = CategoryTheory.Limits.colimit (F ⋙ HomologicalComplex.eval C c n)

full_name: HomologicalComplex.isColimitCoconeOfHasColimitEval
code: /-- The cocone `coconeOfHasLimitEval F` is colimit. -/
noncomputable def isColimitCoconeOfHasColimitEval : IsColimit (coconeOfHasColimitEval F) 
goal: CategoryTheory.Limits.IsColimit (HomologicalComplex.coconeOfHasColimitEval F)

full_name: HomologicalComplex.instHasColimit
code: instance : HasColimit F 
goal: CategoryTheory.Limits.HasColimit F

full_name: HomologicalComplex.instPreservesColimitEval
code: noncomputable instance (n : ι) : PreservesColimit F (eval C c n) 
goal: CategoryTheory.Limits.PreservesColimit F (HomologicalComplex.eval C c n)

full_name: HomologicalComplex.instHasColimitsOfShape
code: instance [HasColimitsOfShape J C] : HasColimitsOfShape J (HomologicalComplex C c) 
goal: CategoryTheory.Limits.HasColimitsOfShape J (HomologicalComplex C c)

full_name: HomologicalComplex.instPreservesColimitsOfShapeEvalOfHasColimitsOfShape
code: noncomputable instance [HasColimitsOfShape J C] (n : ι) :
  PreservesColimitsOfShape J (eval C c n) 
goal: CategoryTheory.Limits.PreservesColimitsOfShape J (HomologicalComplex.eval C c n)

full_name: HomologicalComplex.instHasFiniteColimits
code: instance [HasFiniteColimits C] : HasFiniteColimits (HomologicalComplex C c) 
goal: CategoryTheory.Limits.HasFiniteColimits (HomologicalComplex C c)

full_name: HomologicalComplex.instPreservesFiniteColimitsEvalOfHasFiniteColimits
code: noncomputable instance [HasFiniteColimits C] (n : ι) :
  PreservesFiniteColimits (eval C c n) 
goal: CategoryTheory.Limits.PreservesFiniteColimits (HomologicalComplex.eval C c n)

full_name: HomologicalComplex.instEpiFOfHasFiniteColimits
code: instance [HasFiniteColimits C] {K L : HomologicalComplex C c} (φ : K ⟶ L) [Epi φ] (n : ι) :
    Epi (φ.f n) 
goal: CategoryTheory.Epi (φ.f n)

full_name: HomologicalComplex.preservesLimitsOfShapeOfEval
code: /-- A functor `D ⥤ HomologicalComplex C c` preserves limits of shape `J`
if for any `i`, `G ⋙ eval C c i` does. -/
def preservesLimitsOfShapeOfEval {D : Type*} [Category D]
    (G : D ⥤ HomologicalComplex C c)
    (_ : ∀ (i : ι), PreservesLimitsOfShape J (G ⋙ eval C c i)) :
    PreservesLimitsOfShape J G 
goal: ((i : ι) → CategoryTheory.Limits.PreservesLimitsOfShape J (G ⋙ HomologicalComplex.eval C c i)) →
  CategoryTheory.Limits.PreservesLimitsOfShape J G

full_name: HomologicalComplex.preservesColimitsOfShapeOfEval
code: /-- A functor `D ⥤ HomologicalComplex C c` preserves colimits of shape `J`
if for any `i`, `G ⋙ eval C c i` does. -/
def preservesColimitsOfShapeOfEval {D : Type*} [Category D]
    (G : D ⥤ HomologicalComplex C c)
    (_ : ∀ (i : ι), PreservesColimitsOfShape J (G ⋙ eval C c i)) :
    PreservesColimitsOfShape J G 
goal: ((i : ι) → CategoryTheory.Limits.PreservesColimitsOfShape J (G ⋙ HomologicalComplex.eval C c i)) →
  CategoryTheory.Limits.PreservesColimitsOfShape J G

full_name: HomologicalComplex.instPreservesLimitsOfShapeSingle
code: noncomputable instance : PreservesLimitsOfShape J (single C c i) 
goal: CategoryTheory.Limits.PreservesLimitsOfShape J (HomologicalComplex.single C c i)

full_name: HomologicalComplex.instPreservesColimitsOfShapeSingle
code: noncomputable instance : PreservesColimitsOfShape J (single C c i) 
goal: CategoryTheory.Limits.PreservesColimitsOfShape J (HomologicalComplex.single C c i)

full_name: HomologicalComplex.instPreservesFiniteLimitsSingle
code: noncomputable instance : PreservesFiniteLimits (single C c i) 
goal: CategoryTheory.Limits.PreservesFiniteLimits (HomologicalComplex.single C c i)

full_name: HomologicalComplex.instPreservesFiniteColimitsSingle
code: noncomputable instance : PreservesFiniteColimits (single C c i) 
goal: CategoryTheory.Limits.PreservesFiniteColimits (HomologicalComplex.single C c i)