File: Mathlib/Algebra/Category/ModuleCat/Presheaf/Colimits.lean

full_name: PresheafOfModules.evaluationJointlyReflectsColimits
code: /-- A cocone in the category `PresheafOfModules R` is colimit if it is so after the application
of the functors `evaluation R X` for all `X`. -/
def evaluationJointlyReflectsColimits (c : Cocone F)
    (hc : ∀ (X : Cᵒᵖ), IsColimit ((evaluation R X).mapCocone c)) : IsColimit c 
goal: CategoryTheory.Limits.IsColimit c

full_name: PresheafOfModules.instHasColimitModuleCatαRingObjOppositeRingCatCompEvaluationRestrictScalarsMap
code: instance {X Y : Cᵒᵖ} (f : X ⟶ Y) :
    HasColimit (F ⋙ evaluation R Y ⋙ (ModuleCat.restrictScalars (R.map f))) 
goal: CategoryTheory.Limits.HasColimit (F ⋙ PresheafOfModules.evaluation R Y ⋙ ModuleCat.restrictScalars (R.map f))

full_name: PresheafOfModules.colimitBundledCore_map
code: /-- Given `F : J ⥤ PresheafOfModules.{v} R`, this is the `BundledCorePresheafOfModules R` which
corresponds to the presheaf of modules which sends `X` to the colimit of `F ⋙ evaluation R X`. -/
@[simps]
noncomputable def colimitBundledCore : BundledCorePresheafOfModules R 
goal: (PresheafOfModules.colimitBundledCore F).map f =
  CategoryTheory.Limits.colimMap (CategoryTheory.whiskerLeft F (PresheafOfModules.restriction R f)) ≫
    (CategoryTheory.preservesColimitIso (ModuleCat.restrictScalars (R.map f))
        (F ⋙ PresheafOfModules.evaluation R Y)).inv

full_name: PresheafOfModules.colimitBundledCore_obj
code: /-- Given `F : J ⥤ PresheafOfModules.{v} R`, this is the `BundledCorePresheafOfModules R` which
corresponds to the presheaf of modules which sends `X` to the colimit of `F ⋙ evaluation R X`. -/
@[simps]
noncomputable def colimitBundledCore : BundledCorePresheafOfModules R 
goal: (PresheafOfModules.colimitBundledCore F).obj X = CategoryTheory.Limits.colimit (F ⋙ PresheafOfModules.evaluation R X)

full_name: PresheafOfModules.colimitBundledCore
code: /-- Given `F : J ⥤ PresheafOfModules.{v} R`, this is the `BundledCorePresheafOfModules R` which
corresponds to the presheaf of modules which sends `X` to the colimit of `F ⋙ evaluation R X`. -/
@[simps]
noncomputable def colimitBundledCore : BundledCorePresheafOfModules R 
goal: BundledCorePresheafOfModules R

full_name: PresheafOfModules.colimitCoconeιApp
code: /-- Given `F : J ⥤ PresheafOfModules.{v} R`, this is the canonical map
`F.obj j ⟶ (colimitBundledCore F).toPresheafOfModules` for all `j : J`. -/
noncomputable def colimitCoconeιApp (j : J) :
    F.obj j ⟶ (colimitBundledCore F).toPresheafOfModules 
goal: F.obj j ⟶ (PresheafOfModules.colimitBundledCore F).toPresheafOfModules

full_name: PresheafOfModules.colimitCoconeιApp_naturality
code: @[reassoc (attr := simp)]
lemma colimitCoconeιApp_naturality {i j : J} (f : i ⟶ j) :
    F.map f ≫ colimitCoconeιApp F j = colimitCoconeιApp F i 
goal: F.map f ≫ PresheafOfModules.colimitCoconeιApp F j = PresheafOfModules.colimitCoconeιApp F i

full_name: PresheafOfModules.colimitCocone
code: /-- The (colimit) cocone for `F : J ⥤ PresheafOfModules.{v} R` that is constructed from
the colimit of `F ⋙ evaluation R X` for all `X`. -/
@[simps]
noncomputable def colimitCocone : Cocone F 
goal: CategoryTheory.Limits.Cocone F

full_name: PresheafOfModules.colimitCocone_pt
code: /-- The (colimit) cocone for `F : J ⥤ PresheafOfModules.{v} R` that is constructed from
the colimit of `F ⋙ evaluation R X` for all `X`. -/
@[simps]
noncomputable def colimitCocone : Cocone F 
goal: (PresheafOfModules.colimitCocone F).pt = (PresheafOfModules.colimitBundledCore F).toPresheafOfModules

full_name: PresheafOfModules.colimitCocone_ι_app
code: /-- The (colimit) cocone for `F : J ⥤ PresheafOfModules.{v} R` that is constructed from
the colimit of `F ⋙ evaluation R X` for all `X`. -/
@[simps]
noncomputable def colimitCocone : Cocone F 
goal: (PresheafOfModules.colimitCocone F).ι.app j = PresheafOfModules.colimitCoconeιApp F j

full_name: PresheafOfModules.isColimitColimitCocone
code: /-- The cocone `colimitCocone F` is colimit for any `F : J ⥤ PresheafOfModules.{v} R`. -/
noncomputable def isColimitColimitCocone : IsColimit (colimitCocone F) 
goal: CategoryTheory.Limits.IsColimit (PresheafOfModules.colimitCocone F)

full_name: PresheafOfModules.hasColimit
code: instance hasColimit : HasColimit F 
goal: CategoryTheory.Limits.HasColimit F

full_name: PresheafOfModules.evaluationPreservesColimit
code: noncomputable instance evaluationPreservesColimit (X : Cᵒᵖ) :
    PreservesColimit F (evaluation R X) 
goal: CategoryTheory.Limits.PreservesColimit F (PresheafOfModules.evaluation R X)

full_name: PresheafOfModules.toPresheafPreservesColimit
code: noncomputable instance toPresheafPreservesColimit :
    PreservesColimit F (toPresheaf R) 
goal: CategoryTheory.Limits.PreservesColimit F (PresheafOfModules.toPresheaf R)

full_name: PresheafOfModules.hasColimitsOfShape
code: instance hasColimitsOfShape : HasColimitsOfShape J (PresheafOfModules.{v} R) 
goal: CategoryTheory.Limits.HasColimitsOfShape J (PresheafOfModules R)

full_name: PresheafOfModules.evaluationPreservesColimitsOfShape
code: noncomputable instance evaluationPreservesColimitsOfShape (X : Cᵒᵖ) :
    PreservesColimitsOfShape J (evaluation R X : PresheafOfModules.{v} R ⥤ _) 
goal: CategoryTheory.Limits.PreservesColimitsOfShape J (PresheafOfModules.evaluation R X)

full_name: PresheafOfModules.toPresheafPreservesColimitsOfShape
code: noncomputable instance toPresheafPreservesColimitsOfShape :
    PreservesColimitsOfShape J (toPresheaf.{v} R) 
goal: CategoryTheory.Limits.PreservesColimitsOfShape J (PresheafOfModules.toPresheaf R)

full_name: PresheafOfModules.Finite.hasFiniteColimits
code: instance hasFiniteColimits : HasFiniteColimits (PresheafOfModules.{v} R) 
goal: CategoryTheory.Limits.HasFiniteColimits (PresheafOfModules R)

full_name: PresheafOfModules.Finite.evaluationPreservesFiniteColimits
code: noncomputable instance evaluationPreservesFiniteColimits (X : Cᵒᵖ) :
    PreservesFiniteColimits (evaluation.{v} R X) 
goal: CategoryTheory.Limits.PreservesFiniteColimits (PresheafOfModules.evaluation R X)

full_name: PresheafOfModules.Finite.toPresheafPreservesFiniteColimits
code: noncomputable instance toPresheafPreservesFiniteColimits :
    PreservesFiniteColimits (toPresheaf R) 
goal: CategoryTheory.Limits.PreservesFiniteColimits (PresheafOfModules.toPresheaf R)