File: Mathlib/CategoryTheory/Limits/Elements.lean

full_name: CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedConeElement'
code: /-- (implementation) A system `(Fi, fi)_i` of elements induces an element in `lim_i A(Fi)`. -/
noncomputable def liftedConeElement' : limit ((F ⋙ π A) ⋙ A) 
goal: CategoryTheory.Limits.limit ((F.comp (CategoryTheory.CategoryOfElements.π A)).comp A)

full_name: CategoryTheory.CategoryOfElements.CreatesLimitsAux.π_liftedConeElement'
code: @[simp]
lemma π_liftedConeElement' (i : I) :
    limit.π ((F ⋙ π A) ⋙ A) i (liftedConeElement' F) = (F.obj i).2 
goal: CategoryTheory.Limits.limit.π ((F.comp (CategoryTheory.CategoryOfElements.π A)).comp A) i
    (CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedConeElement' F) =
  (F.obj i).snd

full_name: CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedConeElement
code: /-- (implementation) A system `(Fi, fi)_i` of elements induces an element in `A(lim_i Fi)`. -/
noncomputable def liftedConeElement : A.obj (limit (F ⋙ π A)) 
goal: A.obj (CategoryTheory.Limits.limit (F.comp (CategoryTheory.CategoryOfElements.π A)))

full_name: CategoryTheory.CategoryOfElements.CreatesLimitsAux.map_lift_mapCone
code: @[simp]
lemma map_lift_mapCone (c : Cone F) :
    A.map (limit.lift (F ⋙ π A) ((π A).mapCone c)) c.pt.snd = liftedConeElement F 
goal: A.map
    (CategoryTheory.Limits.limit.lift (F.comp (CategoryTheory.CategoryOfElements.π A))
      ((CategoryTheory.CategoryOfElements.π A).mapCone c))
    c.pt.snd =
  CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedConeElement F

full_name: CategoryTheory.CategoryOfElements.CreatesLimitsAux.map_π_liftedConeElement
code: @[simp]
lemma map_π_liftedConeElement (i : I) :
    A.map (limit.π (F ⋙ π A) i) (liftedConeElement F) = (F.obj i).snd 
goal: A.map (CategoryTheory.Limits.limit.π (F.comp (CategoryTheory.CategoryOfElements.π A)) i)
    (CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedConeElement F) =
  (F.obj i).snd

full_name: CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedCone_π_app_coe
code: /-- (implementation) The constructured limit cone. -/
@[simps]
noncomputable def liftedCone : Cone F 
goal: ↑((CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedCone F).π.app i) =
  CategoryTheory.Limits.limit.π (F.comp (CategoryTheory.CategoryOfElements.π A)) i

full_name: CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedCone_pt_fst
code: /-- (implementation) The constructured limit cone. -/
@[simps]
noncomputable def liftedCone : Cone F 
goal: (CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedCone F).pt.fst =
  CategoryTheory.Limits.limit (F.comp (CategoryTheory.CategoryOfElements.π A))

full_name: CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedCone
code: /-- (implementation) The constructured limit cone. -/
@[simps]
noncomputable def liftedCone : Cone F 
goal: CategoryTheory.Limits.Cone F

full_name: CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedCone_pt_snd
code: /-- (implementation) The constructured limit cone. -/
@[simps]
noncomputable def liftedCone : Cone F 
goal: (CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedCone F).pt.snd =
  CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedConeElement F

full_name: CategoryTheory.CategoryOfElements.CreatesLimitsAux.isValidLift
code: /-- (implementation) The constructed limit cone is a lift of the limit cone in `C`. -/
noncomputable def isValidLift : (π A).mapCone (liftedCone F) ≅ limit.cone (F ⋙ π A) 
goal: (CategoryTheory.CategoryOfElements.π A).mapCone (CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedCone F) ≅
  CategoryTheory.Limits.limit.cone (F.comp (CategoryTheory.CategoryOfElements.π A))

full_name: CategoryTheory.CategoryOfElements.CreatesLimitsAux.isLimit
code: /-- (implementation) The constuctured limit cone is a limit cone. -/
noncomputable def isLimit : IsLimit (liftedCone F) 
goal: CategoryTheory.Limits.IsLimit (CategoryTheory.CategoryOfElements.CreatesLimitsAux.liftedCone F)

full_name: CategoryTheory.CategoryOfElements.instCreatesLimitElementsπ
code: noncomputable instance (F : I ⥤ A.Elements) : CreatesLimit F (π A) 
goal: CategoryTheory.CreatesLimit F (CategoryTheory.CategoryOfElements.π A)

full_name: CategoryTheory.CategoryOfElements.instCreatesLimitsOfShapeElementsπ
code: noncomputable instance : CreatesLimitsOfShape I (π A) 
goal: CategoryTheory.CreatesLimitsOfShape I (CategoryTheory.CategoryOfElements.π A)

full_name: CategoryTheory.CategoryOfElements.instHasLimitsOfShapeElements
code: instance : HasLimitsOfShape I A.Elements 
goal: CategoryTheory.Limits.HasLimitsOfShape I A.Elements