File: Mathlib/CategoryTheory/Limits/Filtered.lean

full_name: CategoryTheory.IsFiltered.iff_nonempty_limit
code: /-- `C` is filtered if and only if for every functor `F : J ⥤ C` from a finite category there is
    some `X : C` such that `lim Hom(F·, X)` is nonempty.

    Lemma 3.1.2 of [Kashiwara2006] -/
theorem IsFiltered.iff_nonempty_limit : IsFiltered C ↔
    ∀ {J : Type v} [SmallCategory J] [FinCategory J] (F : J ⥤ C),
      ∃ (X : C), Nonempty (limit (F.op ⋙ yoneda.obj X)) 
goal: CategoryTheory.IsFiltered C ↔
  ∀ {J : Type v} [inst : CategoryTheory.SmallCategory J] [inst_1 : CategoryTheory.FinCategory J] (F : J ⥤ C),
    ∃ X, Nonempty (CategoryTheory.Limits.limit (F.op ⋙ CategoryTheory.yoneda.obj X))

full_name: CategoryTheory.IsCofiltered.iff_nonempty_limit
code: /-- `C` is cofiltered if and only if for every functor `F : J ⥤ C` from a finite category there is
    some `X : C` such that `lim Hom(X, F·)` is nonempty. -/
theorem IsCofiltered.iff_nonempty_limit : IsCofiltered C ↔
    ∀ {J : Type v} [SmallCategory J] [FinCategory J] (F : J ⥤ C),
      ∃ (X : C), Nonempty (limit (F ⋙ coyoneda.obj (op X))) 
goal: CategoryTheory.IsCofiltered C ↔
  ∀ {J : Type v} [inst : CategoryTheory.SmallCategory J] [inst_1 : CategoryTheory.FinCategory J] (F : J ⥤ C),
    ∃ X, Nonempty (CategoryTheory.Limits.limit (F ⋙ CategoryTheory.coyoneda.obj (Opposite.op X)))

full_name: CategoryTheory.Limits.HasCofilteredLimitsOfSize
code: /-- Class for having all cofiltered limits of a given size. -/
@[pp_with_univ]
class HasCofilteredLimitsOfSize : Prop where
  /-- For all filtered types of size `w`, we have limits -/
  HasLimitsOfShape : ∀ (I : Type w) [Category.{w'} I] [IsCofiltered I], HasLimitsOfShape I C
goal: Prop

full_name: CategoryTheory.Limits.HasCofilteredLimitsOfSize.HasLimitsOfShape
code: /-- Class for having all cofiltered limits of a given size. -/
@[pp_with_univ]
class HasCofilteredLimitsOfSize : Prop where
  /-- For all filtered types of size `w`, we have limits -/
  HasLimitsOfShape : ∀ (I : Type w) [Category.{w'} I] [IsCofiltered I], HasLimitsOfShape I C
goal: CategoryTheory.Limits.HasLimitsOfShape I C

full_name: CategoryTheory.Limits.HasFilteredColimitsOfSize
code: /-- Class for having all filtered colimits of a given size. -/
@[pp_with_univ]
class HasFilteredColimitsOfSize : Prop where
  /-- For all filtered types of a size `w`, we have colimits -/
  HasColimitsOfShape : ∀ (I : Type w) [Category.{w'} I] [IsFiltered I], HasColimitsOfShape I C
goal: Prop

full_name: CategoryTheory.Limits.HasFilteredColimitsOfSize.HasColimitsOfShape
code: /-- Class for having all filtered colimits of a given size. -/
@[pp_with_univ]
class HasFilteredColimitsOfSize : Prop where
  /-- For all filtered types of a size `w`, we have colimits -/
  HasColimitsOfShape : ∀ (I : Type w) [Category.{w'} I] [IsFiltered I], HasColimitsOfShape I C
goal: CategoryTheory.Limits.HasColimitsOfShape I C

full_name: CategoryTheory.Limits.hasLimitsOfShape_of_has_cofiltered_limits
code: instance (priority := 100) hasLimitsOfShape_of_has_cofiltered_limits
    [HasCofilteredLimitsOfSize.{w', w} C] (I : Type w) [Category.{w'} I] [IsCofiltered I] :
    HasLimitsOfShape I C 
goal: CategoryTheory.Limits.HasLimitsOfShape I C

full_name: CategoryTheory.Limits.hasColimitsOfShape_of_has_filtered_colimits
code: instance (priority := 100) hasColimitsOfShape_of_has_filtered_colimits
    [HasFilteredColimitsOfSize.{w', w} C] (I : Type w) [Category.{w'} I] [IsFiltered I] :
    HasColimitsOfShape I C 
goal: CategoryTheory.Limits.HasColimitsOfShape I C