File: Mathlib/CategoryTheory/Limits/Constructions/Filtered.lean

full_name: CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset
code: /-- If `C` has finite coproducts, a functor `Discrete α ⥤ C` lifts to a functor
    `Finset (Discrete α) ⥤ C` by taking coproducts. -/
@[simps!]
def liftToFinset [HasFiniteCoproducts C] (F : Discrete α ⥤ C) : Finset (Discrete α) ⥤ C 
goal: Finset (CategoryTheory.Discrete α) ⥤ C

full_name: CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset_map
code: /-- If `C` has finite coproducts, a functor `Discrete α ⥤ C` lifts to a functor
    `Finset (Discrete α) ⥤ C` by taking coproducts. -/
@[simps!]
def liftToFinset [HasFiniteCoproducts C] (F : Discrete α ⥤ C) : Finset (Discrete α) ⥤ C 
goal: ∀ {x Y : Finset (CategoryTheory.Discrete α)} (h : x ⟶ Y),
  (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset F).map h =
    CategoryTheory.Limits.Sigma.desc fun y => CategoryTheory.Limits.Sigma.ι (fun x => F.obj ↑x) ⟨↑y, ⋯⟩

full_name: CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset_obj
code: /-- If `C` has finite coproducts, a functor `Discrete α ⥤ C` lifts to a functor
    `Finset (Discrete α) ⥤ C` by taking coproducts. -/
@[simps!]
def liftToFinset [HasFiniteCoproducts C] (F : Discrete α ⥤ C) : Finset (Discrete α) ⥤ C 
goal: (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset F).obj s = ∐ fun x => F.obj ↑x

full_name: CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_cocone_ι_app
code: /-- If `C` has finite coproducts and filtered colimits, we can construct arbitrary coproducts by
    taking the colimit of the diagram formed by the coproducts of finite sets over the indexing
    type. -/
@[simps!]
def liftToFinsetColimitCocone [HasFiniteCoproducts C] [HasFilteredColimitsOfSize.{w, w} C]
    (F : Discrete α ⥤ C) : ColimitCocone F 
goal: (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone F).cocone.ι.app j =
  CategoryTheory.Limits.Sigma.ι (fun x => F.obj ↑x) ⟨j, ⋯⟩ ≫
    CategoryTheory.Limits.colimit.ι (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset F) {j}

full_name: CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_cocone_pt
code: /-- If `C` has finite coproducts and filtered colimits, we can construct arbitrary coproducts by
    taking the colimit of the diagram formed by the coproducts of finite sets over the indexing
    type. -/
@[simps!]
def liftToFinsetColimitCocone [HasFiniteCoproducts C] [HasFilteredColimitsOfSize.{w, w} C]
    (F : Discrete α ⥤ C) : ColimitCocone F 
goal: (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone F).cocone.pt =
  CategoryTheory.Limits.colimit (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset F)

full_name: CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_isColimit_desc
code: /-- If `C` has finite coproducts and filtered colimits, we can construct arbitrary coproducts by
    taking the colimit of the diagram formed by the coproducts of finite sets over the indexing
    type. -/
@[simps!]
def liftToFinsetColimitCocone [HasFiniteCoproducts C] [HasFilteredColimitsOfSize.{w, w} C]
    (F : Discrete α ⥤ C) : ColimitCocone F 
goal: (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone F).isColimit.desc s =
  CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset F)
    { pt := s.pt, ι := { app := fun t => CategoryTheory.Limits.Sigma.desc fun x => s.ι.app ↑x, naturality := ⋯ } }

full_name: CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone
code: /-- If `C` has finite coproducts and filtered colimits, we can construct arbitrary coproducts by
    taking the colimit of the diagram formed by the coproducts of finite sets over the indexing
    type. -/
@[simps!]
def liftToFinsetColimitCocone [HasFiniteCoproducts C] [HasFilteredColimitsOfSize.{w, w} C]
    (F : Discrete α ⥤ C) : ColimitCocone F 
goal: CategoryTheory.Limits.ColimitCocone F

full_name: CategoryTheory.Limits.hasCoproducts_of_finite_and_filtered
code: theorem hasCoproducts_of_finite_and_filtered [HasFiniteCoproducts C]
    [HasFilteredColimitsOfSize.{w, w} C] : HasCoproducts.{w} C 
goal: CategoryTheory.Limits.HasCoproducts C

full_name: CategoryTheory.Limits.has_colimits_of_finite_and_filtered
code: theorem has_colimits_of_finite_and_filtered [HasFiniteColimits C]
    [HasFilteredColimitsOfSize.{w, w} C] : HasColimitsOfSize.{w, w} C 
goal: CategoryTheory.Limits.HasColimitsOfSize.{w, w, v, u} C

full_name: CategoryTheory.Limits.hasProducts_of_finite_and_cofiltered
code: theorem hasProducts_of_finite_and_cofiltered [HasFiniteProducts C]
    [HasCofilteredLimitsOfSize.{w, w} C] : HasProducts.{w} C 
goal: CategoryTheory.Limits.HasProducts C

full_name: CategoryTheory.Limits.has_limits_of_finite_and_cofiltered
code: theorem has_limits_of_finite_and_cofiltered [HasFiniteLimits C]
    [HasCofilteredLimitsOfSize.{w, w} C] : HasLimitsOfSize.{w, w} C 
goal: CategoryTheory.Limits.HasLimitsOfSize.{w, w, v, u} C