Constructing colimits from finite colimits and filtered colimits

We construct colimits of size w from finite colimits and filtered colimits of size w. Since w-sized colimits are constructed from coequalizers and w-sized coproducts, it suffices to construct w-sized coproducts from finite coproducts and w-sized filtered colimits.

The idea is simple: to construct coproducts of shape α, we take the colimit of the filtered diagram of all coproducts of finite subsets of α.

We also deduce the dual statement by invoking the original statement in Cᵒᵖ.

@[simp]

theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset_obj {C : Type u} [CategoryTheory.Category.{v, u} C] {α : Type w} [CategoryTheory.Limits.HasFiniteCoproducts C] (F : CategoryTheory.Functor (CategoryTheory.Discrete α) C) (s : Finset (CategoryTheory.Discrete α)) : (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset F).obj s = ∐ fun x => F.obj ↑x

@[simp]

theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset_map {C : Type u} [CategoryTheory.Category.{v, u} C] {α : Type w} [CategoryTheory.Limits.HasFiniteCoproducts C] (F : CategoryTheory.Functor (CategoryTheory.Discrete α) C) : ∀ {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) { val := ↑y, property := (_ : ↑y ∈ Y) }

def CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset {C : Type u} [CategoryTheory.Category.{v, u} C] {α : Type w} [CategoryTheory.Limits.HasFiniteCoproducts C] (F : CategoryTheory.Functor (CategoryTheory.Discrete α) C) : CategoryTheory.Functor (Finset (CategoryTheory.Discrete α)) C

If C has finite coproducts, a functor Discrete α ⥤ C lifts to a functor Finset (Discrete α) ⥤ C by taking coproducts.

@[simp]

theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_cocone_ι_app {C : Type u} [CategoryTheory.Category.{v, u} C] {α : Type w} [CategoryTheory.Limits.HasFiniteCoproducts C] [CategoryTheory.Limits.HasFilteredColimitsOfSize.{w, w, v, u} C] (F : CategoryTheory.Functor (CategoryTheory.Discrete α) C) (j : CategoryTheory.Discrete α) : (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone F).cocone.ι.app j = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Sigma.ι (fun x => F.obj ↑x) { val := j, property := (_ : j ∈ {j}) }) (CategoryTheory.Limits.colimit.ι (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset F) {j})

@[simp]

theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_cocone_pt {C : Type u} [CategoryTheory.Category.{v, u} C] {α : Type w} [CategoryTheory.Limits.HasFiniteCoproducts C] [CategoryTheory.Limits.HasFilteredColimitsOfSize.{w, w, v, u} C] (F : CategoryTheory.Functor (CategoryTheory.Discrete α) C) : (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone F).cocone.pt = CategoryTheory.Limits.colimit (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset F)

@[simp]

theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_isColimit_desc {C : Type u} [CategoryTheory.Category.{v, u} C] {α : Type w} [CategoryTheory.Limits.HasFiniteCoproducts C] [CategoryTheory.Limits.HasFilteredColimitsOfSize.{w, w, v, u} C] (F : CategoryTheory.Functor (CategoryTheory.Discrete α) C) (s : CategoryTheory.Limits.Cocone F) : CategoryTheory.Limits.IsColimit.desc (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone F).isColimit s = CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset F) { pt := s.pt, ι := CategoryTheory.NatTrans.mk fun t => CategoryTheory.Limits.Sigma.desc fun x => s.ι.app ↑x }

def CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone {C : Type u} [CategoryTheory.Category.{v, u} C] {α : Type w} [CategoryTheory.Limits.HasFiniteCoproducts C] [CategoryTheory.Limits.HasFilteredColimitsOfSize.{w, w, v, u} C] (F : CategoryTheory.Functor (CategoryTheory.Discrete α) C) : CategoryTheory.Limits.ColimitCocone F

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.

theorem CategoryTheory.Limits.hasCoproducts_of_finite_and_filtered {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteCoproducts C] [CategoryTheory.Limits.HasFilteredColimitsOfSize.{w, w, v, u} C] : CategoryTheory.Limits.HasCoproducts C

theorem CategoryTheory.Limits.has_colimits_of_finite_and_filtered {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteColimits C] [CategoryTheory.Limits.HasFilteredColimitsOfSize.{w, w, v, u} C] : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v, u} C

theorem CategoryTheory.Limits.hasProducts_of_finite_and_cofiltered {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasCofilteredLimitsOfSize.{w, w, v, u} C] : CategoryTheory.Limits.HasProducts C

theorem CategoryTheory.Limits.has_limits_of_finite_and_cofiltered {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasFiniteLimits C] [CategoryTheory.Limits.HasCofilteredLimitsOfSize.{w, w, v, u} C] : CategoryTheory.Limits.HasLimitsOfSize.{w, w, v, u} C