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.

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

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

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theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetObj_map {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteCoproducts C] (F : Functor (Discrete α) C) {x✝ Y : Finset (Discrete α)} (h : x✝ ⟶ Y) : (liftToFinsetObj F).map h = Sigma.desc fun (y : { x : Discrete α // x ∈ x✝ }) => Sigma.ι (fun (x : { x : Discrete α // x ∈ Y }) => F.obj ↑x) ⟨↑y, ⋯⟩

@[simp]

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

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

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theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_cocone_ι_app {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteCoproducts C] [HasColimitsOfShape (Finset (Discrete α)) C] (F : Functor (Discrete α) C) (j : Discrete α) : (liftToFinsetColimitCocone F).cocone.ι.app j = CategoryStruct.comp (Sigma.ι (fun (x : { x : Discrete α // x ∈ {j} }) => F.obj ↑x) ⟨j, ⋯⟩) (colimit.ι (liftToFinsetObj F) {j})

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

@[simp]

theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_isColimit_desc {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteCoproducts C] [HasColimitsOfShape (Finset (Discrete α)) C] (F : Functor (Discrete α) C) (s : Cocone F) : (liftToFinsetColimitCocone F).isColimit.desc s = colimit.desc (liftToFinsetObj F) { pt := s.pt, ι := { app := fun (x : Finset (Discrete α)) => Sigma.desc fun (x_1 : { x_1 : Discrete α // x_1 ∈ x }) => s.ι.app ↑x_1, naturality := ⋯ } }

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

The functor taking a functor Discrete α ⥤ C to a functor Finset (Discrete α) ⥤ C by taking coproducts.

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

@[simp]

theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset_obj_map (C : Type u) [Category.{v, u} C] (α : Type w) [HasFiniteCoproducts C] (F : Functor (Discrete α) C) {x✝ Y : Finset (Discrete α)} (h : x✝ ⟶ Y) : ((liftToFinset C α).obj F).map h = Sigma.desc fun (y : { x : Discrete α // x ∈ x✝ }) => Sigma.ι (fun (x : { x : Discrete α // x ∈ Y }) => F.obj ↑x) ⟨↑y, ⋯⟩

@[simp]

theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset_map_app (C : Type u) [Category.{v, u} C] (α : Type w) [HasFiniteCoproducts C] {X✝ Y✝ : Functor (Discrete α) C} (β : X✝ ⟶ Y✝) (x✝ : Finset (Discrete α)) : ((liftToFinset C α).map β).app x✝ = Sigma.map fun (x : { x : Discrete α // x ∈ x✝ }) => β.app ↑x

def CategoryTheory.Limits.CoproductsFromFiniteFiltered.finiteSubcoproductsCocone {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteCoproducts C] (f : α → C) [HasCoproduct f] : Cocone (liftToFinsetObj (Discrete.functor f))

The converse of the construction in liftToFinsetColimitCocone: we can form a cocone on the coproduct of f whose legs are the coproducts over the finite subsets of α.

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theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.finiteSubcoproductsCocone_pt {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteCoproducts C] (f : α → C) [HasCoproduct f] : (finiteSubcoproductsCocone f).pt = ∐ f

@[simp]

theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.finiteSubcoproductsCocone_ι_app {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteCoproducts C] (f : α → C) [HasCoproduct f] (S : Finset (Discrete α)) : (finiteSubcoproductsCocone f).ι.app S = Sigma.desc fun (s : { x : Discrete α // x ∈ S }) => Sigma.ι f (↑s).as

def CategoryTheory.Limits.CoproductsFromFiniteFiltered.isColimitFiniteSubproductsCocone {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteCoproducts C] (f : α → C) [HasColimitsOfShape (Finset (Discrete α)) C] [HasCoproduct f] : IsColimit (finiteSubcoproductsCocone f)

The cocone finiteSubcoproductsCocone is a colimit cocone.

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

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

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

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

theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimIso_aux {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteCoproducts C] [HasColimitsOfShape (Finset (Discrete α)) C] [HasColimitsOfShape (Discrete α) C] (F : Functor (Discrete α) C) {J : Finset (Discrete α)} (j : { x : Discrete α // x ∈ J }) : CategoryStruct.comp (Sigma.ι (fun (x : { x : Discrete α // x ∈ J }) => F.obj ↑x) j) (CategoryStruct.comp (colimit.ι (liftToFinsetObj F) J) (colimit.isoColimitCocone (liftToFinsetColimitCocone F)).inv) = colimit.ι F ↑j

Helper construction for liftToFinsetColimIso.

theorem CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimIso_aux_assoc {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteCoproducts C] [HasColimitsOfShape (Finset (Discrete α)) C] [HasColimitsOfShape (Discrete α) C] (F : Functor (Discrete α) C) {J : Finset (Discrete α)} (j : { x : Discrete α // x ∈ J }) {Z : C} (h : colimit F ⟶ Z) : CategoryStruct.comp (Sigma.ι (fun (x : { x : Discrete α // x ∈ J }) => F.obj ↑x) j) (CategoryStruct.comp (colimit.ι (liftToFinsetObj F) J) (CategoryStruct.comp (colimit.isoColimitCocone (liftToFinsetColimitCocone F)).inv h)) = CategoryStruct.comp (colimit.ι F ↑j) h

def CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimIso {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteCoproducts C] [HasColimitsOfShape (Finset (Discrete α)) C] [HasColimitsOfShape (Discrete α) C] : (liftToFinset C α).comp colim ≅ colim

The liftToFinset functor, precomposed with forming a colimit, is a coproduct on the original functor.

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def CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetEvaluationIso {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteCoproducts C] (I : Finset (Discrete α)) : (liftToFinset C α).comp ((evaluation (Finset (Discrete α)) C).obj I) ≅ ((Functor.whiskeringLeft (Discrete { x : Discrete α // x ∈ I }) (Discrete α) C).obj (Discrete.functor fun (x : { x : Discrete α // x ∈ I }) => ↑x)).comp colim

liftToFinset, when composed with the evaluation functor, results in the whiskering composed with colim.

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def CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetObj {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteProducts C] (F : Functor (Discrete α) C) : Functor (Finset (Discrete α))ᵒᵖ C

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

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theorem CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetObj_map {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteProducts C] (F : Functor (Discrete α) C) {Y x✝ : (Finset (Discrete α))ᵒᵖ} (h : Y ⟶ x✝) : (liftToFinsetObj F).map h = Pi.lift fun (y : { x : Discrete α // x ∈ Opposite.unop x✝ }) => Pi.π (fun (x : { x : Discrete α // x ∈ Opposite.unop Y }) => F.obj ↑x) ⟨↑y, ⋯⟩

@[simp]

theorem CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetObj_obj {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteProducts C] (F : Functor (Discrete α) C) (s : (Finset (Discrete α))ᵒᵖ) : (liftToFinsetObj F).obj s = ∏ᶜ fun (x : { x : Discrete α // x ∈ Opposite.unop s }) => F.obj ↑x

def CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteProducts C] [HasLimitsOfShape (Finset (Discrete α))ᵒᵖ C] (F : Functor (Discrete α) C) : LimitCone 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.

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theorem CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_isLimit_lift {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteProducts C] [HasLimitsOfShape (Finset (Discrete α))ᵒᵖ C] (F : Functor (Discrete α) C) (s : Cone F) : (liftToFinsetLimitCone F).isLimit.lift s = limit.lift (liftToFinsetObj F) { pt := s.pt, π := { app := fun (x : (Finset (Discrete α))ᵒᵖ) => Pi.lift fun (x_1 : { x_1 : Discrete α // x_1 ∈ Opposite.unop x }) => s.π.app ↑x_1, naturality := ⋯ } }

@[simp]

theorem CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_cone_pt {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteProducts C] [HasLimitsOfShape (Finset (Discrete α))ᵒᵖ C] (F : Functor (Discrete α) C) : (liftToFinsetLimitCone F).cone.pt = limit (liftToFinsetObj F)

@[simp]

theorem CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_cone_π_app {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteProducts C] [HasLimitsOfShape (Finset (Discrete α))ᵒᵖ C] (F : Functor (Discrete α) C) (j : Discrete α) : (liftToFinsetLimitCone F).cone.π.app j = CategoryStruct.comp (limit.π (liftToFinsetObj F) (Opposite.op {j})) (Pi.π (fun (x : { x : Discrete α // x ∈ {j} }) => F.obj ↑x) ⟨j, ⋯⟩)

def CategoryTheory.Limits.ProductsFromFiniteCofiltered.finiteSubproductsCone {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteProducts C] (f : α → C) [HasProduct f] : Cone (liftToFinsetObj (Discrete.functor f))

The converse of the construction in liftToFinsetLimitCone: we can form a cone on the product of f whose legs are the products over the finite subsets of α.

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theorem CategoryTheory.Limits.ProductsFromFiniteCofiltered.finiteSubproductsCone_π_app {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteProducts C] (f : α → C) [HasProduct f] (S : (Finset (Discrete α))ᵒᵖ) : (finiteSubproductsCone f).π.app S = Pi.lift fun (s : { x : Discrete α // x ∈ Opposite.unop S }) => Pi.π f (↑s).as

@[simp]

theorem CategoryTheory.Limits.ProductsFromFiniteCofiltered.finiteSubproductsCone_pt {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteProducts C] (f : α → C) [HasProduct f] : (finiteSubproductsCone f).pt = ∏ᶜ f

def CategoryTheory.Limits.ProductsFromFiniteCofiltered.isLimitFiniteSubproductsCone {C : Type u} [Category.{v, u} C] {α : Type w} [HasFiniteProducts C] (f : α → C) [HasLimitsOfShape (Finset (Discrete α))ᵒᵖ C] [HasProduct f] : IsLimit (finiteSubproductsCone f)

The cone finiteSubproductsCone is a limit cone.

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def CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinset (C : Type u) [Category.{v, u} C] (α : Type w) [HasFiniteProducts C] : Functor (Functor (Discrete α) C) (Functor (Finset (Discrete α))ᵒᵖ C)

The functor taking a functor Discrete α ⥤ C to a functor Finset (Discrete α) ⥤ C by taking coproducts.

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theorem CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinset_map_app (C : Type u) [Category.{v, u} C] (α : Type w) [HasFiniteProducts C] {X✝ Y✝ : Functor (Discrete α) C} (β : X✝ ⟶ Y✝) (x✝ : (Finset (Discrete α))ᵒᵖ) : ((liftToFinset C α).map β).app x✝ = Pi.map fun (x : { x : Discrete α // x ∈ Opposite.unop x✝ }) => β.app ↑x

@[simp]

theorem CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinset_obj_map (C : Type u) [Category.{v, u} C] (α : Type w) [HasFiniteProducts C] (F : Functor (Discrete α) C) {Y x✝ : (Finset (Discrete α))ᵒᵖ} (h : Y ⟶ x✝) : ((liftToFinset C α).obj F).map h = Pi.lift fun (y : { x : Discrete α // x ∈ Opposite.unop x✝ }) => Pi.π (fun (x : { x : Discrete α // x ∈ Opposite.unop Y }) => F.obj ↑x) ⟨↑y, ⋯⟩

@[simp]

theorem CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinset_obj_obj (C : Type u) [Category.{v, u} C] (α : Type w) [HasFiniteProducts C] (F : Functor (Discrete α) C) (s : (Finset (Discrete α))ᵒᵖ) : ((liftToFinset C α).obj F).obj s = ∏ᶜ fun (x : { x : Discrete α // x ∈ Opposite.unop s }) => F.obj ↑x

def CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimIso (C : Type u) [Category.{v, u} C] (α : Type w) [HasFiniteProducts C] [HasLimitsOfShape (Finset (Discrete α))ᵒᵖ C] [HasLimitsOfShape (Discrete α) C] : (liftToFinset C α).comp lim ≅ lim

The liftToFinset functor, precomposed with forming a colimit, is a coproduct on the original functor.

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def CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetEvaluationIso (C : Type u) [Category.{v, u} C] (α : Type w) [HasFiniteProducts C] (I : Finset (Discrete α)) : (liftToFinset C α).comp ((evaluation (Finset (Discrete α))ᵒᵖ C).obj (Opposite.op I)) ≅ ((Functor.whiskeringLeft (Discrete { x : Discrete α // x ∈ I }) (Discrete α) C).obj (Discrete.functor fun (x : { x : Discrete α // x ∈ I }) => ↑x)).comp lim

liftToFinset, when composed with the evaluation functor, results in the whiskering composed with colim.

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