Connected limits

A connected limit is a limit whose shape is a connected category.

We show that constant functors from a connected category have a limit and a colimit, give examples of connected categories, and prove that the functor given by (X × -) preserves any connected limit. That is, any limit of shape J where J is a connected category is preserved by the functor (X × -).

def CategoryTheory.Limits.constCone (J : Type u₁) [Category.{v₁, u₁} J] {C : Type u₂} [Category.{v₂, u₂} C] (X : C) : Cone ((Functor.const J).obj X)

The obvious cone of a constant functor.

Equations

• CategoryTheory.Limits.constCone J X = { pt := X, π := CategoryTheory.CategoryStruct.id ((CategoryTheory.Functor.const J).obj X) }

@[simp]

theorem CategoryTheory.Limits.constCone_pt (J : Type u₁) [Category.{v₁, u₁} J] {C : Type u₂} [Category.{v₂, u₂} C] (X : C) : (constCone J X).pt = X

@[simp]

theorem CategoryTheory.Limits.constCone_π (J : Type u₁) [Category.{v₁, u₁} J] {C : Type u₂} [Category.{v₂, u₂} C] (X : C) : (constCone J X).π = CategoryStruct.id ((Functor.const J).obj X)

def CategoryTheory.Limits.constCocone (J : Type u₁) [Category.{v₁, u₁} J] {C : Type u₂} [Category.{v₂, u₂} C] (X : C) : Cocone ((Functor.const J).obj X)

The obvious cocone of a constant functor.

Equations

• CategoryTheory.Limits.constCocone J X = { pt := X, ι := CategoryTheory.CategoryStruct.id ((CategoryTheory.Functor.const J).obj X) }

@[simp]

theorem CategoryTheory.Limits.constCocone_ι (J : Type u₁) [Category.{v₁, u₁} J] {C : Type u₂} [Category.{v₂, u₂} C] (X : C) : (constCocone J X).ι = CategoryStruct.id ((Functor.const J).obj X)

@[simp]

theorem CategoryTheory.Limits.constCocone_pt (J : Type u₁) [Category.{v₁, u₁} J] {C : Type u₂} [Category.{v₂, u₂} C] (X : C) : (constCocone J X).pt = X

def CategoryTheory.Limits.isLimitConstCone (J : Type u₁) [Category.{v₁, u₁} J] {C : Type u₂} [Category.{v₂, u₂} C] (X : C) [IsConnected J] : IsLimit (constCone J X)

When J is a connected category, the limit of a constant functor J ⥤ C with value X : C identifies to X.

Equations

• One or more equations did not get rendered due to their size.

def CategoryTheory.Limits.isColimitConstCocone (J : Type u₁) [Category.{v₁, u₁} J] {C : Type u₂} [Category.{v₂, u₂} C] (X : C) [IsConnected J] : IsColimit (constCocone J X)

When J is a connected category, the colimit of a constant functor J ⥤ C with value X : C identifies to X.

Equations

• One or more equations did not get rendered due to their size.

instance CategoryTheory.widePullbackShape_connected (J : Type v₁) : IsConnected (Limits.WidePullbackShape J)

instance CategoryTheory.widePushoutShape_connected (J : Type v₁) : IsConnected (Limits.WidePushoutShape J)

instance CategoryTheory.parallelPairInhabited : Inhabited Limits.WalkingParallelPair

Equations

• CategoryTheory.parallelPairInhabited = { default := CategoryTheory.Limits.WalkingParallelPair.one }

instance CategoryTheory.parallel_pair_connected : IsConnected Limits.WalkingParallelPair

def CategoryTheory.ProdPreservesConnectedLimits.γ₂ {C : Type u₂} [Category.{v₂, u₂} C] [Limits.HasBinaryProducts C] {J : Type v₂} [SmallCategory J] {K : Functor J C} (X : C) : K.comp (Limits.prod.functor.obj X) ⟶ K

(Impl). The obvious natural transformation from (X × K -) to K.

Equations

• CategoryTheory.ProdPreservesConnectedLimits.γ₂ X = { app := fun (x : J) => CategoryTheory.Limits.prod.snd, naturality := ⋯ }

@[simp]

theorem CategoryTheory.ProdPreservesConnectedLimits.γ₂_app {C : Type u₂} [Category.{v₂, u₂} C] [Limits.HasBinaryProducts C] {J : Type v₂} [SmallCategory J] {K : Functor J C} (X : C) (x✝ : J) : (γ₂ X).app x✝ = Limits.prod.snd

def CategoryTheory.ProdPreservesConnectedLimits.γ₁ {C : Type u₂} [Category.{v₂, u₂} C] [Limits.HasBinaryProducts C] {J : Type v₂} [SmallCategory J] {K : Functor J C} (X : C) : K.comp (Limits.prod.functor.obj X) ⟶ (Functor.const J).obj X

(Impl). The obvious natural transformation from (X × K -) to X

Equations

• CategoryTheory.ProdPreservesConnectedLimits.γ₁ X = { app := fun (x : J) => CategoryTheory.Limits.prod.fst, naturality := ⋯ }

@[simp]

theorem CategoryTheory.ProdPreservesConnectedLimits.γ₁_app {C : Type u₂} [Category.{v₂, u₂} C] [Limits.HasBinaryProducts C] {J : Type v₂} [SmallCategory J] {K : Functor J C} (X : C) (x✝ : J) : (γ₁ X).app x✝ = Limits.prod.fst

def CategoryTheory.ProdPreservesConnectedLimits.forgetCone {C : Type u₂} [Category.{v₂, u₂} C] [Limits.HasBinaryProducts C] {J : Type v₂} [SmallCategory J] {X : C} {K : Functor J C} (s : Limits.Cone (K.comp (Limits.prod.functor.obj X))) : Limits.Cone K

(Impl). Given a cone for (X × K -), produce a cone for K using the natural transformation γ₂

Equations

• CategoryTheory.ProdPreservesConnectedLimits.forgetCone s = { pt := s.pt, π := CategoryTheory.CategoryStruct.comp s.π (CategoryTheory.ProdPreservesConnectedLimits.γ₂ X) }

@[simp]

theorem CategoryTheory.ProdPreservesConnectedLimits.forgetCone_π {C : Type u₂} [Category.{v₂, u₂} C] [Limits.HasBinaryProducts C] {J : Type v₂} [SmallCategory J] {X : C} {K : Functor J C} (s : Limits.Cone (K.comp (Limits.prod.functor.obj X))) : (forgetCone s).π = CategoryStruct.comp s.π (γ₂ X)

@[simp]

theorem CategoryTheory.ProdPreservesConnectedLimits.forgetCone_pt {C : Type u₂} [Category.{v₂, u₂} C] [Limits.HasBinaryProducts C] {J : Type v₂} [SmallCategory J] {X : C} {K : Functor J C} (s : Limits.Cone (K.comp (Limits.prod.functor.obj X))) : (forgetCone s).pt = s.pt

theorem CategoryTheory.prod_preservesConnectedLimits {C : Type u₂} [Category.{v₂, u₂} C] [Limits.HasBinaryProducts C] {J : Type v₂} [SmallCategory J] [IsConnected J] (X : C) : Limits.PreservesLimitsOfShape J (Limits.prod.functor.obj X)

The functor (X × -) preserves any connected limit. Note that this functor does not preserve the two most obvious disconnected limits - that is, (X × -) does not preserve products or terminal object, eg (X ⨯ A) ⨯ (X ⨯ B) is not isomorphic to X ⨯ (A ⨯ B) and X ⨯ 1 is not isomorphic to 1.