Limit properties relating to the (co)yoneda embedding.

We calculate the colimit of Y ↦ (X ⟶ Y), which is just PUnit. (This is used in characterising cofinal functors.)

We also show the (co)yoneda embeddings preserve limits and jointly reflect them.

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theorem CategoryTheory.Coyoneda.colimitCocone_ι_app {C : Type v} [CategoryTheory.SmallCategory C] (X : Cᵒᵖ) (X_1 : C) (a : (CategoryTheory.coyoneda.toPrefunctor.obj X).toPrefunctor.obj X_1) : (CategoryTheory.Coyoneda.colimitCocone X).ι.app X_1 a = id PUnit.unit

@[simp]

theorem CategoryTheory.Coyoneda.colimitCocone_pt {C : Type v} [CategoryTheory.SmallCategory C] (X : Cᵒᵖ) : (CategoryTheory.Coyoneda.colimitCocone X).pt = PUnit.{v + 1}

def CategoryTheory.Coyoneda.colimitCocone {C : Type v} [CategoryTheory.SmallCategory C] (X : Cᵒᵖ) : CategoryTheory.Limits.Cocone (CategoryTheory.coyoneda.toPrefunctor.obj X)

The colimit cocone over coyoneda.obj X, with cocone point PUnit.

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@[simp]

theorem CategoryTheory.Coyoneda.colimitCoconeIsColimit_desc {C : Type v} [CategoryTheory.SmallCategory C] (X : Cᵒᵖ) (s : CategoryTheory.Limits.Cocone (CategoryTheory.coyoneda.toPrefunctor.obj X)) : ∀ (x : (CategoryTheory.Coyoneda.colimitCocone X).pt), (CategoryTheory.Coyoneda.colimitCoconeIsColimit X).desc s x = s.ι.app X.unop (CategoryTheory.CategoryStruct.id X.unop)

def CategoryTheory.Coyoneda.colimitCoconeIsColimit {C : Type v} [CategoryTheory.SmallCategory C] (X : Cᵒᵖ) : CategoryTheory.Limits.IsColimit (CategoryTheory.Coyoneda.colimitCocone X)

The proposed colimit cocone over coyoneda.obj X is a colimit cocone.

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instance CategoryTheory.Coyoneda.instHasColimitTypeTypesObjOppositeToQuiverToCategoryStructOppositeFunctorToQuiverToCategoryStructCategoryToPrefunctorCoyoneda {C : Type v} [CategoryTheory.SmallCategory C] (X : Cᵒᵖ) : CategoryTheory.Limits.HasColimit (CategoryTheory.coyoneda.toPrefunctor.obj X)

Equations

• (_ : CategoryTheory.Limits.HasColimit (CategoryTheory.coyoneda.toPrefunctor.obj X)) = (_ : CategoryTheory.Limits.HasColimit (CategoryTheory.coyoneda.toPrefunctor.obj X))

noncomputable def CategoryTheory.Coyoneda.colimitCoyonedaIso {C : Type v} [CategoryTheory.SmallCategory C] (X : Cᵒᵖ) : CategoryTheory.Limits.colimit (CategoryTheory.coyoneda.toPrefunctor.obj X) ≅ PUnit.{v + 1}

The colimit of coyoneda.obj X is isomorphic to PUnit.

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instance CategoryTheory.yonedaPreservesLimits {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) : CategoryTheory.Limits.PreservesLimits (CategoryTheory.yoneda.toPrefunctor.obj X)

The yoneda embedding yoneda.obj X : Cᵒᵖ ⥤ Type v for X : C preserves limits.

Equations

• CategoryTheory.yonedaPreservesLimits X = CategoryTheory.Limits.PreservesLimitsOfSize.mk

instance CategoryTheory.coyonedaPreservesLimits {C : Type u} [CategoryTheory.Category.{v, u} C] (X : Cᵒᵖ) : CategoryTheory.Limits.PreservesLimits (CategoryTheory.coyoneda.toPrefunctor.obj X)

The coyoneda embedding coyoneda.obj X : C ⥤ Type v for X : Cᵒᵖ preserves limits.

Equations

• CategoryTheory.coyonedaPreservesLimits X = CategoryTheory.Limits.PreservesLimitsOfSize.mk

def CategoryTheory.yonedaJointlyReflectsLimits {C : Type u} [CategoryTheory.Category.{v, u} C] (J : Type w) [CategoryTheory.SmallCategory J] (K : CategoryTheory.Functor J Cᵒᵖ) (c : CategoryTheory.Limits.Cone K) (t : (X : C) → CategoryTheory.Limits.IsLimit ((CategoryTheory.yoneda.toPrefunctor.obj X).mapCone c)) : CategoryTheory.Limits.IsLimit c

The yoneda embeddings jointly reflect limits.

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def CategoryTheory.coyonedaJointlyReflectsLimits {C : Type u} [CategoryTheory.Category.{v, u} C] (J : Type w) [CategoryTheory.SmallCategory J] (K : CategoryTheory.Functor J C) (c : CategoryTheory.Limits.Cone K) (t : (X : Cᵒᵖ) → CategoryTheory.Limits.IsLimit ((CategoryTheory.coyoneda.toPrefunctor.obj X).mapCone c)) : CategoryTheory.Limits.IsLimit c

The coyoneda embeddings jointly reflect limits.

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instance CategoryTheory.yonedaFunctorPreservesLimits {D : Type u} [CategoryTheory.SmallCategory D] : CategoryTheory.Limits.PreservesLimits CategoryTheory.yoneda

Equations

• CategoryTheory.yonedaFunctorPreservesLimits = CategoryTheory.Limits.preservesLimitsOfEvaluation CategoryTheory.yoneda fun (K : Dᵒᵖ) => let_fun this := inferInstance; this

instance CategoryTheory.coyonedaFunctorPreservesLimits {D : Type u} [CategoryTheory.SmallCategory D] : CategoryTheory.Limits.PreservesLimits CategoryTheory.coyoneda

Equations

• CategoryTheory.coyonedaFunctorPreservesLimits = CategoryTheory.Limits.preservesLimitsOfEvaluation CategoryTheory.coyoneda fun (K : D) => let_fun this := inferInstance; this

instance CategoryTheory.yonedaFunctorReflectsLimits {D : Type u} [CategoryTheory.SmallCategory D] : CategoryTheory.Limits.ReflectsLimits CategoryTheory.yoneda

Equations

• CategoryTheory.yonedaFunctorReflectsLimits = inferInstance

instance CategoryTheory.coyonedaFunctorReflectsLimits {D : Type u} [CategoryTheory.SmallCategory D] : CategoryTheory.Limits.ReflectsLimits CategoryTheory.coyoneda

Equations

• CategoryTheory.coyonedaFunctorReflectsLimits = inferInstance