The Yoneda embedding

The Yoneda embedding as a functor yoneda : C ⥤ (Cᵒᵖ ⥤ Type v₁), along with an instance that it is FullyFaithful.

Also the Yoneda lemma, yonedaLemma : (yoneda_pairing C) ≅ (yoneda_evaluation C).

References

• Stacks: Opposite Categories and the Yoneda Lemma

@[simp]

theorem CategoryTheory.yoneda_obj_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : C) : ∀ {X_1 Y : Cᵒᵖ} (f : X_1 ⟶ Y) (g : Opposite.unop X_1 ⟶ X), (CategoryTheory.yoneda.obj X).map f g = CategoryTheory.CategoryStruct.comp f.unop g

@[simp]

theorem CategoryTheory.yoneda_map_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : ∀ {X Y : C} (f : X ⟶ Y) (Y_1 : Cᵒᵖ) (g : ((fun (X : C) => { obj := fun (Y : Cᵒᵖ) => Opposite.unop Y ⟶ X, map := fun {X_1 Y : Cᵒᵖ} (f : X_1 ⟶ Y) (g : (fun (Y : Cᵒᵖ) => Opposite.unop Y ⟶ X) X_1) => CategoryTheory.CategoryStruct.comp f.unop g, map_id := ⋯, map_comp := ⋯ }) X).obj Y_1), (CategoryTheory.yoneda.map f).app Y_1 g = CategoryTheory.CategoryStruct.comp g f

@[simp]

theorem CategoryTheory.yoneda_obj_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : C) (Y : Cᵒᵖ) : (CategoryTheory.yoneda.obj X).obj Y = (Opposite.unop Y ⟶ X)

def CategoryTheory.yoneda {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Functor C (CategoryTheory.Functor Cᵒᵖ (Type v₁))

The Yoneda embedding, as a functor from C into presheaves on C.

See https://stacks.math.columbia.edu/tag/001O.

Equations

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@[simp]

theorem CategoryTheory.coyoneda_obj_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : Cᵒᵖ) (Y : C) : (CategoryTheory.coyoneda.obj X).obj Y = (Opposite.unop X ⟶ Y)

@[simp]

theorem CategoryTheory.coyoneda_map_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : ∀ {X Y : Cᵒᵖ} (f : X ⟶ Y) (Y_1 : C) (g : ((fun (X : Cᵒᵖ) => { obj := fun (Y : C) => Opposite.unop X ⟶ Y, map := fun {X_1 Y : C} (f : X_1 ⟶ Y) (g : (fun (Y : C) => Opposite.unop X ⟶ Y) X_1) => CategoryTheory.CategoryStruct.comp g f, map_id := ⋯, map_comp := ⋯ }) X).obj Y_1), (CategoryTheory.coyoneda.map f).app Y_1 g = CategoryTheory.CategoryStruct.comp f.unop g

@[simp]

theorem CategoryTheory.coyoneda_obj_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : Cᵒᵖ) : ∀ {X_1 Y : C} (f : X_1 ⟶ Y) (g : Opposite.unop X ⟶ X_1), (CategoryTheory.coyoneda.obj X).map f g = CategoryTheory.CategoryStruct.comp g f

def CategoryTheory.coyoneda {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Functor Cᵒᵖ (CategoryTheory.Functor C (Type v₁))

The co-Yoneda embedding, as a functor from Cᵒᵖ into co-presheaves on C.

Equations

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theorem CategoryTheory.Yoneda.obj_map_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : Opposite.op X ⟶ Opposite.op Y) : (CategoryTheory.yoneda.obj X).map f (CategoryTheory.CategoryStruct.id X) = (CategoryTheory.yoneda.map f.unop).app (Opposite.op Y) (CategoryTheory.CategoryStruct.id Y)

@[simp]

theorem CategoryTheory.Yoneda.naturality {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (α : CategoryTheory.yoneda.obj X ⟶ CategoryTheory.yoneda.obj Y) {Z : C} {Z' : C} (f : Z ⟶ Z') (h : Z' ⟶ X) : CategoryTheory.CategoryStruct.comp f (α.app (Opposite.op Z') h) = α.app (Opposite.op Z) (CategoryTheory.CategoryStruct.comp f h)

def CategoryTheory.Yoneda.fullyFaithful {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.yoneda.FullyFaithful

The Yoneda embedding is fully faithful.

Equations

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theorem CategoryTheory.Yoneda.fullyFaithful_preimage {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : CategoryTheory.yoneda.obj X ⟶ CategoryTheory.yoneda.obj Y) : CategoryTheory.Yoneda.fullyFaithful.preimage f = f.app (Opposite.op X) (CategoryTheory.CategoryStruct.id X)

instance CategoryTheory.Yoneda.yoneda_full {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.yoneda.Full

The Yoneda embedding is full.

See https://stacks.math.columbia.edu/tag/001P.

Equations

• ⋯ = ⋯

instance CategoryTheory.Yoneda.yoneda_faithful {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.yoneda.Faithful

The Yoneda embedding is faithful.

See https://stacks.math.columbia.edu/tag/001P.

Equations

• ⋯ = ⋯

def CategoryTheory.Yoneda.ext {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : C) (Y : C) (p : {Z : C} → (Z ⟶ X) → (Z ⟶ Y)) (q : {Z : C} → (Z ⟶ Y) → (Z ⟶ X)) (h₁ : ∀ {Z : C} (f : Z ⟶ X), q (p f) = f) (h₂ : ∀ {Z : C} (f : Z ⟶ Y), p (q f) = f) (n : ∀ {Z Z' : C} (f : Z' ⟶ Z) (g : Z ⟶ X), p (CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp f (p g)) : X ≅ Y

Extensionality via Yoneda. The typical usage would be

-- Goal is `X ≅ Y`
apply yoneda.ext,
-- Goals are now functions `(Z ⟶ X) → (Z ⟶ Y)`, `(Z ⟶ Y) → (Z ⟶ X)`, and the fact that these
-- functions are inverses and natural in `Z`.

Equations

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

theorem CategoryTheory.Yoneda.isIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.IsIso (CategoryTheory.yoneda.map f)] : CategoryTheory.IsIso f

If yoneda.map f is an isomorphism, so was f.

@[simp]

theorem CategoryTheory.Coyoneda.naturality {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (α : CategoryTheory.coyoneda.obj X ⟶ CategoryTheory.coyoneda.obj Y) {Z : C} {Z' : C} (f : Z' ⟶ Z) (h : Opposite.unop X ⟶ Z') : CategoryTheory.CategoryStruct.comp (α.app Z' h) f = α.app Z (CategoryTheory.CategoryStruct.comp h f)

def CategoryTheory.Coyoneda.fullyFaithful {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.coyoneda.FullyFaithful

The co-Yoneda embedding is fully faithful.

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theorem CategoryTheory.Coyoneda.fullyFaithful_preimage {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : CategoryTheory.coyoneda.obj X ⟶ CategoryTheory.coyoneda.obj Y) : CategoryTheory.Coyoneda.fullyFaithful.preimage f = Quiver.Hom.op (f.app (Opposite.unop X) (CategoryTheory.CategoryStruct.id (Opposite.unop X)))

def CategoryTheory.Coyoneda.preimage {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : CategoryTheory.coyoneda.obj X ⟶ CategoryTheory.coyoneda.obj Y) : X ⟶ Y

The morphism X ⟶ Y corresponding to a natural transformation coyoneda.obj X ⟶ coyoneda.obj Y.

Equations

• CategoryTheory.Coyoneda.preimage f = Quiver.Hom.op (f.app (Opposite.unop X) (CategoryTheory.CategoryStruct.id (Opposite.unop X)))

instance CategoryTheory.Coyoneda.coyoneda_full {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.coyoneda.Full

Equations

• ⋯ = ⋯

instance CategoryTheory.Coyoneda.coyoneda_faithful {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.coyoneda.Faithful

Equations

• ⋯ = ⋯

theorem CategoryTheory.Coyoneda.isIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : X ⟶ Y) [CategoryTheory.IsIso (CategoryTheory.coyoneda.map f)] : CategoryTheory.IsIso f

If coyoneda.map f is an isomorphism, so was f.

def CategoryTheory.Coyoneda.punitIso : CategoryTheory.coyoneda.obj (Opposite.op PUnit.{v₁ + 1} ) ≅ CategoryTheory.Functor.id (Type v₁)

The identity functor on Type is isomorphic to the coyoneda functor coming from PUnit.

Equations

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

@[simp]

theorem CategoryTheory.Coyoneda.objOpOp_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : C) (X : Cᵒᵖ) : ∀ (a : (CategoryTheory.coyoneda.obj (Opposite.op (Opposite.op X✝))).obj X), (CategoryTheory.Coyoneda.objOpOp X✝).hom.app X a = (CategoryTheory.opEquiv (Opposite.op X✝) X) a

@[simp]

theorem CategoryTheory.Coyoneda.objOpOp_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : C) (X : Cᵒᵖ) : ∀ (a : (CategoryTheory.yoneda.obj X✝).obj X), (CategoryTheory.Coyoneda.objOpOp X✝).inv.app X a = (CategoryTheory.opEquiv (Opposite.op X✝) X).symm a

def CategoryTheory.Coyoneda.objOpOp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : C) : CategoryTheory.coyoneda.obj (Opposite.op (Opposite.op X)) ≅ CategoryTheory.yoneda.obj X

Taking the unop of morphisms is a natural isomorphism.

Equations

• CategoryTheory.Coyoneda.objOpOp X = CategoryTheory.NatIso.ofComponents (fun (x : Cᵒᵖ) => (CategoryTheory.opEquiv (Opposite.unop (Opposite.op (Opposite.op X))) x).toIso) ⋯

class CategoryTheory.Functor.Representable {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor Cᵒᵖ (Type v₁)) : Prop

A functor F : Cᵒᵖ ⥤ Type v₁ is representable if there is object X so F ≅ yoneda.obj X.

See https://stacks.math.columbia.edu/tag/001Q.

• has_representation : ∃ (X : C), Nonempty (CategoryTheory.yoneda.obj X ≅ F)

  Hom(-,X) ≅ F via f

theorem CategoryTheory.Functor.Representable.has_representation {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {F : CategoryTheory.Functor Cᵒᵖ (Type v₁)} [self : F.Representable] : ∃ (X : C), Nonempty (CategoryTheory.yoneda.obj X ≅ F)

Hom(-,X) ≅ F via f

instance CategoryTheory.Functor.instRepresentableObjOppositeTypeYoneda {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} : (CategoryTheory.yoneda.obj X).Representable

Equations

• ⋯ = ⋯

class CategoryTheory.Functor.Corepresentable {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor C (Type v₁)) : Prop

A functor F : C ⥤ Type v₁ is corepresentable if there is object X so F ≅ coyoneda.obj X.

See https://stacks.math.columbia.edu/tag/001Q.

• has_corepresentation : ∃ (X : Cᵒᵖ), Nonempty (CategoryTheory.coyoneda.obj X ≅ F)

  Hom(X,-) ≅ F via f

theorem CategoryTheory.Functor.Corepresentable.has_corepresentation {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {F : CategoryTheory.Functor C (Type v₁)} [self : F.Corepresentable] : ∃ (X : Cᵒᵖ), Nonempty (CategoryTheory.coyoneda.obj X ≅ F)

Hom(X,-) ≅ F via f

instance CategoryTheory.Functor.instCorepresentableObjOppositeTypeCoyoneda {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} : (CategoryTheory.coyoneda.obj X).Corepresentable

Equations

• ⋯ = ⋯

noncomputable def CategoryTheory.Functor.reprX {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor Cᵒᵖ (Type v₁)) [hF : F.Representable] : C

The representing object for the representable functor F.

Equations

• F.reprX = ⋯.choose

noncomputable def CategoryTheory.Functor.reprW {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor Cᵒᵖ (Type v₁)) [hF : F.Representable] : CategoryTheory.yoneda.obj F.reprX ≅ F

An isomorphism between a representable F and a functor of the form C(-, F.reprX). Note the components F.reprW.app X definitionally have type (X.unop ⟶ F.repr_X) ≅ F.obj X.

Equations

• F.reprW = ⋯.some

noncomputable def CategoryTheory.Functor.reprx {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor Cᵒᵖ (Type v₁)) [hF : F.Representable] : F.obj (Opposite.op F.reprX)

The representing element for the representable functor F, sometimes called the universal element of the functor.

Equations

• F.reprx = F.reprW.hom.app (Opposite.op F.reprX) (CategoryTheory.CategoryStruct.id F.reprX)

theorem CategoryTheory.Functor.reprW_app_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor Cᵒᵖ (Type v₁)) [hF : F.Representable] (X : Cᵒᵖ) (f : Opposite.unop X ⟶ F.reprX) : (F.reprW.app X).hom f = F.map f.op F.reprx

noncomputable def CategoryTheory.Functor.coreprX {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor C (Type v₁)) [hF : F.Corepresentable] : C

The representing object for the corepresentable functor F.

Equations

• F.coreprX = Opposite.unop ⋯.choose

noncomputable def CategoryTheory.Functor.coreprW {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor C (Type v₁)) [hF : F.Corepresentable] : CategoryTheory.coyoneda.obj (Opposite.op F.coreprX) ≅ F

An isomorphism between a corepresnetable F and a functor of the form C(F.corepr X, -). Note the components F.coreprW.app X definitionally have type F.corepr_X ⟶ X ≅ F.obj X.

Equations

• F.coreprW = ⋯.some

noncomputable def CategoryTheory.Functor.coreprx {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor C (Type v₁)) [hF : F.Corepresentable] : F.obj F.coreprX

The representing element for the corepresentable functor F, sometimes called the universal element of the functor.

Equations

• F.coreprx = F.coreprW.hom.app F.coreprX (CategoryTheory.CategoryStruct.id F.coreprX)

theorem CategoryTheory.Functor.coreprW_app_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor C (Type v₁)) [hF : F.Corepresentable] (X : C) (f : F.coreprX ⟶ X) : (F.coreprW.app X).hom f = F.map f F.coreprx

theorem CategoryTheory.representable_of_natIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor Cᵒᵖ (Type v₁)) {G : CategoryTheory.Functor Cᵒᵖ (Type v₁)} (i : F ≅ G) [F.Representable] : G.Representable

theorem CategoryTheory.corepresentable_of_natIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor C (Type v₁)) {G : CategoryTheory.Functor C (Type v₁)} (i : F ≅ G) [F.Corepresentable] : G.Corepresentable

instance CategoryTheory.instCorepresentableIdType : (CategoryTheory.Functor.id (Type v₁)).Corepresentable

Equations

• CategoryTheory.instCorepresentableIdType = ⋯

instance CategoryTheory.prodCategoryInstance1 (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Category.{max u₁ v₁, max u₁ (max (v₁ + 1) u₁) v₁} (CategoryTheory.Functor Cᵒᵖ (Type v₁) × Cᵒᵖ)

Equations

• CategoryTheory.prodCategoryInstance1 C = CategoryTheory.prod (CategoryTheory.Functor Cᵒᵖ (Type v₁)) Cᵒᵖ

instance CategoryTheory.prodCategoryInstance2 (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Category.{max u₁ v₁, max (max (max (v₁ + 1) u₁) v₁) u₁} (Cᵒᵖ × CategoryTheory.Functor Cᵒᵖ (Type v₁))

Equations

• CategoryTheory.prodCategoryInstance2 C = CategoryTheory.prod Cᵒᵖ (CategoryTheory.Functor Cᵒᵖ (Type v₁))

def CategoryTheory.yonedaEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {F : CategoryTheory.Functor Cᵒᵖ (Type v₁)} : (CategoryTheory.yoneda.obj X ⟶ F) ≃ F.obj (Opposite.op X)

We have a type-level equivalence between natural transformations from the yoneda embedding and elements of F.obj X, without any universe switching.

Equations

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

theorem CategoryTheory.yonedaEquiv_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {F : CategoryTheory.Functor Cᵒᵖ (Type v₁)} (f : CategoryTheory.yoneda.obj X ⟶ F) : CategoryTheory.yonedaEquiv f = f.app (Opposite.op X) (CategoryTheory.CategoryStruct.id X)

@[simp]

theorem CategoryTheory.yonedaEquiv_symm_app_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {F : CategoryTheory.Functor Cᵒᵖ (Type v₁)} (x : F.obj (Opposite.op X)) (Y : Cᵒᵖ) (f : Opposite.unop Y ⟶ X) : (CategoryTheory.yonedaEquiv.symm x).app Y f = F.map f.op x

theorem CategoryTheory.yonedaEquiv_naturality {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} {F : CategoryTheory.Functor Cᵒᵖ (Type v₁)} (f : CategoryTheory.yoneda.obj X ⟶ F) (g : Y ⟶ X) : F.map g.op (CategoryTheory.yonedaEquiv f) = CategoryTheory.yonedaEquiv (CategoryTheory.CategoryStruct.comp (CategoryTheory.yoneda.map g) f)

theorem CategoryTheory.yonedaEquiv_naturality' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} {F : CategoryTheory.Functor Cᵒᵖ (Type v₁)} (f : CategoryTheory.yoneda.obj (Opposite.unop X) ⟶ F) (g : X ⟶ Y) : F.map g (CategoryTheory.yonedaEquiv f) = CategoryTheory.yonedaEquiv (CategoryTheory.CategoryStruct.comp (CategoryTheory.yoneda.map g.unop) f)

theorem CategoryTheory.yonedaEquiv_comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {F : CategoryTheory.Functor Cᵒᵖ (Type v₁)} {G : CategoryTheory.Functor Cᵒᵖ (Type v₁)} (α : CategoryTheory.yoneda.obj X ⟶ F) (β : F ⟶ G) : CategoryTheory.yonedaEquiv (CategoryTheory.CategoryStruct.comp α β) = β.app (Opposite.op X) (CategoryTheory.yonedaEquiv α)

theorem CategoryTheory.yonedaEquiv_yoneda_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : X ⟶ Y) : CategoryTheory.yonedaEquiv (CategoryTheory.yoneda.map f) = f

theorem CategoryTheory.yonedaEquiv_symm_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : X ⟶ Y) {F : CategoryTheory.Functor Cᵒᵖ (Type v₁)} (t : F.obj X) : CategoryTheory.yonedaEquiv.symm (F.map f t) = CategoryTheory.CategoryStruct.comp (CategoryTheory.yoneda.map f.unop) (CategoryTheory.yonedaEquiv.symm t)

theorem CategoryTheory.hom_ext_yoneda {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {P : CategoryTheory.Functor Cᵒᵖ (Type v₁)} {Q : CategoryTheory.Functor Cᵒᵖ (Type v₁)} {f : P ⟶ Q} {g : P ⟶ Q} (h : ∀ (X : C) (p : CategoryTheory.yoneda.obj X ⟶ P), CategoryTheory.CategoryStruct.comp p f = CategoryTheory.CategoryStruct.comp p g) : f = g

Two morphisms of presheaves of types P ⟶ Q coincide if the precompositions with morphisms yoneda.obj X ⟶ P agree.

def CategoryTheory.yonedaEvaluation (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Functor (Cᵒᵖ × CategoryTheory.Functor Cᵒᵖ (Type v₁)) (Type (max u₁ v₁))

The "Yoneda evaluation" functor, which sends X : Cᵒᵖ and F : Cᵒᵖ ⥤ Type to F.obj X, functorially in both X and F.

Equations

• CategoryTheory.yonedaEvaluation C = (CategoryTheory.evaluationUncurried Cᵒᵖ (Type v₁)).comp CategoryTheory.uliftFunctor.{u₁, v₁}

@[simp]

theorem CategoryTheory.yonedaEvaluation_map_down (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (P : Cᵒᵖ × CategoryTheory.Functor Cᵒᵖ (Type v₁)) (Q : Cᵒᵖ × CategoryTheory.Functor Cᵒᵖ (Type v₁)) (α : P ⟶ Q) (x : (CategoryTheory.yonedaEvaluation C).obj P) : ((CategoryTheory.yonedaEvaluation C).map α x).down = α.2.app Q.1 (P.2.map α.1 x.down)

def CategoryTheory.yonedaPairing (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Functor (Cᵒᵖ × CategoryTheory.Functor Cᵒᵖ (Type v₁)) (Type (max u₁ v₁))

The "Yoneda pairing" functor, which sends X : Cᵒᵖ and F : Cᵒᵖ ⥤ Type to yoneda.op.obj X ⟶ F, functorially in both X and F.

Equations

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

theorem CategoryTheory.yonedaPairingExt (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ × CategoryTheory.Functor Cᵒᵖ (Type v₁)} {x : (CategoryTheory.yonedaPairing C).obj X} {y : (CategoryTheory.yonedaPairing C).obj X} (w : ∀ (Y : Cᵒᵖ), x.app Y = y.app Y) : x = y

@[simp]

theorem CategoryTheory.yonedaPairing_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (P : Cᵒᵖ × CategoryTheory.Functor Cᵒᵖ (Type v₁)) (Q : Cᵒᵖ × CategoryTheory.Functor Cᵒᵖ (Type v₁)) (α : P ⟶ Q) (β : (CategoryTheory.yonedaPairing C).obj P) : (CategoryTheory.yonedaPairing C).map α β = CategoryTheory.CategoryStruct.comp (CategoryTheory.yoneda.map α.1.unop) (CategoryTheory.CategoryStruct.comp β α.2)

def CategoryTheory.yonedaCompUliftFunctorEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor Cᵒᵖ (Type (max v₁ w))) (X : C) : ((CategoryTheory.yoneda.obj X).comp CategoryTheory.uliftFunctor.{w, v₁} ⟶ F) ≃ F.obj (Opposite.op X)

A bijection (yoneda.obj X ⋙ uliftFunctor ⟶ F) ≃ F.obj (op X) which is a variant of yonedaEquiv with heterogeneous universes.

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def CategoryTheory.yonedaLemma (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.yonedaPairing C ≅ CategoryTheory.yonedaEvaluation C

The Yoneda lemma asserts that the Yoneda pairing (X : Cᵒᵖ, F : Cᵒᵖ ⥤ Type) ↦ (yoneda.obj (unop X) ⟶ F) is naturally isomorphic to the evaluation (X, F) ↦ F.obj X.

See https://stacks.math.columbia.edu/tag/001P.

Equations

• CategoryTheory.yonedaLemma C = CategoryTheory.NatIso.ofComponents (fun (X : Cᵒᵖ × CategoryTheory.Functor Cᵒᵖ (Type v₁)) => (CategoryTheory.yonedaEquiv.trans Equiv.ulift.symm).toIso) ⋯

def CategoryTheory.curriedYonedaLemma {C : Type u₁} [CategoryTheory.SmallCategory C] : CategoryTheory.yoneda.op.comp CategoryTheory.coyoneda ≅ CategoryTheory.evaluation Cᵒᵖ (Type u₁)

The curried version of yoneda lemma when C is small.

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def CategoryTheory.largeCurriedYonedaLemma {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.yoneda.op.comp CategoryTheory.coyoneda ≅ (CategoryTheory.evaluation Cᵒᵖ (Type v₁)).comp ((CategoryTheory.whiskeringRight (CategoryTheory.Functor Cᵒᵖ (Type v₁)) (Type v₁) (Type (max u₁ v₁))).obj CategoryTheory.uliftFunctor.{u₁, v₁} )

The curried version of the Yoneda lemma.

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def CategoryTheory.yonedaOpCompYonedaObj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (P : CategoryTheory.Functor Cᵒᵖ (Type v₁)) : CategoryTheory.yoneda.op.comp (CategoryTheory.yoneda.obj P) ≅ P.comp CategoryTheory.uliftFunctor.{u₁, v₁}

Version of the Yoneda lemma where the presheaf is fixed but the argument varies.

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def CategoryTheory.curriedYonedaLemma' {C : Type u₁} [CategoryTheory.SmallCategory C] : CategoryTheory.yoneda.comp ((CategoryTheory.whiskeringLeft Cᵒᵖ (CategoryTheory.Functor Cᵒᵖ (Type u₁))ᵒᵖ (Type u₁)).obj CategoryTheory.yoneda.op) ≅ CategoryTheory.Functor.id (CategoryTheory.Functor Cᵒᵖ (Type u₁))

The curried version of yoneda lemma when C is small.

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theorem CategoryTheory.isIso_of_yoneda_map_bijective {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : X ⟶ Y) (hf : ∀ (T : C), Function.Bijective fun (x : T ⟶ X) => CategoryTheory.CategoryStruct.comp x f) : CategoryTheory.IsIso f

def CategoryTheory.coyonedaEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {F : CategoryTheory.Functor C (Type v₁)} : (CategoryTheory.coyoneda.obj (Opposite.op X) ⟶ F) ≃ F.obj X

We have a type-level equivalence between natural transformations from the coyoneda embedding and elements of F.obj X.unop, without any universe switching.

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theorem CategoryTheory.coyonedaEquiv_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {F : CategoryTheory.Functor C (Type v₁)} (f : CategoryTheory.coyoneda.obj (Opposite.op X) ⟶ F) : CategoryTheory.coyonedaEquiv f = f.app X (CategoryTheory.CategoryStruct.id X)

@[simp]

theorem CategoryTheory.coyonedaEquiv_symm_app_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {F : CategoryTheory.Functor C (Type v₁)} (x : F.obj X) (Y : C) (f : X ⟶ Y) : (CategoryTheory.coyonedaEquiv.symm x).app Y f = F.map f x

theorem CategoryTheory.coyonedaEquiv_naturality {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} {F : CategoryTheory.Functor C (Type v₁)} (f : CategoryTheory.coyoneda.obj (Opposite.op X) ⟶ F) (g : X ⟶ Y) : F.map g (CategoryTheory.coyonedaEquiv f) = CategoryTheory.coyonedaEquiv (CategoryTheory.CategoryStruct.comp (CategoryTheory.coyoneda.map g.op) f)

theorem CategoryTheory.coyonedaEquiv_comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {F : CategoryTheory.Functor C (Type v₁)} {G : CategoryTheory.Functor C (Type v₁)} (α : CategoryTheory.coyoneda.obj (Opposite.op X) ⟶ F) (β : F ⟶ G) : CategoryTheory.coyonedaEquiv (CategoryTheory.CategoryStruct.comp α β) = β.app X (CategoryTheory.coyonedaEquiv α)

theorem CategoryTheory.coyonedaEquiv_coyoneda_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : X ⟶ Y) : CategoryTheory.coyonedaEquiv (CategoryTheory.coyoneda.map f.op) = f

theorem CategoryTheory.coyonedaEquiv_symm_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : X ⟶ Y) {F : CategoryTheory.Functor C (Type v₁)} (t : F.obj X) : CategoryTheory.coyonedaEquiv.symm (F.map f t) = CategoryTheory.CategoryStruct.comp (CategoryTheory.coyoneda.map f.op) (CategoryTheory.coyonedaEquiv.symm t)

def CategoryTheory.coyonedaEvaluation (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Functor (C × CategoryTheory.Functor C (Type v₁)) (Type (max u₁ v₁))

The "Coyoneda evaluation" functor, which sends X : C and F : C ⥤ Type to F.obj X, functorially in both X and F.

Equations

• CategoryTheory.coyonedaEvaluation C = (CategoryTheory.evaluationUncurried C (Type v₁)).comp CategoryTheory.uliftFunctor.{u₁, v₁}

@[simp]

theorem CategoryTheory.coyonedaEvaluation_map_down (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (P : C × CategoryTheory.Functor C (Type v₁)) (Q : C × CategoryTheory.Functor C (Type v₁)) (α : P ⟶ Q) (x : (CategoryTheory.coyonedaEvaluation C).obj P) : ((CategoryTheory.coyonedaEvaluation C).map α x).down = α.2.app Q.1 (P.2.map α.1 x.down)

def CategoryTheory.coyonedaPairing (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Functor (C × CategoryTheory.Functor C (Type v₁)) (Type (max u₁ v₁))

The "Coyoneda pairing" functor, which sends X : C and F : C ⥤ Type to coyoneda.rightOp.obj X ⟶ F, functorially in both X and F.

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theorem CategoryTheory.coyonedaPairingExt (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] {X : C × CategoryTheory.Functor C (Type v₁)} {x : (CategoryTheory.coyonedaPairing C).obj X} {y : (CategoryTheory.coyonedaPairing C).obj X} (w : ∀ (Y : C), x.app Y = y.app Y) : x = y

@[simp]

theorem CategoryTheory.coyonedaPairing_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (P : C × CategoryTheory.Functor C (Type v₁)) (Q : C × CategoryTheory.Functor C (Type v₁)) (α : P ⟶ Q) (β : (CategoryTheory.coyonedaPairing C).obj P) : (CategoryTheory.coyonedaPairing C).map α β = CategoryTheory.CategoryStruct.comp (CategoryTheory.coyoneda.map α.1.op) (CategoryTheory.CategoryStruct.comp β α.2)

def CategoryTheory.coyonedaCompUliftFunctorEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (F : CategoryTheory.Functor C (Type (max v₁ w))) (X : Cᵒᵖ) : ((CategoryTheory.coyoneda.obj X).comp CategoryTheory.uliftFunctor.{w, v₁} ⟶ F) ≃ F.obj (Opposite.unop X)

A bijection (coyoneda.obj X ⋙ uliftFunctor ⟶ F) ≃ F.obj (unop X) which is a variant of coyonedaEquiv with heterogeneous universes.

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def CategoryTheory.coyonedaLemma (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.coyonedaPairing C ≅ CategoryTheory.coyonedaEvaluation C

The Coyoneda lemma asserts that the Coyoneda pairing (X : C, F : C ⥤ Type) ↦ (coyoneda.obj X ⟶ F) is naturally isomorphic to the evaluation (X, F) ↦ F.obj X.

See https://stacks.math.columbia.edu/tag/001P.

Equations

• CategoryTheory.coyonedaLemma C = CategoryTheory.NatIso.ofComponents (fun (X : C × CategoryTheory.Functor C (Type v₁)) => (CategoryTheory.coyonedaEquiv.trans Equiv.ulift.symm).toIso) ⋯

def CategoryTheory.curriedCoyonedaLemma {C : Type u₁} [CategoryTheory.SmallCategory C] : CategoryTheory.coyoneda.rightOp.comp CategoryTheory.coyoneda ≅ CategoryTheory.evaluation C (Type u₁)

The curried version of coyoneda lemma when C is small.

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def CategoryTheory.largeCurriedCoyonedaLemma {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.coyoneda.rightOp.comp CategoryTheory.coyoneda ≅ (CategoryTheory.evaluation C (Type v₁)).comp ((CategoryTheory.whiskeringRight (CategoryTheory.Functor C (Type v₁)) (Type v₁) (Type (max u₁ v₁))).obj CategoryTheory.uliftFunctor.{u₁, v₁} )

The curried version of the Coyoneda lemma.

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def CategoryTheory.coyonedaCompYonedaObj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (P : CategoryTheory.Functor C (Type v₁)) : CategoryTheory.coyoneda.rightOp.comp (CategoryTheory.yoneda.obj P) ≅ P.comp CategoryTheory.uliftFunctor.{u₁, v₁}

Version of the Coyoneda lemma where the presheaf is fixed but the argument varies.

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def CategoryTheory.curriedCoyonedaLemma' {C : Type u₁} [CategoryTheory.SmallCategory C] : CategoryTheory.yoneda.comp ((CategoryTheory.whiskeringLeft C (CategoryTheory.Functor C (Type u₁))ᵒᵖ (Type u₁)).obj CategoryTheory.coyoneda.rightOp) ≅ CategoryTheory.Functor.id (CategoryTheory.Functor C (Type u₁))

The curried version of coyoneda lemma when C is small.

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theorem CategoryTheory.isIso_of_coyoneda_map_bijective {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : X ⟶ Y) (hf : ∀ (T : C), Function.Bijective fun (x : Y ⟶ T) => CategoryTheory.CategoryStruct.comp f x) : CategoryTheory.IsIso f

def CategoryTheory.yonedaMap {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u_1} [CategoryTheory.Category.{v₁, u_1} D] (F : CategoryTheory.Functor C D) (X : C) : CategoryTheory.yoneda.obj X ⟶ F.op.comp (CategoryTheory.yoneda.obj (F.obj X))

The natural transformation yoneda.obj X ⟶ F.op ⋙ yoneda.obj (F.obj X) when F : C ⥤ D and X : C.

Equations

• CategoryTheory.yonedaMap F X = { app := fun (X_1 : Cᵒᵖ) (f : (CategoryTheory.yoneda.obj X).obj X_1) => F.map f, naturality := ⋯ }

@[simp]

theorem CategoryTheory.yonedaMap_app_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u_1} [CategoryTheory.Category.{v₁, u_1} D] (F : CategoryTheory.Functor C D) {Y : C} {X : Cᵒᵖ} (f : Opposite.unop X ⟶ Y) : (CategoryTheory.yonedaMap F Y).app X f = F.map f