Kan extensions

The basic definitions for Kan extensions of functors is introduced in this file. Part of API is parallel to the definitions for bicategories (see CategoryTheory.Bicategory.Kan.IsKan). (The bicategory API cannot be used directly here because it would not allow the universe polymorphism which is necessary for some applications.)

Given a natural transformation α : L ⋙ F' ⟶ F, we define the property F'.IsRightKanExtension α which expresses that (F', α) is a right Kan extension of F along L, i.e. that it is a terminal object in a category RightExtension L F of costructured arrows. The condition F'.IsLeftKanExtension α for α : F ⟶ L ⋙ F' is defined similarly.

We also introduce typeclasses HasRightKanExtension L F and HasLeftKanExtension L F which assert the existence of a right or left Kan extension, and chosen Kan extensions are obtained as leftKanExtension L F and rightKanExtension L F.

TODO (@joelriou)

• define left/right derived functors as particular cases of Kan extensions

References

• https://ncatlab.org/nlab/show/Kan+extension

@[reducible, inline]

abbrev CategoryTheory.Functor.RightExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) : Type (max (max (max (max u_2 u_3) u_6) u_7) u_1 u_6)

Given two functors L : C ⥤ D and F : C ⥤ H, this is the category of functors F' : H ⥤ D equipped with a natural transformation L ⋙ F' ⟶ F.

Equations

• L.RightExtension F = CategoryTheory.CostructuredArrow ((CategoryTheory.whiskeringLeft C D H).obj L) F

@[reducible, inline]

abbrev CategoryTheory.Functor.LeftExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) : Type (max (max (max (max u_2 u_3) u_6) u_7) u_1 u_6)

Given two functors L : C ⥤ D and F : C ⥤ H, this is the category of functors F' : H ⥤ D equipped with a natural transformation F ⟶ L ⋙ F'.

Equations

• L.LeftExtension F = CategoryTheory.StructuredArrow F ((CategoryTheory.whiskeringLeft C D H).obj L)

@[simp]

theorem CategoryTheory.Functor.RightExtension.mk_hom {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) : (CategoryTheory.Functor.RightExtension.mk F' α).hom = α

@[simp]

theorem CategoryTheory.Functor.RightExtension.mk_left {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) : (CategoryTheory.Functor.RightExtension.mk F' α).left = F'

@[simp]

theorem CategoryTheory.Functor.RightExtension.mk_right_as {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) : (CategoryTheory.Functor.RightExtension.mk F' α).right.as = PUnit.unit

def CategoryTheory.Functor.RightExtension.mk {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) : L.RightExtension F

Constructor for objects of the category Functor.RightExtension L F.

Equations

• CategoryTheory.Functor.RightExtension.mk F' α = CategoryTheory.CostructuredArrow.mk α

@[simp]

theorem CategoryTheory.Functor.LeftExtension.mk_right {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') : (CategoryTheory.Functor.LeftExtension.mk F' α).right = F'

@[simp]

theorem CategoryTheory.Functor.LeftExtension.mk_left_as {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') : (CategoryTheory.Functor.LeftExtension.mk F' α).left.as = PUnit.unit

@[simp]

theorem CategoryTheory.Functor.LeftExtension.mk_hom {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') : (CategoryTheory.Functor.LeftExtension.mk F' α).hom = α

def CategoryTheory.Functor.LeftExtension.mk {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') : L.LeftExtension F

Constructor for objects of the category Functor.LeftExtension L F.

Equations

• CategoryTheory.Functor.LeftExtension.mk F' α = CategoryTheory.StructuredArrow.mk α

class CategoryTheory.Functor.IsRightKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) : Prop

Given α : L ⋙ F' ⟶ F, the property F'.IsRightKanExtension α asserts that (F', α) is a terminal object in the category RightExtension L F, i.e. that (F', α) is a right Kan extension of F along L.

• nonempty_isUniversal : Nonempty (CategoryTheory.CostructuredArrow.IsUniversal (CategoryTheory.Functor.RightExtension.mk F' α))

theorem CategoryTheory.Functor.IsRightKanExtension.nonempty_isUniversal {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] {F' : CategoryTheory.Functor D H} {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {α : L.comp F' ⟶ F} [self : F'.IsRightKanExtension α] : Nonempty (CategoryTheory.CostructuredArrow.IsUniversal (CategoryTheory.Functor.RightExtension.mk F' α))

noncomputable def CategoryTheory.Functor.isUniversalOfIsRightKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) [F'.IsRightKanExtension α] : CategoryTheory.CostructuredArrow.IsUniversal (CategoryTheory.Functor.RightExtension.mk F' α)

If (F', α) is a right Kan extension of F along L, then (F', α) is a terminal object in the category RightExtension L F.

Equations

• F'.isUniversalOfIsRightKanExtension α = ⋯.some

noncomputable def CategoryTheory.Functor.liftOfIsRightKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) [F'.IsRightKanExtension α] (G : CategoryTheory.Functor D H) (β : L.comp G ⟶ F) : G ⟶ F'

If (F', α) is a right Kan extension of F along L and β : L ⋙ G ⟶ F is a natural transformation, this is the induced morphism G ⟶ F'.

Equations

• F'.liftOfIsRightKanExtension α G β = (F'.isUniversalOfIsRightKanExtension α).lift (CategoryTheory.Functor.RightExtension.mk G β)

theorem CategoryTheory.Functor.liftOfIsRightKanExtension_fac {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) [F'.IsRightKanExtension α] (G : CategoryTheory.Functor D H) (β : L.comp G ⟶ F) : CategoryTheory.CategoryStruct.comp (CategoryTheory.whiskerLeft L (F'.liftOfIsRightKanExtension α G β)) α = β

@[simp]

theorem CategoryTheory.Functor.liftOfIsRightKanExtension_fac_app_assoc {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) [F'.IsRightKanExtension α] (G : CategoryTheory.Functor D H) (β : L.comp G ⟶ F) (X : C) {Z : H} (h : F.obj X ⟶ Z) : CategoryTheory.CategoryStruct.comp ((F'.liftOfIsRightKanExtension α G β).app (L.obj X)) (CategoryTheory.CategoryStruct.comp (α.app X) h) = CategoryTheory.CategoryStruct.comp (β.app X) h

@[simp]

theorem CategoryTheory.Functor.liftOfIsRightKanExtension_fac_app {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) [F'.IsRightKanExtension α] (G : CategoryTheory.Functor D H) (β : L.comp G ⟶ F) (X : C) : CategoryTheory.CategoryStruct.comp ((F'.liftOfIsRightKanExtension α G β).app (L.obj X)) (α.app X) = β.app X

theorem CategoryTheory.Functor.hom_ext_of_isRightKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) [F'.IsRightKanExtension α] {G : CategoryTheory.Functor D H} (γ₁ : G ⟶ F') (γ₂ : G ⟶ F') (hγ : CategoryTheory.CategoryStruct.comp (CategoryTheory.whiskerLeft L γ₁) α = CategoryTheory.CategoryStruct.comp (CategoryTheory.whiskerLeft L γ₂) α) : γ₁ = γ₂

theorem CategoryTheory.Functor.isRightKanExtension_of_iso {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] {F' : CategoryTheory.Functor D H} {F'' : CategoryTheory.Functor D H} (e : F' ≅ F'') {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) (α' : L.comp F'' ⟶ F) (comm : CategoryTheory.CategoryStruct.comp (CategoryTheory.whiskerLeft L e.hom) α' = α) [F'.IsRightKanExtension α] : F''.IsRightKanExtension α'

theorem CategoryTheory.Functor.isRightKanExtension_iff_of_iso {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] {F' : CategoryTheory.Functor D H} {F'' : CategoryTheory.Functor D H} (e : F' ≅ F'') {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) (α' : L.comp F'' ⟶ F) (comm : CategoryTheory.CategoryStruct.comp (CategoryTheory.whiskerLeft L e.hom) α' = α) : F'.IsRightKanExtension α ↔ F''.IsRightKanExtension α'

class CategoryTheory.Functor.IsLeftKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') : Prop

Given α : F ⟶ L ⋙ F', the property F'.IsLeftKanExtension α asserts that (F', α) is an initial object in the category LeftExtension L F, i.e. that (F', α) is a left Kan extension of F along L.

• nonempty_isUniversal : Nonempty (CategoryTheory.StructuredArrow.IsUniversal (CategoryTheory.Functor.LeftExtension.mk F' α))

theorem CategoryTheory.Functor.IsLeftKanExtension.nonempty_isUniversal {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] {F' : CategoryTheory.Functor D H} {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {α : F ⟶ L.comp F'} [self : F'.IsLeftKanExtension α] : Nonempty (CategoryTheory.StructuredArrow.IsUniversal (CategoryTheory.Functor.LeftExtension.mk F' α))

noncomputable def CategoryTheory.Functor.isUniversalOfIsLeftKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α] : CategoryTheory.StructuredArrow.IsUniversal (CategoryTheory.Functor.LeftExtension.mk F' α)

If (F', α) is a left Kan extension of F along L, then (F', α) is an initial object in the category LeftExtension L F.

Equations

• F'.isUniversalOfIsLeftKanExtension α = ⋯.some

noncomputable def CategoryTheory.Functor.descOfIsLeftKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) : F' ⟶ G

If (F', α) is a left Kan extension of F along L and β : F ⟶ L ⋙ G is a natural transformation, this is the induced morphism F' ⟶ G.

Equations

• F'.descOfIsLeftKanExtension α G β = (F'.isUniversalOfIsLeftKanExtension α).desc (CategoryTheory.Functor.LeftExtension.mk G β)

@[simp]

theorem CategoryTheory.Functor.descOfIsLeftKanExtension_fac_assoc {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) {Z : CategoryTheory.Functor C H} (h : L.comp G ⟶ Z) : CategoryTheory.CategoryStruct.comp α (CategoryTheory.CategoryStruct.comp (CategoryTheory.whiskerLeft L (F'.descOfIsLeftKanExtension α G β)) h) = CategoryTheory.CategoryStruct.comp β h

@[simp]

theorem CategoryTheory.Functor.descOfIsLeftKanExtension_fac {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) : CategoryTheory.CategoryStruct.comp α (CategoryTheory.whiskerLeft L (F'.descOfIsLeftKanExtension α G β)) = β

@[simp]

theorem CategoryTheory.Functor.descOfIsLeftKanExtension_fac_app_assoc {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) (X : C) {Z : H} (h : G.obj (L.obj X) ⟶ Z) : CategoryTheory.CategoryStruct.comp (α.app X) (CategoryTheory.CategoryStruct.comp ((F'.descOfIsLeftKanExtension α G β).app (L.obj X)) h) = CategoryTheory.CategoryStruct.comp (β.app X) h

@[simp]

theorem CategoryTheory.Functor.descOfIsLeftKanExtension_fac_app {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) (X : C) : CategoryTheory.CategoryStruct.comp (α.app X) ((F'.descOfIsLeftKanExtension α G β).app (L.obj X)) = β.app X

theorem CategoryTheory.Functor.hom_ext_of_isLeftKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α] {G : CategoryTheory.Functor D H} (γ₁ : F' ⟶ G) (γ₂ : F' ⟶ G) (hγ : CategoryTheory.CategoryStruct.comp α (CategoryTheory.whiskerLeft L γ₁) = CategoryTheory.CategoryStruct.comp α (CategoryTheory.whiskerLeft L γ₂)) : γ₁ = γ₂

noncomputable def CategoryTheory.Functor.homEquivOfIsLeftKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α] (G : CategoryTheory.Functor D H) : (F' ⟶ G) ≃ (F ⟶ L.comp G)

If (F', α) is a left Kan extension of F along L, then this is the induced bijection (F' ⟶ G) ≃ (F ⟶ L ⋙ G) for all G.

Equations

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

theorem CategoryTheory.Functor.isLeftKanExtension_of_iso {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] {F' : CategoryTheory.Functor D H} {F'' : CategoryTheory.Functor D H} (e : F' ≅ F'') {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') (α' : F ⟶ L.comp F'') (comm : CategoryTheory.CategoryStruct.comp α (CategoryTheory.whiskerLeft L e.hom) = α') [F'.IsLeftKanExtension α] : F''.IsLeftKanExtension α'

theorem CategoryTheory.Functor.isLeftKanExtension_iff_of_iso {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] {F' : CategoryTheory.Functor D H} {F'' : CategoryTheory.Functor D H} (e : F' ≅ F'') {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') (α' : F ⟶ L.comp F'') (comm : CategoryTheory.CategoryStruct.comp α (CategoryTheory.whiskerLeft L e.hom) = α') : F'.IsLeftKanExtension α ↔ F''.IsLeftKanExtension α'

@[simp]

theorem CategoryTheory.Functor.leftKanExtensionUnique_inv {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α] (F'' : CategoryTheory.Functor D H) (α' : F ⟶ L.comp F'') [F''.IsLeftKanExtension α'] : (F'.leftKanExtensionUnique α F'' α').inv = F''.descOfIsLeftKanExtension α' F' α

@[simp]

theorem CategoryTheory.Functor.leftKanExtensionUnique_hom {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α] (F'' : CategoryTheory.Functor D H) (α' : F ⟶ L.comp F'') [F''.IsLeftKanExtension α'] : (F'.leftKanExtensionUnique α F'' α').hom = F'.descOfIsLeftKanExtension α F'' α'

noncomputable def CategoryTheory.Functor.leftKanExtensionUnique {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α] (F'' : CategoryTheory.Functor D H) (α' : F ⟶ L.comp F'') [F''.IsLeftKanExtension α'] : F' ≅ F''

Two left Kan extensions are (canonically) isomorphic.

Equations

• F'.leftKanExtensionUnique α F'' α' = { hom := F'.descOfIsLeftKanExtension α F'' α', inv := F''.descOfIsLeftKanExtension α' F' α, hom_inv_id := ⋯, inv_hom_id := ⋯ }

theorem CategoryTheory.Functor.isLeftKanExtension_iff_isIso {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] {F' : CategoryTheory.Functor D H} {F'' : CategoryTheory.Functor D H} (φ : F' ⟶ F'') {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') (α' : F ⟶ L.comp F'') (comm : CategoryTheory.CategoryStruct.comp α (CategoryTheory.whiskerLeft L φ) = α') [F'.IsLeftKanExtension α] : F''.IsLeftKanExtension α' ↔ CategoryTheory.IsIso φ

@[reducible, inline]

abbrev CategoryTheory.Functor.HasRightKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) : Prop

This property HasRightKanExtension L F holds when the functor F has a right Kan extension along L.

Equations

• L.HasRightKanExtension F = CategoryTheory.Limits.HasTerminal (L.RightExtension F)

theorem CategoryTheory.Functor.HasRightKanExtension.mk {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : L.comp F' ⟶ F) [F'.IsRightKanExtension α] : L.HasRightKanExtension F

@[reducible, inline]

abbrev CategoryTheory.Functor.HasLeftKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) : Prop

This property HasLeftKanExtension L F holds when the functor F has a left Kan extension along L.

Equations

• L.HasLeftKanExtension F = CategoryTheory.Limits.HasInitial (L.LeftExtension F)

theorem CategoryTheory.Functor.HasLeftKanExtension.mk {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α] : L.HasLeftKanExtension F

noncomputable def CategoryTheory.Functor.rightKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasRightKanExtension F] : CategoryTheory.Functor D H

A chosen right Kan extension when [HasRightKanExtension L F] holds.

Equations

• L.rightKanExtension F = (⊤_ L.RightExtension F).left

noncomputable def CategoryTheory.Functor.rightKanExtensionCounit {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasRightKanExtension F] : L.comp (L.rightKanExtension F) ⟶ F

The counit of the chosen right Kan extension rightKanExtension L F.

Equations

• L.rightKanExtensionCounit F = (⊤_ L.RightExtension F).hom

instance CategoryTheory.Functor.instIsRightKanExtensionRightKanExtensionRightKanExtensionCounit {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasRightKanExtension F] : (L.rightKanExtension F).IsRightKanExtension (L.rightKanExtensionCounit F)

Equations

• ⋯ = ⋯

theorem CategoryTheory.Functor.rightKanExtension_hom_ext {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasRightKanExtension F] {G : CategoryTheory.Functor D H} (γ₁ : G ⟶ L.rightKanExtension F) (γ₂ : G ⟶ L.rightKanExtension F) (hγ : CategoryTheory.CategoryStruct.comp (CategoryTheory.whiskerLeft L γ₁) (L.rightKanExtensionCounit F) = CategoryTheory.CategoryStruct.comp (CategoryTheory.whiskerLeft L γ₂) (L.rightKanExtensionCounit F)) : γ₁ = γ₂

noncomputable def CategoryTheory.Functor.leftKanExtension {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasLeftKanExtension F] : CategoryTheory.Functor D H

A chosen left Kan extension when [HasLeftKanExtension L F] holds.

Equations

• L.leftKanExtension F = (⊥_ L.LeftExtension F).right

noncomputable def CategoryTheory.Functor.leftKanExtensionUnit {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasLeftKanExtension F] : F ⟶ L.comp (L.leftKanExtension F)

The unit of the chosen left Kan extension leftKanExtension L F.

Equations

• L.leftKanExtensionUnit F = (⊥_ L.LeftExtension F).hom

instance CategoryTheory.Functor.instIsLeftKanExtensionLeftKanExtensionLeftKanExtensionUnit {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasLeftKanExtension F] : (L.leftKanExtension F).IsLeftKanExtension (L.leftKanExtensionUnit F)

Equations

• ⋯ = ⋯

theorem CategoryTheory.Functor.leftKanExtension_hom_ext {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_7, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_5, u_3} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasLeftKanExtension F] {G : CategoryTheory.Functor D H} (γ₁ : L.leftKanExtension F ⟶ G) (γ₂ : L.leftKanExtension F ⟶ G) (hγ : CategoryTheory.CategoryStruct.comp (L.leftKanExtensionUnit F) (CategoryTheory.whiskerLeft L γ₁) = CategoryTheory.CategoryStruct.comp (L.leftKanExtensionUnit F) (CategoryTheory.whiskerLeft L γ₂)) : γ₁ = γ₂

noncomputable def CategoryTheory.Functor.rightExtensionEquivalenceOfPostcomp₁ {C : Type u_1} {H : Type u_2} {D : Type u_3} {D' : Type u_4} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] [CategoryTheory.Category.{u_8, u_4} D'] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (e : D ≌ D') : (L.comp e.functor).RightExtension F ≌ L.RightExtension F

The equivalence of categories RightExtension (L ⋙ e.functor) F ≌ RightExtension L F when e is an equivalence.

Equations

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

theorem CategoryTheory.Functor.hasRightExtension_iff_postcomp₁ {C : Type u_1} {H : Type u_2} {D : Type u_3} {D' : Type u_4} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] [CategoryTheory.Category.{u_8, u_4} D'] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (e : D ≌ D') : L.HasRightKanExtension F ↔ (L.comp e.functor).HasRightKanExtension F

noncomputable def CategoryTheory.Functor.leftExtensionEquivalenceOfPostcomp₁ {C : Type u_1} {H : Type u_2} {D : Type u_3} {D' : Type u_4} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] [CategoryTheory.Category.{u_8, u_4} D'] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (e : D ≌ D') : (L.comp e.functor).LeftExtension F ≌ L.LeftExtension F

The equivalence of categories LeftExtension (L ⋙ e.functor) F ≌ LeftExtension L F when e is an equivalence.

Equations

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

theorem CategoryTheory.Functor.hasLeftExtension_iff_postcomp₁ {C : Type u_1} {H : Type u_2} {D : Type u_3} {D' : Type u_4} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] [CategoryTheory.Category.{u_8, u_4} D'] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (e : D ≌ D') : L.HasLeftKanExtension F ↔ (L.comp e.functor).HasLeftKanExtension F

def CategoryTheory.Functor.rightExtensionEquivalenceOfIso₁ {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) : L.RightExtension F ≌ L'.RightExtension F

The equivalence RightExtension L F ≌ RightExtension L' F induced by a natural isomorphism L ≅ L'.

Equations

• CategoryTheory.Functor.rightExtensionEquivalenceOfIso₁ iso₁ F = CategoryTheory.CostructuredArrow.mapNatIso ((CategoryTheory.whiskeringLeft C D H).mapIso iso₁)

theorem CategoryTheory.Functor.hasRightExtension_iff_of_iso₁ {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) : L.HasRightKanExtension F ↔ L'.HasRightKanExtension F

def CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) : L.LeftExtension F ≌ L'.LeftExtension F

The equivalence LeftExtension L F ≌ LeftExtension L' F induced by a natural isomorphism L ≅ L'.

Equations

• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F = CategoryTheory.StructuredArrow.mapNatIso ((CategoryTheory.whiskeringLeft C D H).mapIso iso₁)

theorem CategoryTheory.Functor.hasLeftExtension_iff_of_iso₁ {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) : L.HasLeftKanExtension F ↔ L'.HasLeftKanExtension F

def CategoryTheory.Functor.rightExtensionEquivalenceOfIso₂ {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) {F : CategoryTheory.Functor C H} {F' : CategoryTheory.Functor C H} (iso₂ : F ≅ F') : L.RightExtension F ≌ L.RightExtension F'

The equivalence RightExtension L F ≌ RightExtension L F' induced by a natural isomorphism F ≅ F'.

Equations

• L.rightExtensionEquivalenceOfIso₂ iso₂ = CategoryTheory.CostructuredArrow.mapIso iso₂

theorem CategoryTheory.Functor.hasRightExtension_iff_of_iso₂ {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) {F : CategoryTheory.Functor C H} {F' : CategoryTheory.Functor C H} (iso₂ : F ≅ F') : L.HasRightKanExtension F ↔ L.HasRightKanExtension F'

def CategoryTheory.Functor.leftExtensionEquivalenceOfIso₂ {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) {F : CategoryTheory.Functor C H} {F' : CategoryTheory.Functor C H} (iso₂ : F ≅ F') : L.LeftExtension F ≌ L.LeftExtension F'

The equivalence LeftExtension L F ≌ LeftExtension L F' induced by a natural isomorphism F ≅ F'.

Equations

• L.leftExtensionEquivalenceOfIso₂ iso₂ = CategoryTheory.StructuredArrow.mapIso iso₂

theorem CategoryTheory.Functor.hasLeftExtension_iff_of_iso₂ {C : Type u_1} {H : Type u_2} {D : Type u_3} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} H] [CategoryTheory.Category.{u_7, u_3} D] (L : CategoryTheory.Functor C D) {F : CategoryTheory.Functor C H} {F' : CategoryTheory.Functor C H} (iso₂ : F ≅ F') : L.HasLeftKanExtension F ↔ L.HasLeftKanExtension F'