Modifications between oplax transformations

A modification Γ between oplax transformations η and θ consists of a family of 2-morphisms Γ.app a : η.app a ⟶ θ.app a, which for all 1-morphisms f : a ⟶ b satisfies the equation (F.map f ◁ app b) ≫ θ.naturality f = η.naturality f ≫ app a ▷ G.map f.

Main definitions

• Modification η θ: modifications between oplax transformations η and θ
• Modification.vcomp η θ: the vertical composition of oplax transformations η and θ
• OplaxTrans.category F G: the category structure on the oplax transformations between F and G

structure CategoryTheory.Oplax.OplaxTrans.Modification {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} (η θ : F ⟶ G) : Type (max u₁ w₂)

A modification Γ between oplax natural transformations η and θ consists of a family of 2-morphisms Γ.app a : η.app a ⟶ θ.app a, which satisfies the equation (F.map f ◁ app b) ≫ θ.naturality f = η.naturality f ≫ (app a ▷ G.map f) for each 1-morphism f : a ⟶ b.

• app (a : B) : η.app a ⟶ θ.app a

  The underlying family of 2-morphisms.

• naturality {a b : B} (f : a ⟶ b) : CategoryStruct.comp (Bicategory.whiskerLeft (F.map f) (self.app b)) (θ.naturality f) = CategoryStruct.comp (η.naturality f) (Bicategory.whiskerRight (self.app a) (G.map f))

  The naturality condition.

theorem CategoryTheory.Oplax.OplaxTrans.Modification.ext_iff {B : Type u₁} {inst✝ : Bicategory B} {C : Type u₂} {inst✝¹ : Bicategory C} {F G : OplaxFunctor B C} {η θ : F ⟶ G} {x y : Modification η θ} : x = y ↔ x.app = y.app

theorem CategoryTheory.Oplax.OplaxTrans.Modification.ext {B : Type u₁} {inst✝ : Bicategory B} {C : Type u₂} {inst✝¹ : Bicategory C} {F G : OplaxFunctor B C} {η θ : F ⟶ G} {x y : Modification η θ} (app : x.app = y.app) : x = y

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.Modification.naturality_assoc {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} {η θ : F ⟶ G} (self : Modification η θ) {a b : B} (f : a ⟶ b) {Z : F.obj a ⟶ G.obj b} (h : CategoryStruct.comp (θ.app a) (G.map f) ⟶ Z) : CategoryStruct.comp (Bicategory.whiskerLeft (F.map f) (self.app b)) (CategoryStruct.comp (θ.naturality f) h) = CategoryStruct.comp (η.naturality f) (CategoryStruct.comp (Bicategory.whiskerRight (self.app a) (G.map f)) h)

def CategoryTheory.Oplax.OplaxTrans.Modification.id {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} (η : F ⟶ G) : Modification η η

The identity modification.

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.Modification.id_app {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} (η : F ⟶ G) (a : B) : (id η).app a = CategoryStruct.id (η.app a)

instance CategoryTheory.Oplax.OplaxTrans.Modification.instInhabited {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} (η : F ⟶ G) : Inhabited (Modification η η)

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.Modification.whiskerLeft_naturality {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} {η θ : F ⟶ G} (Γ : Modification η θ) {b c : B} {a' : C} (f : a' ⟶ F.obj b) (g : b ⟶ c) : CategoryStruct.comp (Bicategory.whiskerLeft f (Bicategory.whiskerLeft (F.map g) (Γ.app c))) (Bicategory.whiskerLeft f (θ.naturality g)) = CategoryStruct.comp (Bicategory.whiskerLeft f (η.naturality g)) (Bicategory.whiskerLeft f (Bicategory.whiskerRight (Γ.app b) (G.map g)))

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.Modification.whiskerLeft_naturality_assoc {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} {η θ : F ⟶ G} (Γ : Modification η θ) {b c : B} {a' : C} (f : a' ⟶ F.obj b) (g : b ⟶ c) {Z : a' ⟶ G.obj c} (h : CategoryStruct.comp f (CategoryStruct.comp (θ.app b) (G.map g)) ⟶ Z) : CategoryStruct.comp (Bicategory.whiskerLeft f (Bicategory.whiskerLeft (F.map g) (Γ.app c))) (CategoryStruct.comp (Bicategory.whiskerLeft f (θ.naturality g)) h) = CategoryStruct.comp (Bicategory.whiskerLeft f (η.naturality g)) (CategoryStruct.comp (Bicategory.whiskerLeft f (Bicategory.whiskerRight (Γ.app b) (G.map g))) h)

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.Modification.whiskerRight_naturality {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} {η θ : F ⟶ G} (Γ : Modification η θ) {a b : B} {a' : C} (f : a ⟶ b) (g : G.obj b ⟶ a') : CategoryStruct.comp (Bicategory.whiskerLeft (F.map f) (Bicategory.whiskerRight (Γ.app b) g)) (CategoryStruct.comp (Bicategory.associator (F.map f) (θ.app b) g).inv (Bicategory.whiskerRight (θ.naturality f) g)) = CategoryStruct.comp (Bicategory.associator (F.map f) (η.app b) g).inv (CategoryStruct.comp (Bicategory.whiskerRight (η.naturality f) g) (Bicategory.whiskerRight (Bicategory.whiskerRight (Γ.app a) (G.map f)) g))

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.Modification.whiskerRight_naturality_assoc {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} {η θ : F ⟶ G} (Γ : Modification η θ) {a b : B} {a' : C} (f : a ⟶ b) (g : G.obj b ⟶ a') {Z : F.obj a ⟶ a'} (h : CategoryStruct.comp (CategoryStruct.comp (θ.app a) (G.map f)) g ⟶ Z) : CategoryStruct.comp (Bicategory.whiskerLeft (F.map f) (Bicategory.whiskerRight (Γ.app b) g)) (CategoryStruct.comp (Bicategory.associator (F.map f) (θ.app b) g).inv (CategoryStruct.comp (Bicategory.whiskerRight (θ.naturality f) g) h)) = CategoryStruct.comp (Bicategory.associator (F.map f) (η.app b) g).inv (CategoryStruct.comp (Bicategory.whiskerRight (η.naturality f) g) (CategoryStruct.comp (Bicategory.whiskerRight (Bicategory.whiskerRight (Γ.app a) (G.map f)) g) h))

def CategoryTheory.Oplax.OplaxTrans.Modification.vcomp {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} {η θ ι : F ⟶ G} (Γ : Modification η θ) (Δ : Modification θ ι) : Modification η ι

Vertical composition of modifications.

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.Modification.vcomp_app {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} {η θ ι : F ⟶ G} (Γ : Modification η θ) (Δ : Modification θ ι) (a : B) : (Γ.vcomp Δ).app a = CategoryStruct.comp (Γ.app a) (Δ.app a)

def CategoryTheory.Oplax.OplaxTrans.category {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F G : OplaxFunctor B C) : Category.{max u₁ w₂, max (max (max u₁ v₁) v₂) w₂} (F ⟶ G)

Category structure on the oplax natural transformations between OplaxFunctors.

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.category_comp {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F G : OplaxFunctor B C) {X✝ Y✝ Z✝ : F ⟶ G} (Γ : Modification X✝ Y✝) (Δ : Modification Y✝ Z✝) : CategoryStruct.comp Γ Δ = Γ.vcomp Δ

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.category_id {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F G : OplaxFunctor B C) (η : F ⟶ G) : CategoryStruct.id η = Modification.id η

theorem CategoryTheory.Oplax.OplaxTrans.ext {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} {α β : F ⟶ G} {m n : α ⟶ β} (w : ∀ (b : B), m.app b = n.app b) : m = n

theorem CategoryTheory.Oplax.OplaxTrans.ext_iff {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} {α β : F ⟶ G} {m n : α ⟶ β} : m = n ↔ ∀ (b : B), m.app b = n.app b

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.Modification.id_app' {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {X : B} {F G : OplaxFunctor B C} (α : F ⟶ G) : (CategoryStruct.id α).app X = CategoryStruct.id (α.app X)

Version of Modification.id_app using category notation

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.Modification.comp_app' {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {X : B} {F G : OplaxFunctor B C} {α β γ : F ⟶ G} (m : α ⟶ β) (n : β ⟶ γ) : (CategoryStruct.comp m n).app X = CategoryStruct.comp (m.app X) (n.app X)

Version of Modification.comp_app using category notation

def CategoryTheory.Oplax.OplaxTrans.ModificationIso.ofComponents {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} {η θ : F ⟶ G} (app : (a : B) → η.app a ≅ θ.app a) (naturality : ∀ {a b : B} (f : a ⟶ b), CategoryStruct.comp (Bicategory.whiskerLeft (F.map f) (app b).hom) (θ.naturality f) = CategoryStruct.comp (η.naturality f) (Bicategory.whiskerRight (app a).hom (G.map f))) : η ≅ θ

Construct a modification isomorphism between oplax natural transformations by giving object level isomorphisms, and checking naturality only in the forward direction.

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.ModificationIso.ofComponents_hom_app {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} {η θ : F ⟶ G} (app : (a : B) → η.app a ≅ θ.app a) (naturality : ∀ {a b : B} (f : a ⟶ b), CategoryStruct.comp (Bicategory.whiskerLeft (F.map f) (app b).hom) (θ.naturality f) = CategoryStruct.comp (η.naturality f) (Bicategory.whiskerRight (app a).hom (G.map f))) (a : B) : (ofComponents app naturality).hom.app a = (app a).hom

@[simp]

theorem CategoryTheory.Oplax.OplaxTrans.ModificationIso.ofComponents_inv_app {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {F G : OplaxFunctor B C} {η θ : F ⟶ G} (app : (a : B) → η.app a ≅ θ.app a) (naturality : ∀ {a b : B} (f : a ⟶ b), CategoryStruct.comp (Bicategory.whiskerLeft (F.map f) (app b).hom) (θ.naturality f) = CategoryStruct.comp (η.naturality f) (Bicategory.whiskerRight (app a).hom (G.map f))) (a : B) : (ofComponents app naturality).inv.app a = (app a).inv