Oplax functors

An oplax functor F between bicategories B and C consists of

• a function between objects F.obj : B ⟶ C,
• a family of functions between 1-morphisms F.map : (a ⟶ b) → (F.obj a ⟶ F.obj b),
• a family of functions between 2-morphisms F.map₂ : (f ⟶ g) → (F.map f ⟶ F.map g),
• a family of 2-morphisms F.mapId a : F.map (𝟙 a) ⟶ 𝟙 (F.obj a),
• a family of 2-morphisms F.mapComp f g : F.map (f ≫ g) ⟶ F.map f ≫ F.map g, and
• certain consistency conditions on them.

Main definitions

• CategoryTheory.OplaxFunctor B C : an oplax functor between bicategories B and C, which we denote by B ⥤ᵒᵖᴸ C.
• CategoryTheory.OplaxFunctor.comp F G : the composition of oplax functors

structure CategoryTheory.OplaxFunctor (B : Type u₁) [Bicategory B] (C : Type u₂) [Bicategory C] extends CategoryTheory.PrelaxFunctor B C : Type (max (max (max (max (max u₁ u₂) v₁) v₂) w₁) w₂)

An oplax functor F between bicategories B and C consists of a function between objects F.obj, a function between 1-morphisms F.map, and a function between 2-morphisms F.map₂.

Unlike functors between categories, F.map do not need to strictly commute with the composition, and do not need to strictly preserve the identity. Instead, there are specified 2-morphisms F.map (𝟙 a) ⟶ 𝟙 (F.obj a) and F.map (f ≫ g) ⟶ F.map f ≫ F.map g.

F.map₂ strictly commute with compositions and preserve the identity. They also preserve the associator, the left unitor, and the right unitor modulo some adjustments of domains and codomains of 2-morphisms.

• obj : B → C
• map {X Y : B} : (X ⟶ Y) → (self.obj X ⟶ self.obj Y)
• map₂ {a b : B} {f g : a ⟶ b} : (f ⟶ g) → (self.map f ⟶ self.map g)
• map₂_id {a b : B} (f : a ⟶ b) : self.map₂ (CategoryStruct.id f) = CategoryStruct.id (self.map f)
• map₂_comp {a b : B} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) : self.map₂ (CategoryStruct.comp η θ) = CategoryStruct.comp (self.map₂ η) (self.map₂ θ)
• mapId (a : B) : self.map (CategoryStruct.id a) ⟶ CategoryStruct.id (self.obj a)

  The 2-morphism underlying the oplax unity constraint.

• mapComp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : self.map (CategoryStruct.comp f g) ⟶ CategoryStruct.comp (self.map f) (self.map g)

  The 2-morphism underlying the oplax functoriality constraint.

• mapComp_naturality_left {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) : CategoryStruct.comp (self.map₂ (Bicategory.whiskerRight η g)) (self.mapComp f' g) = CategoryStruct.comp (self.mapComp f g) (Bicategory.whiskerRight (self.map₂ η) (self.map g))

  Naturality of the oplax functoriality constraint, on the left.

• mapComp_naturality_right {a b c : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') : CategoryStruct.comp (self.map₂ (Bicategory.whiskerLeft f η)) (self.mapComp f g') = CategoryStruct.comp (self.mapComp f g) (Bicategory.whiskerLeft (self.map f) (self.map₂ η))

  Naturality of the lax functoriality constraint, on the right.

• map₂_associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : CategoryStruct.comp (self.map₂ (Bicategory.associator f g h).hom) (CategoryStruct.comp (self.mapComp f (CategoryStruct.comp g h)) (Bicategory.whiskerLeft (self.map f) (self.mapComp g h))) = CategoryStruct.comp (self.mapComp (CategoryStruct.comp f g) h) (CategoryStruct.comp (Bicategory.whiskerRight (self.mapComp f g) (self.map h)) (Bicategory.associator (self.map f) (self.map g) (self.map h)).hom)

  Oplax associativity.

• map₂_leftUnitor {a b : B} (f : a ⟶ b) : self.map₂ (Bicategory.leftUnitor f).hom = CategoryStruct.comp (self.mapComp (CategoryStruct.id a) f) (CategoryStruct.comp (Bicategory.whiskerRight (self.mapId a) (self.map f)) (Bicategory.leftUnitor (self.map f)).hom)

  Oplax left unity.

• map₂_rightUnitor {a b : B} (f : a ⟶ b) : self.map₂ (Bicategory.rightUnitor f).hom = CategoryStruct.comp (self.mapComp f (CategoryStruct.id b)) (CategoryStruct.comp (Bicategory.whiskerLeft (self.map f) (self.mapId b)) (Bicategory.rightUnitor (self.map f)).hom)

  Oplax right unity.

def CategoryTheory.Bicategory.«term_⥤ᵒᵖᴸ_» : Lean.TrailingParserDescr

Notation for a pseudofunctor between bicategories.

Equations

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

@[simp]

theorem CategoryTheory.OplaxFunctor.mapComp_naturality_right_app {B : Type u_1} [Bicategory B] (self : OplaxFunctor B Cat) {a b c : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (X : ↑(self.obj a)) : CategoryStruct.comp ((self.map₂ (Bicategory.whiskerLeft f η)).toNatTrans.app X) ((self.mapComp f g').toNatTrans.app X) = CategoryStruct.comp ((self.mapComp f g).toNatTrans.app X) ((self.map₂ η).toNatTrans.app ((self.map f).toFunctor.obj X))

Naturality of the lax functoriality constraint, on the right.

@[simp]

theorem CategoryTheory.OplaxFunctor.map₂_associator_app {B : Type u_1} [Bicategory B] (self : OplaxFunctor B Cat) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (X : ↑(self.obj a)) : CategoryStruct.comp ((self.map₂ (Bicategory.associator f g h).hom).toNatTrans.app X) (CategoryStruct.comp ((self.mapComp f (CategoryStruct.comp g h)).toNatTrans.app X) ((self.mapComp g h).toNatTrans.app ((self.map f).toFunctor.obj X))) = CategoryStruct.comp ((self.mapComp (CategoryStruct.comp f g) h).toNatTrans.app X) ((self.map h).toFunctor.map ((self.mapComp f g).toNatTrans.app X))

Oplax associativity.

@[simp]

theorem CategoryTheory.OplaxFunctor.mapComp_naturality_left_app {B : Type u_1} [Bicategory B] (self : OplaxFunctor B Cat) {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (X : ↑(self.obj a)) : CategoryStruct.comp ((self.map₂ (Bicategory.whiskerRight η g)).toNatTrans.app X) ((self.mapComp f' g).toNatTrans.app X) = CategoryStruct.comp ((self.mapComp f g).toNatTrans.app X) ((self.map g).toFunctor.map ((self.map₂ η).toNatTrans.app X))

Naturality of the oplax functoriality constraint, on the left.

@[simp]

theorem CategoryTheory.OplaxFunctor.mapComp_naturality_left_assoc {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (self : OplaxFunctor B C) {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) {Z : self.obj a ⟶ self.obj c} (h : CategoryStruct.comp (self.map f') (self.map g) ⟶ Z) : CategoryStruct.comp (self.map₂ (Bicategory.whiskerRight η g)) (CategoryStruct.comp (self.mapComp f' g) h) = CategoryStruct.comp (self.mapComp f g) (CategoryStruct.comp (Bicategory.whiskerRight (self.map₂ η) (self.map g)) h)

Naturality of the oplax functoriality constraint, on the left.

@[simp]

theorem CategoryTheory.OplaxFunctor.mapComp_naturality_right_assoc {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (self : OplaxFunctor B C) {a b c : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') {Z : self.obj a ⟶ self.obj c} (h : CategoryStruct.comp (self.map f) (self.map g') ⟶ Z) : CategoryStruct.comp (self.map₂ (Bicategory.whiskerLeft f η)) (CategoryStruct.comp (self.mapComp f g') h) = CategoryStruct.comp (self.mapComp f g) (CategoryStruct.comp (Bicategory.whiskerLeft (self.map f) (self.map₂ η)) h)

Naturality of the lax functoriality constraint, on the right.

@[simp]

theorem CategoryTheory.OplaxFunctor.map₂_associator_app_assoc {B : Type u_1} [Bicategory B] (self : OplaxFunctor B Cat) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (X : ↑(self.obj a)) {Z : ↑(self.obj d)} (h✝ : (self.map h).toFunctor.obj ((self.map g).toFunctor.obj ((self.map f).toFunctor.obj X)) ⟶ Z) : CategoryStruct.comp ((self.map₂ (Bicategory.associator f g h).hom).toNatTrans.app X) (CategoryStruct.comp ((self.mapComp f (CategoryStruct.comp g h)).toNatTrans.app X) (CategoryStruct.comp ((self.mapComp g h).toNatTrans.app ((self.map f).toFunctor.obj X)) h✝)) = CategoryStruct.comp ((self.mapComp (CategoryStruct.comp f g) h).toNatTrans.app X) (CategoryStruct.comp ((self.map h).toFunctor.map ((self.mapComp f g).toNatTrans.app X)) h✝)

Oplax associativity.

@[simp]

theorem CategoryTheory.OplaxFunctor.map₂_associator_assoc {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (self : OplaxFunctor B C) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) {Z : self.obj a ⟶ self.obj d} (h✝ : CategoryStruct.comp (self.map f) (CategoryStruct.comp (self.map g) (self.map h)) ⟶ Z) : CategoryStruct.comp (self.map₂ (Bicategory.associator f g h).hom) (CategoryStruct.comp (self.mapComp f (CategoryStruct.comp g h)) (CategoryStruct.comp (Bicategory.whiskerLeft (self.map f) (self.mapComp g h)) h✝)) = CategoryStruct.comp (self.mapComp (CategoryStruct.comp f g) h) (CategoryStruct.comp (Bicategory.whiskerRight (self.mapComp f g) (self.map h)) (CategoryStruct.comp (Bicategory.associator (self.map f) (self.map g) (self.map h)).hom h✝))

Oplax associativity.

@[simp]

theorem CategoryTheory.OplaxFunctor.mapComp_naturality_right_app_assoc {B : Type u_1} [Bicategory B] (self : OplaxFunctor B Cat) {a b c : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (X : ↑(self.obj a)) {Z : ↑(self.obj c)} (h : (self.map g').toFunctor.obj ((self.map f).toFunctor.obj X) ⟶ Z) : CategoryStruct.comp ((self.map₂ (Bicategory.whiskerLeft f η)).toNatTrans.app X) (CategoryStruct.comp ((self.mapComp f g').toNatTrans.app X) h) = CategoryStruct.comp ((self.mapComp f g).toNatTrans.app X) (CategoryStruct.comp ((self.map₂ η).toNatTrans.app ((self.map f).toFunctor.obj X)) h)

Naturality of the lax functoriality constraint, on the right.

@[simp]

theorem CategoryTheory.OplaxFunctor.mapComp_naturality_left_app_assoc {B : Type u_1} [Bicategory B] (self : OplaxFunctor B Cat) {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (X : ↑(self.obj a)) {Z : ↑(self.obj c)} (h : (self.map g).toFunctor.obj ((self.map f').toFunctor.obj X) ⟶ Z) : CategoryStruct.comp ((self.map₂ (Bicategory.whiskerRight η g)).toNatTrans.app X) (CategoryStruct.comp ((self.mapComp f' g).toNatTrans.app X) h) = CategoryStruct.comp ((self.mapComp f g).toNatTrans.app X) (CategoryStruct.comp ((self.map g).toFunctor.map ((self.map₂ η).toNatTrans.app X)) h)

Naturality of the oplax functoriality constraint, on the left.

theorem CategoryTheory.OplaxFunctor.map₂_rightUnitor_app {B : Type u_1} [Bicategory B] (self : OplaxFunctor B Cat) {a b : B} (f : a ⟶ b) (X : ↑(self.obj a)) : (self.map₂ (Bicategory.rightUnitor f).hom).toNatTrans.app X = CategoryStruct.comp ((self.mapComp f (CategoryStruct.id b)).toNatTrans.app X) ((self.mapId b).toNatTrans.app ((self.map f).toFunctor.obj X))

Oplax right unity.

theorem CategoryTheory.OplaxFunctor.map₂_leftUnitor_app {B : Type u_1} [Bicategory B] (self : OplaxFunctor B Cat) {a b : B} (f : a ⟶ b) (X : ↑(self.obj a)) : (self.map₂ (Bicategory.leftUnitor f).hom).toNatTrans.app X = CategoryStruct.comp ((self.mapComp (CategoryStruct.id a) f).toNatTrans.app X) ((self.map f).toFunctor.map ((self.mapId a).toNatTrans.app X))

Oplax left unity.

theorem CategoryTheory.OplaxFunctor.map₂_leftUnitor_assoc {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (self : OplaxFunctor B C) {a b : B} (f : a ⟶ b) {Z : self.obj a ⟶ self.obj b} (h : self.map f ⟶ Z) : CategoryStruct.comp (self.map₂ (Bicategory.leftUnitor f).hom) h = CategoryStruct.comp (self.mapComp (CategoryStruct.id a) f) (CategoryStruct.comp (Bicategory.whiskerRight (self.mapId a) (self.map f)) (CategoryStruct.comp (Bicategory.leftUnitor (self.map f)).hom h))

Oplax left unity.

theorem CategoryTheory.OplaxFunctor.map₂_leftUnitor_app_assoc {B : Type u_1} [Bicategory B] (self : OplaxFunctor B Cat) {a b : B} (f : a ⟶ b) (X : ↑(self.obj a)) {Z : ↑(self.obj b)} (h : (self.map f).toFunctor.obj X ⟶ Z) : CategoryStruct.comp ((self.map₂ (Bicategory.leftUnitor f).hom).toNatTrans.app X) h = CategoryStruct.comp ((self.mapComp (CategoryStruct.id a) f).toNatTrans.app X) (CategoryStruct.comp ((self.map f).toFunctor.map ((self.mapId a).toNatTrans.app X)) h)

Oplax left unity.

theorem CategoryTheory.OplaxFunctor.map₂_rightUnitor_assoc {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (self : OplaxFunctor B C) {a b : B} (f : a ⟶ b) {Z : self.obj a ⟶ self.obj b} (h : self.map f ⟶ Z) : CategoryStruct.comp (self.map₂ (Bicategory.rightUnitor f).hom) h = CategoryStruct.comp (self.mapComp f (CategoryStruct.id b)) (CategoryStruct.comp (Bicategory.whiskerLeft (self.map f) (self.mapId b)) (CategoryStruct.comp (Bicategory.rightUnitor (self.map f)).hom h))

Oplax right unity.

theorem CategoryTheory.OplaxFunctor.map₂_rightUnitor_app_assoc {B : Type u_1} [Bicategory B] (self : OplaxFunctor B Cat) {a b : B} (f : a ⟶ b) (X : ↑(self.obj a)) {Z : ↑(self.obj b)} (h : (self.map f).toFunctor.obj X ⟶ Z) : CategoryStruct.comp ((self.map₂ (Bicategory.rightUnitor f).hom).toNatTrans.app X) h = CategoryStruct.comp ((self.mapComp f (CategoryStruct.id b)).toNatTrans.app X) (CategoryStruct.comp ((self.mapId b).toNatTrans.app ((self.map f).toFunctor.obj X)) h)

Oplax right unity.

theorem CategoryTheory.OplaxFunctor.mapComp_assoc_right {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : CategoryStruct.comp (F.mapComp f (CategoryStruct.comp g h)) (Bicategory.whiskerLeft (F.map f) (F.mapComp g h)) = CategoryStruct.comp (F.map₂ (Bicategory.associator f g h).inv) (CategoryStruct.comp (F.mapComp (CategoryStruct.comp f g) h) (CategoryStruct.comp (Bicategory.whiskerRight (F.mapComp f g) (F.map h)) (Bicategory.associator (F.map f) (F.map g) (F.map h)).hom))

theorem CategoryTheory.OplaxFunctor.mapComp_assoc_right_app {B : Type u_1} [Bicategory B] (F : OplaxFunctor B Cat) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (X : ↑(F.obj a)) : CategoryStruct.comp ((F.mapComp f (CategoryStruct.comp g h)).toNatTrans.app X) ((F.mapComp g h).toNatTrans.app ((F.map f).toFunctor.obj X)) = CategoryStruct.comp ((F.map₂ (Bicategory.associator f g h).inv).toNatTrans.app X) (CategoryStruct.comp ((F.mapComp (CategoryStruct.comp f g) h).toNatTrans.app X) ((F.map h).toFunctor.map ((F.mapComp f g).toNatTrans.app X)))

theorem CategoryTheory.OplaxFunctor.mapComp_assoc_right_app_assoc {B : Type u_1} [Bicategory B] (F : OplaxFunctor B Cat) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (X : ↑(F.obj a)) {Z : ↑(F.obj d)} (h✝ : (F.map h).toFunctor.obj ((F.map g).toFunctor.obj ((F.map f).toFunctor.obj X)) ⟶ Z) : CategoryStruct.comp ((F.mapComp f (CategoryStruct.comp g h)).toNatTrans.app X) (CategoryStruct.comp ((F.mapComp g h).toNatTrans.app ((F.map f).toFunctor.obj X)) h✝) = CategoryStruct.comp ((F.map₂ (Bicategory.associator f g h).inv).toNatTrans.app X) (CategoryStruct.comp ((F.mapComp (CategoryStruct.comp f g) h).toNatTrans.app X) (CategoryStruct.comp ((F.map h).toFunctor.map ((F.mapComp f g).toNatTrans.app X)) h✝))

theorem CategoryTheory.OplaxFunctor.mapComp_assoc_right_assoc {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) {Z : F.obj a ⟶ F.obj d} (h✝ : CategoryStruct.comp (F.map f) (CategoryStruct.comp (F.map g) (F.map h)) ⟶ Z) : CategoryStruct.comp (F.mapComp f (CategoryStruct.comp g h)) (CategoryStruct.comp (Bicategory.whiskerLeft (F.map f) (F.mapComp g h)) h✝) = CategoryStruct.comp (F.map₂ (Bicategory.associator f g h).inv) (CategoryStruct.comp (F.mapComp (CategoryStruct.comp f g) h) (CategoryStruct.comp (Bicategory.whiskerRight (F.mapComp f g) (F.map h)) (CategoryStruct.comp (Bicategory.associator (F.map f) (F.map g) (F.map h)).hom h✝)))

theorem CategoryTheory.OplaxFunctor.mapComp_assoc_left {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : CategoryStruct.comp (F.mapComp (CategoryStruct.comp f g) h) (Bicategory.whiskerRight (F.mapComp f g) (F.map h)) = CategoryStruct.comp (F.map₂ (Bicategory.associator f g h).hom) (CategoryStruct.comp (F.mapComp f (CategoryStruct.comp g h)) (CategoryStruct.comp (Bicategory.whiskerLeft (F.map f) (F.mapComp g h)) (Bicategory.associator (F.map f) (F.map g) (F.map h)).inv))

theorem CategoryTheory.OplaxFunctor.mapComp_assoc_left_app {B : Type u_1} [Bicategory B] (F : OplaxFunctor B Cat) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (X : ↑(F.obj a)) : CategoryStruct.comp ((F.mapComp (CategoryStruct.comp f g) h).toNatTrans.app X) ((F.map h).toFunctor.map ((F.mapComp f g).toNatTrans.app X)) = CategoryStruct.comp ((F.map₂ (Bicategory.associator f g h).hom).toNatTrans.app X) (CategoryStruct.comp ((F.mapComp f (CategoryStruct.comp g h)).toNatTrans.app X) ((F.mapComp g h).toNatTrans.app ((F.map f).toFunctor.obj X)))

theorem CategoryTheory.OplaxFunctor.mapComp_assoc_left_assoc {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) {Z : F.obj a ⟶ F.obj d} (h✝ : CategoryStruct.comp (CategoryStruct.comp (F.map f) (F.map g)) (F.map h) ⟶ Z) : CategoryStruct.comp (F.mapComp (CategoryStruct.comp f g) h) (CategoryStruct.comp (Bicategory.whiskerRight (F.mapComp f g) (F.map h)) h✝) = CategoryStruct.comp (F.map₂ (Bicategory.associator f g h).hom) (CategoryStruct.comp (F.mapComp f (CategoryStruct.comp g h)) (CategoryStruct.comp (Bicategory.whiskerLeft (F.map f) (F.mapComp g h)) (CategoryStruct.comp (Bicategory.associator (F.map f) (F.map g) (F.map h)).inv h✝)))

theorem CategoryTheory.OplaxFunctor.mapComp_assoc_left_app_assoc {B : Type u_1} [Bicategory B] (F : OplaxFunctor B Cat) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (X : ↑(F.obj a)) {Z : ↑(F.obj d)} (h✝ : (F.map h).toFunctor.obj ((F.map g).toFunctor.obj ((F.map f).toFunctor.obj X)) ⟶ Z) : CategoryStruct.comp ((F.mapComp (CategoryStruct.comp f g) h).toNatTrans.app X) (CategoryStruct.comp ((F.map h).toFunctor.map ((F.mapComp f g).toNatTrans.app X)) h✝) = CategoryStruct.comp ((F.map₂ (Bicategory.associator f g h).hom).toNatTrans.app X) (CategoryStruct.comp ((F.mapComp f (CategoryStruct.comp g h)).toNatTrans.app X) (CategoryStruct.comp ((F.mapComp g h).toNatTrans.app ((F.map f).toFunctor.obj X)) h✝))

theorem CategoryTheory.OplaxFunctor.mapComp_id_left {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) {a b : B} (f : a ⟶ b) : CategoryStruct.comp (F.mapComp (CategoryStruct.id a) f) (Bicategory.whiskerRight (F.mapId a) (F.map f)) = CategoryStruct.comp (F.map₂ (Bicategory.leftUnitor f).hom) (Bicategory.leftUnitor (F.map f)).inv

theorem CategoryTheory.OplaxFunctor.mapComp_id_left_assoc {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) {a b : B} (f : a ⟶ b) {Z : F.obj a ⟶ F.obj b} (h : CategoryStruct.comp (CategoryStruct.id (F.obj a)) (F.map f) ⟶ Z) : CategoryStruct.comp (F.mapComp (CategoryStruct.id a) f) (CategoryStruct.comp (Bicategory.whiskerRight (F.mapId a) (F.map f)) h) = CategoryStruct.comp (F.map₂ (Bicategory.leftUnitor f).hom) (CategoryStruct.comp (Bicategory.leftUnitor (F.map f)).inv h)

theorem CategoryTheory.OplaxFunctor.mapComp_id_right {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) {a b : B} (f : a ⟶ b) : CategoryStruct.comp (F.mapComp f (CategoryStruct.id b)) (Bicategory.whiskerLeft (F.map f) (F.mapId b)) = CategoryStruct.comp (F.map₂ (Bicategory.rightUnitor f).hom) (Bicategory.rightUnitor (F.map f)).inv

theorem CategoryTheory.OplaxFunctor.mapComp_id_right_assoc {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) {a b : B} (f : a ⟶ b) {Z : F.obj a ⟶ F.obj b} (h : CategoryStruct.comp (F.map f) (CategoryStruct.id (F.obj b)) ⟶ Z) : CategoryStruct.comp (F.mapComp f (CategoryStruct.id b)) (CategoryStruct.comp (Bicategory.whiskerLeft (F.map f) (F.mapId b)) h) = CategoryStruct.comp (F.map₂ (Bicategory.rightUnitor f).hom) (CategoryStruct.comp (Bicategory.rightUnitor (F.map f)).inv h)

def CategoryTheory.OplaxFunctor.id (B : Type u₁) [Bicategory B] : OplaxFunctor B B

The identity oplax functor.

Equations

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

@[simp]

theorem CategoryTheory.OplaxFunctor.id_toPrelaxFunctor (B : Type u₁) [Bicategory B] : (id B).toPrelaxFunctor = PrelaxFunctor.id B

@[simp]

theorem CategoryTheory.OplaxFunctor.id_mapComp (B : Type u₁) [Bicategory B] {a✝ b✝ c✝ : B} (f : a✝ ⟶ b✝) (g : b✝ ⟶ c✝) : (id B).mapComp f g = CategoryStruct.id (CategoryStruct.comp f g)

@[simp]

theorem CategoryTheory.OplaxFunctor.id_mapId (B : Type u₁) [Bicategory B] (a : B) : (id B).mapId a = CategoryStruct.id (CategoryStruct.id a)

instance CategoryTheory.OplaxFunctor.instInhabited {B : Type u₁} [Bicategory B] : Inhabited (OplaxFunctor B B)

Equations

• CategoryTheory.OplaxFunctor.instInhabited = { default := CategoryTheory.OplaxFunctor.id B }

def CategoryTheory.OplaxFunctor.mapId' {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) {b : B} (f : b ⟶ b) (hf : f = CategoryStruct.id b := by cat_disch) : F.map f ⟶ CategoryStruct.id (F.obj b)

More flexible variant of mapId. (See the file Bicategory.Functor.Strict for applications to strict bicategories.)

Equations

• F.mapId' f hf = CategoryTheory.CategoryStruct.comp (F.map₂ (CategoryTheory.eqToHom ⋯)) (F.mapId b)

theorem CategoryTheory.OplaxFunctor.mapId'_eq_mapId {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) (b : B) : F.mapId' (CategoryStruct.id b) ⋯ = F.mapId b

def CategoryTheory.OplaxFunctor.mapComp' {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) {b₀ b₁ b₂ : B} (f : b₀ ⟶ b₁) (g : b₁ ⟶ b₂) (fg : b₀ ⟶ b₂) (h : CategoryStruct.comp f g = fg := by cat_disch) : F.map fg ⟶ CategoryStruct.comp (F.map f) (F.map g)

More flexible variant of mapComp. (See Bicategory.Functor.Strict for applications to strict bicategories.)

Equations

• F.mapComp' f g fg h = CategoryTheory.CategoryStruct.comp (F.map₂ (CategoryTheory.eqToHom ⋯)) (F.mapComp f g)

theorem CategoryTheory.OplaxFunctor.mapComp'_eq_mapComp {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) {b₀ b₁ b₂ : B} (f : b₀ ⟶ b₁) (g : b₁ ⟶ b₂) : F.mapComp' f g (CategoryStruct.comp f g) ⋯ = F.mapComp f g

def CategoryTheory.OplaxFunctor.comp {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] {D : Type u₃} [Bicategory D] (F : OplaxFunctor B C) (G : OplaxFunctor C D) : OplaxFunctor B D

Composition of oplax functors.

Equations

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

structure CategoryTheory.OplaxFunctor.PseudoCore {B : Type u₁} [Bicategory B] {C : Type u₂} [Bicategory C] (F : OplaxFunctor B C) : Type (max (max u₁ v₁) w₂)

A structure on an oplax functor that promotes an oplax functor to a pseudofunctor.

See Pseudofunctor.mkOfOplax.

• mapIdIso (a : B) : F.map (CategoryStruct.id a) ≅ CategoryStruct.id (F.obj a)

  The isomorphism giving rise to the oplax unity constraint

• mapCompIso {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : F.map (CategoryStruct.comp f g) ≅ CategoryStruct.comp (F.map f) (F.map g)

  The isomorphism giving rise to the oplax functoriality constraint

• mapIdIso_hom {a : B} : (self.mapIdIso a).hom = F.mapId a

  mapIdIso gives rise to the oplax unity constraint

• mapCompIso_hom {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : (self.mapCompIso f g).hom = F.mapComp f g

  mapCompIso gives rise to the oplax functoriality constraint