Strict bicategories

A bicategory is called Strict if the left unitors, the right unitors, and the associators are isomorphisms given by equalities.

Implementation notes

In the literature of category theory, a strict bicategory (usually called a strict 2-category) is often defined as a bicategory whose left unitors, right unitors, and associators are identities. We cannot use this definition directly here since the types of 2-morphisms depend on 1-morphisms. For this reason, we use eqToIso, which gives isomorphisms from equalities, instead of identities.

class CategoryTheory.Bicategory.Strict (B : Type u) [CategoryTheory.Bicategory B] : Prop

• id_comp : ∀ {a b : B} (f : a ⟶ b), CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id a) f = f

  Identity morphisms are left identities for composition.

• comp_id : ∀ {a b : B} (f : a ⟶ b), CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.id b) = f

  Identity morphisms are right identities for composition.

• assoc : ∀ {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d), CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f g) h = CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp g h)

  Composition in a bicategory is associative.

• leftUnitor_eqToIso : ∀ {a b : B} (f : a ⟶ b), CategoryTheory.Bicategory.leftUnitor f = CategoryTheory.eqToIso (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id a) f = f)

  The left unitors are given by equalities

• rightUnitor_eqToIso : ∀ {a b : B} (f : a ⟶ b), CategoryTheory.Bicategory.rightUnitor f = CategoryTheory.eqToIso (_ : CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.id b) = f)

  The right unitors are given by equalities

• associator_eqToIso : ∀ {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d), CategoryTheory.Bicategory.associator f g h = CategoryTheory.eqToIso (_ : CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f g) h = CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp g h))

  The associators are given by equalities

A bicategory is called Strict if the left unitors, the right unitors, and the associators are isomorphisms given by equalities.

instance CategoryTheory.StrictBicategory.category (B : Type u) [CategoryTheory.Bicategory B] [CategoryTheory.Bicategory.Strict B] : CategoryTheory.Category.{v, u} B

Category structure on a strict bicategory

Equations

• CategoryTheory.StrictBicategory.category B = CategoryTheory.Category.mk

@[simp]

theorem CategoryTheory.Bicategory.whiskerLeft_eqToHom {B : Type u} [CategoryTheory.Bicategory B] {a : B} {b : B} {c : B} (f : a ⟶ b) {g : b ⟶ c} {h : b ⟶ c} (η : g = h) : CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.eqToHom η) = CategoryTheory.eqToHom (_ : CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp f h)

@[simp]

theorem CategoryTheory.Bicategory.eqToHom_whiskerRight {B : Type u} [CategoryTheory.Bicategory B] {a : B} {b : B} {c : B} {f : a ⟶ b} {g : a ⟶ b} (η : f = g) (h : b ⟶ c) : CategoryTheory.Bicategory.whiskerRight (CategoryTheory.eqToHom η) h = CategoryTheory.eqToHom (_ : CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g h)