Locally discrete bicategories

A category C can be promoted to a strict bicategory LocallyDiscrete C. The objects and the 1-morphisms in LocallyDiscrete C are the same as the objects and the morphisms, respectively, in C, and the 2-morphisms in LocallyDiscrete C are the equalities between 1-morphisms. In other words, the category consisting of the 1-morphisms between each pair of objects X and Y in LocallyDiscrete C is defined as the discrete category associated with the type X ⟶ Y.

structure CategoryTheory.LocallyDiscrete (C : Type u) : Type u

A wrapper for promoting any category to a bicategory, with the only 2-morphisms being equalities.

• as : C

  A wrapper for promoting any category to a bicategory, with the only 2-morphisms being equalities.

theorem CategoryTheory.LocallyDiscrete.ext_iff {C : Type u} {x y : LocallyDiscrete C} : x = y ↔ x.as = y.as

theorem CategoryTheory.LocallyDiscrete.ext {C : Type u} {x y : LocallyDiscrete C} (as : x.as = y.as) : x = y

@[simp]

theorem CategoryTheory.LocallyDiscrete.mk_as {C : Type u} (a : LocallyDiscrete C) : { as := a.as } = a

def CategoryTheory.LocallyDiscrete.locallyDiscreteEquiv {C : Type u} : LocallyDiscrete C ≃ C

LocallyDiscrete C is equivalent to the original type C.

@[simp]

theorem CategoryTheory.LocallyDiscrete.locallyDiscreteEquiv_apply {C : Type u} (self : LocallyDiscrete C) : locallyDiscreteEquiv self = self.as

@[simp]

theorem CategoryTheory.LocallyDiscrete.locallyDiscreteEquiv_symm_apply_as {C : Type u} (as : C) : (locallyDiscreteEquiv.symm as).as = as

@[implicit_reducible]

instance CategoryTheory.LocallyDiscrete.instDecidableEq {C : Type u} [DecidableEq C] : DecidableEq (LocallyDiscrete C)

@[implicit_reducible]

instance CategoryTheory.LocallyDiscrete.instInhabited {C : Type u} [Inhabited C] : Inhabited (LocallyDiscrete C)

@[implicit_reducible]

instance CategoryTheory.LocallyDiscrete.categoryStruct {C : Type u} [CategoryStruct.{v, u} C] : CategoryStruct.{v, u} (LocallyDiscrete C)

@[simp]

theorem CategoryTheory.LocallyDiscrete.id_as {C : Type u} [CategoryStruct.{v, u} C] (a : LocallyDiscrete C) : (CategoryStruct.id a).as = CategoryStruct.id a.as

@[simp]

theorem CategoryTheory.LocallyDiscrete.comp_as {C : Type u} [CategoryStruct.{v, u} C] {a b c : LocallyDiscrete C} (f : a ⟶ b) (g : b ⟶ c) : (CategoryStruct.comp f g).as = CategoryStruct.comp f.as g.as

@[implicit_reducible, instance 900]

instance CategoryTheory.LocallyDiscrete.homSmallCategory {C : Type u} [CategoryStruct.{v, u} C] (a b : LocallyDiscrete C) : SmallCategory (a ⟶ b)

instance CategoryTheory.LocallyDiscrete.subsingleton2Hom {C : Type u} [CategoryStruct.{v, u} C] {a b : LocallyDiscrete C} (f g : a ⟶ b) : Subsingleton (f ⟶ g)

This instance is used to see through the synonym a ⟶ b = Discrete (a.as ⟶ b.as).

theorem CategoryTheory.LocallyDiscrete.eq_of_hom {C : Type u} [CategoryStruct.{v, u} C] {X Y : LocallyDiscrete C} {f g : X ⟶ Y} (η : f ⟶ g) : f = g

Extract the equation from a 2-morphism in a locally discrete 2-category.

@[implicit_reducible]

instance CategoryTheory.locallyDiscreteBicategory (C : Type u) [Category.{v, u} C] : Bicategory (LocallyDiscrete C)

The locally discrete bicategory on a category is a bicategory in which the objects and the 1-morphisms are the same as those in the underlying category, and the 2-morphisms are the equalities between 1-morphisms.

instance CategoryTheory.locallyDiscreteBicategory.strict (C : Type u) [Category.{v, u} C] : Bicategory.Strict (LocallyDiscrete C)

A locally discrete bicategory is strict.

@[reducible, inline]

abbrev CategoryTheory.Bicategory.IsLocallyDiscrete (B : Type u_1) [Bicategory B] : Prop

A bicategory is locally discrete if the categories of 1-morphisms are discrete.

instance CategoryTheory.Bicategory.instIsLocallyDiscreteLocallyDiscrete (C : Type u_1) [Category.{v_1, u_1} C] : IsLocallyDiscrete (LocallyDiscrete C)

instance CategoryTheory.Bicategory.instStrictOfIsLocallyDiscrete (B : Type u_1) [Bicategory B] [IsLocallyDiscrete B] : Strict B

def Quiver.Hom.toLoc {C : Type u} [CategoryTheory.CategoryStruct.{v, u} C] {a b : C} (f : a ⟶ b) : { as := a } ⟶ { as := b }

The 1-morphism in LocallyDiscrete C associated to a given morphism f : a ⟶ b in C

@[simp]

theorem Quiver.Hom.toLoc_as {C : Type u} [CategoryTheory.CategoryStruct.{v, u} C] {a b : C} (f : a ⟶ b) : f.toLoc.as = f

@[simp]

theorem Quiver.Hom.id_toLoc {C : Type u} [CategoryTheory.CategoryStruct.{v, u} C] (a : C) : (CategoryTheory.CategoryStruct.id a).toLoc = CategoryTheory.CategoryStruct.id { as := a }

@[simp]

theorem Quiver.Hom.comp_toLoc {C : Type u} [CategoryTheory.CategoryStruct.{v, u} C] {a b c : C} (f : a ⟶ b) (g : b ⟶ c) : (CategoryTheory.CategoryStruct.comp f g).toLoc = CategoryTheory.CategoryStruct.comp f.toLoc g.toLoc

@[simp]

theorem CategoryTheory.LocallyDiscrete.eqToHom_toLoc {C : Type u} [Category.{v, u} C] {a b : C} (h : a = b) : (eqToHom h).toLoc = eqToHom ⋯

theorem CategoryTheory.CommSq.toLoc {C : Type u_1} [Category.{u_2, u_1} C] {X₁ X₂ X₃ X₄ : C} {t : X₁ ⟶ X₂} {l : X₁ ⟶ X₃} {r : X₂ ⟶ X₄} {b : X₃ ⟶ X₄} (h : CommSq t l r b) : CommSq t.toLoc l.toLoc r.toLoc b.toLoc