Codiscrete categories

We define Codiscrete A as an alias for the type A , and use this type alias to provide a Category instance whose Hom type are Unit types.

Codiscrete.functor promotes a function f : C → A (for any category C) to a functor f : C ⥤ Codiscrete A.

Similarly, Codiscrete.natTrans and Codiscrete.natIso promote I-indexed families of morphisms, or I-indexed families of isomorphisms to natural transformations or natural isomorphism.

We define functorToCat : Type u ⥤ Cat.{0,u} which sends a type to the codiscrete category and show it is right adjoint to Cat.objects.

structure CategoryTheory.Codiscrete (α : Type u) : Type u

A wrapper for promoting any type to a category, with a unique morphisms between any two objects of the category.

• as : α

  A wrapper for promoting any type to a category, with a unique morphisms between any two objects of the category.

theorem CategoryTheory.Codiscrete.ext {α : Type u} {x y : CategoryTheory.Codiscrete α} (as : x.as = y.as) : x = y

@[simp]

theorem CategoryTheory.Codiscrete.mk_as {α : Type u} (X : CategoryTheory.Codiscrete α) : { as := X.as } = X

def CategoryTheory.codiscreteEquiv {α : Type u} : CategoryTheory.Codiscrete α ≃ α

Codiscrete α is equivalent to the original type α.

Equations

• CategoryTheory.codiscreteEquiv = { toFun := CategoryTheory.Codiscrete.as, invFun := CategoryTheory.Codiscrete.mk, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem CategoryTheory.codiscreteEquiv_apply {α : Type u} (self : CategoryTheory.Codiscrete α) : CategoryTheory.codiscreteEquiv self = self.as

@[simp]

theorem CategoryTheory.codiscreteEquiv_symm_apply_as {α : Type u} (as : α) : (CategoryTheory.codiscreteEquiv.symm as).as = as

instance CategoryTheory.instDecidableEqCodiscrete {α : Type u} [DecidableEq α] : DecidableEq (CategoryTheory.Codiscrete α)

Equations

• CategoryTheory.instDecidableEqCodiscrete = CategoryTheory.codiscreteEquiv.decidableEq

instance CategoryTheory.Codiscrete.instCategory (A : Type u_1) : CategoryTheory.Category.{0, u_1} (CategoryTheory.Codiscrete A)

Equations

• CategoryTheory.Codiscrete.instCategory A = CategoryTheory.Category.mk ⋯ ⋯ ⋯

def CategoryTheory.Codiscrete.functor {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type w} (F : C → A) : CategoryTheory.Functor C (CategoryTheory.Codiscrete A)

Any function C → A lifts to a functor C ⥤ Codiscrete A.

Equations

• CategoryTheory.Codiscrete.functor F = { obj := CategoryTheory.Codiscrete.mk ∘ F, map := fun {X Y : C} (x : X ⟶ Y) => PUnit.unit, map_id := ⋯, map_comp := ⋯ }

def CategoryTheory.Codiscrete.invFunctor {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type w} (F : CategoryTheory.Functor C (CategoryTheory.Codiscrete A)) : C → A

The underlying function C → A of a functor C ⥤ Codiscrete A.

Equations

• CategoryTheory.Codiscrete.invFunctor F = CategoryTheory.Codiscrete.as ∘ F.obj

def CategoryTheory.Codiscrete.natTrans {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type w} {F G : CategoryTheory.Functor C (CategoryTheory.Codiscrete A)} : F ⟶ G

Given two functors to a codiscrete category, there is a trivial natural transformation.

Equations

• CategoryTheory.Codiscrete.natTrans = { app := fun (x : C) => PUnit.unit, naturality := ⋯ }

def CategoryTheory.Codiscrete.natIso {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type w} {F G : CategoryTheory.Functor C (CategoryTheory.Codiscrete A)} : F ≅ G

Given two functors into a codiscrete category, the trivial natural transformation is an natural isomorphism.

Equations

• CategoryTheory.Codiscrete.natIso = { hom := CategoryTheory.Codiscrete.natTrans, inv := CategoryTheory.Codiscrete.natTrans, hom_inv_id := ⋯, inv_hom_id := ⋯ }

def CategoryTheory.Codiscrete.natIsoFunctor {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type w} {F : CategoryTheory.Functor C (CategoryTheory.Codiscrete A)} : F ≅ CategoryTheory.Codiscrete.functor (CategoryTheory.Codiscrete.as ∘ F.obj)

Every functor F to a codiscrete category is naturally isomorphic {(actually, equal)} to Codiscrete.as ∘ F.obj.

Equations

• CategoryTheory.Codiscrete.natIsoFunctor = CategoryTheory.Iso.refl F

@[simp]

theorem CategoryTheory.Codiscrete.natIsoFunctor_inv_app {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type w} {F : CategoryTheory.Functor C (CategoryTheory.Codiscrete A)} (X : C) : CategoryTheory.Codiscrete.natIsoFunctor.inv.app X = CategoryTheory.CategoryStruct.id (F.obj X)

@[simp]

theorem CategoryTheory.Codiscrete.natIsoFunctor_hom_app {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type w} {F : CategoryTheory.Functor C (CategoryTheory.Codiscrete A)} (X : C) : CategoryTheory.Codiscrete.natIsoFunctor.hom.app X = CategoryTheory.CategoryStruct.id (F.obj X)

def CategoryTheory.Codiscrete.functorOfFun {A : Type u_1} {B : Type u_2} (f : A → B) : CategoryTheory.Functor (CategoryTheory.Codiscrete A) (CategoryTheory.Codiscrete B)

A function induces a functor between codiscrete categories.

Equations

• CategoryTheory.Codiscrete.functorOfFun f = CategoryTheory.Codiscrete.functor (f ∘ CategoryTheory.Codiscrete.as)

def CategoryTheory.Codiscrete.oppositeEquivalence (A : Type u_1) : (CategoryTheory.Codiscrete A)ᵒᵖ ≌ CategoryTheory.Codiscrete A

A codiscrete category is equivalent to its opposite category.

Equations

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

def CategoryTheory.Codiscrete.functorToCat : CategoryTheory.Functor (Type u) CategoryTheory.Cat

Codiscrete.functorToCat turns a type into a codiscrete category

Equations

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

def CategoryTheory.Codiscrete.equivFunctorToCodiscrete {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type w} : (C → A) ≃ CategoryTheory.Functor C (CategoryTheory.Codiscrete A)

For a category C and type A, there is an equivalence between functions objects.obj C ⟶ A and functors C ⥤ Codiscrete A.

Equations

• CategoryTheory.Codiscrete.equivFunctorToCodiscrete = { toFun := CategoryTheory.Codiscrete.functor, invFun := CategoryTheory.Codiscrete.invFunctor, left_inv := ⋯, right_inv := ⋯ }

def CategoryTheory.Codiscrete.adj : CategoryTheory.Cat.objects ⊣ CategoryTheory.Codiscrete.functorToCat

The functor that turns a type into a codiscrete category is right adjoint to the objects functor.

Equations

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

def CategoryTheory.Codiscrete.unitApp (C : Type u) [CategoryTheory.Category.{v, u} C] : CategoryTheory.Functor C (CategoryTheory.Codiscrete C)

Components of the unit of the adjunction Cat.objects ⊣ Codiscrete.functorToCat.

Equations

• CategoryTheory.Codiscrete.unitApp C = CategoryTheory.Codiscrete.functor id

def CategoryTheory.Codiscrete.counitApp (A : Type u) : CategoryTheory.Codiscrete A → A

Components of the counit of the adjunction Cat.objects ⊣ Codiscrete.functorToCat

Equations

• CategoryTheory.Codiscrete.counitApp A = CategoryTheory.Codiscrete.as

theorem CategoryTheory.Codiscrete.adj_unit_app (X : CategoryTheory.Cat) : CategoryTheory.Codiscrete.adj.unit.app X = CategoryTheory.Codiscrete.unitApp ↑X

theorem CategoryTheory.Codiscrete.adj_counit_app (A : Type u) : CategoryTheory.Codiscrete.adj.counit.app A = CategoryTheory.Codiscrete.counitApp A

theorem CategoryTheory.Codiscrete.left_triangle_components (C : Type u) [CategoryTheory.Category.{v, u} C] : CategoryTheory.Codiscrete.counitApp C ∘ (CategoryTheory.Codiscrete.unitApp C).obj = id

Left triangle equality of the adjunction Cat.objects ⊣ Codiscrete.functorToCat, as a universe polymorphic statement.

theorem CategoryTheory.Codiscrete.right_triangle_components (X : Type u) : (CategoryTheory.Codiscrete.unitApp (CategoryTheory.Codiscrete X)).comp (CategoryTheory.Codiscrete.functorOfFun (CategoryTheory.Codiscrete.counitApp X)) = CategoryTheory.Functor.id (CategoryTheory.Codiscrete X)

Right triangle equality of the adjunction Cat.objects ⊣ Codiscrete.functorToCat, stated using a composition of functors.