The category of bounded orders #

This defines BddOrd, the category of bounded orders.

source

structure BddOrdextends PartOrd :

Type (u_1 + 1)

The category of bounded orders with monotone functions.

carrier : Type u_1
str : PartialOrder ↑self.toPartOrd
isBoundedOrder : BoundedOrder ↑self.toPartOrd

Instances For

source

instance BddOrd.instCoeSortType :

CoeSort BddOrd (Type u_1)

Equations

source

@[reducible, inline]

abbrev BddOrd.of (X : Type u_1) [PartialOrder X] [BoundedOrder X] :

BddOrd

Construct a bundled BddOrd from the underlying type and typeclass.

Equations

Instances For

source

structure BddOrd.Hom (X Y : BddOrd) :

Type u

The type of morphisms in BddOrd R.

hom' : BoundedOrderHom ↑X.toPartOrd ↑Y.toPartOrd
The underlying BoundedOrderHom.

Instances For

source

theorem BddOrd.Hom.ext {X Y : BddOrd} {x y : X.Hom Y} (hom' : x.hom' = y.hom') :

x = y

source

theorem BddOrd.Hom.ext_iff {X Y : BddOrd} {x y : X.Hom Y} :

x = y ↔ x.hom' = y.hom'

source

instance BddOrd.instCategory :

CategoryTheory.Category.{u, u + 1} BddOrd

Equations

source

instance BddOrd.instConcreteCategoryBoundedOrderHomCarrier :

CategoryTheory.ConcreteCategory BddOrd fun (x1 x2 : BddOrd) => BoundedOrderHom ↑x1.toPartOrd ↑x2.toPartOrd

Equations

source

@[reducible, inline]

abbrev BddOrd.Hom.hom {X Y : BddOrd} (f : X.Hom Y) :

BoundedOrderHom ↑X.toPartOrd ↑Y.toPartOrd

Turn a morphism in BddOrd back into a BoundedOrderHom.

Equations

Instances For

source

@[reducible, inline]

abbrev BddOrd.ofHom {X Y : Type u} [PartialOrder X] [BoundedOrder X] [PartialOrder Y] [BoundedOrder Y] (f : BoundedOrderHom X Y) :

of X ⟶ of Y

Typecheck a BoundedOrderHom as a morphism in BddOrd.

Equations

Instances For

source

def BddOrd.Hom.Simps.hom (X Y : BddOrd) (f : X.Hom Y) :

BoundedOrderHom ↑X.toPartOrd ↑Y.toPartOrd

Use the ConcreteCategory.hom projection for @[simps] lemmas.

Equations

Instances For

The results below duplicate the ConcreteCategory simp lemmas, but we can keep them for dsimp.

source

@[simp]

theorem BddOrd.coe_id {X : BddOrd} :

⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) = id

source

@[simp]

theorem BddOrd.coe_comp {X Y Z : BddOrd} {f : X ⟶ Y} {g : Y ⟶ Z} :

⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.comp f g)) = ⇑(CategoryTheory.ConcreteCategory.hom g) ∘ ⇑(CategoryTheory.ConcreteCategory.hom f)

source

@[simp]

theorem BddOrd.forget_map {X Y : BddOrd} (f : X ⟶ Y) :

(CategoryTheory.forget BddOrd).map f = ⇑(CategoryTheory.ConcreteCategory.hom f)

source

theorem BddOrd.ext {X Y : BddOrd} {f g : X ⟶ Y} (w : ∀ (x : ↑X.toPartOrd), (CategoryTheory.ConcreteCategory.hom f) x = (CategoryTheory.ConcreteCategory.hom g) x) :

f = g

source

theorem BddOrd.ext_iff {X Y : BddOrd} {f g : X ⟶ Y} :

f = g ↔ ∀ (x : ↑X.toPartOrd), (CategoryTheory.ConcreteCategory.hom f) x = (CategoryTheory.ConcreteCategory.hom g) x

source

theorem BddOrd.coe_of (X : Type u) [PartialOrder X] [BoundedOrder X] :

↑(of X).toPartOrd = X

source

@[simp]

theorem BddOrd.hom_id {X : BddOrd} :

Hom.hom (CategoryTheory.CategoryStruct.id X) = BoundedOrderHom.id ↑X.toPartOrd

source

theorem BddOrd.id_apply (X : BddOrd) (x : ↑X.toPartOrd) :

(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) x = x

source

@[simp]

theorem BddOrd.hom_comp {X Y Z : BddOrd} (f : X ⟶ Y) (g : Y ⟶ Z) :

Hom.hom (CategoryTheory.CategoryStruct.comp f g) = (Hom.hom g).comp (Hom.hom f)

source

theorem BddOrd.comp_apply {X Y Z : BddOrd} (f : X ⟶ Y) (g : Y ⟶ Z) (x : ↑X.toPartOrd) :

(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.comp f g)) x = (CategoryTheory.ConcreteCategory.hom g) ((CategoryTheory.ConcreteCategory.hom f) x)

source

theorem BddOrd.hom_ext {X Y : BddOrd} {f g : X ⟶ Y} (hf : Hom.hom f = Hom.hom g) :

f = g

source

theorem BddOrd.hom_ext_iff {X Y : BddOrd} {f g : X ⟶ Y} :

f = g ↔ Hom.hom f = Hom.hom g

source

@[simp]

theorem BddOrd.hom_ofHom {X Y : Type u} [PartialOrder X] [BoundedOrder X] [PartialOrder Y] [BoundedOrder Y] (f : BoundedOrderHom X Y) :

Hom.hom (ofHom f) = f

source

@[simp]

theorem BddOrd.ofHom_hom {X Y : BddOrd} (f : X ⟶ Y) :

ofHom (Hom.hom f) = f

source

@[simp]

theorem BddOrd.ofHom_id {X : Type u} [PartialOrder X] [BoundedOrder X] :

ofHom (BoundedOrderHom.id X) = CategoryTheory.CategoryStruct.id (of X)

source

@[simp]

theorem BddOrd.ofHom_comp {X Y Z : Type u} [PartialOrder X] [BoundedOrder X] [PartialOrder Y] [BoundedOrder Y] [PartialOrder Z] [BoundedOrder Z] (f : BoundedOrderHom X Y) (g : BoundedOrderHom Y Z) :

ofHom (g.comp f) = CategoryTheory.CategoryStruct.comp (ofHom f) (ofHom g)

source

theorem BddOrd.ofHom_apply {X Y : Type u} [PartialOrder X] [BoundedOrder X] [PartialOrder Y] [BoundedOrder Y] (f : BoundedOrderHom X Y) (x : X) :

(CategoryTheory.ConcreteCategory.hom (ofHom f)) x = f x

source

theorem BddOrd.inv_hom_apply {X Y : BddOrd} (e : X ≅ Y) (x : ↑X.toPartOrd) :

(CategoryTheory.ConcreteCategory.hom e.inv) ((CategoryTheory.ConcreteCategory.hom e.hom) x) = x

source

theorem BddOrd.hom_inv_apply {X Y : BddOrd} (e : X ≅ Y) (s : ↑Y.toPartOrd) :

(CategoryTheory.ConcreteCategory.hom e.hom) ((CategoryTheory.ConcreteCategory.hom e.inv) s) = s

source

instance BddOrd.instInhabited :

Inhabited BddOrd

Equations

source

instance BddOrd.hasForgetToPartOrd :

CategoryTheory.HasForget₂ BddOrd PartOrd

Equations

source

instance BddOrd.hasForgetToBipointed :

CategoryTheory.HasForget₂ BddOrd Bipointed

Equations

source

def BddOrd.dual :

CategoryTheory.Functor BddOrd BddOrd

OrderDual as a functor.

Equations

Instances For

source

@[simp]

theorem BddOrd.dual_map {X✝ Y✝ : BddOrd} (f : X✝ ⟶ Y✝) :

dual.map f = ofHom (BoundedOrderHom.dual (Hom.hom f))

source

def BddOrd.Iso.mk {α β : BddOrd} (e : ↑α.toPartOrd ≃o ↑β.toPartOrd) :

α ≅ β

Constructs an equivalence between bounded orders from an order isomorphism between them.

Equations

Instances For

source

@[simp]

theorem BddOrd.Iso.mk_inv {α β : BddOrd} (e : ↑α.toPartOrd ≃o ↑β.toPartOrd) :

(mk e).inv = ofHom ↑e.symm

source

@[simp]

theorem BddOrd.Iso.mk_hom {α β : BddOrd} (e : ↑α.toPartOrd ≃o ↑β.toPartOrd) :

(mk e).hom = ofHom ↑e

source

def BddOrd.dualEquiv :

BddOrd ≌ BddOrd

The equivalence between BddOrd and itself induced by OrderDual both ways.

Equations

Instances For

source

@[simp]

theorem BddOrd.dualEquiv_inverse :

dualEquiv.inverse = dual

source

@[simp]

theorem BddOrd.dualEquiv_functor :

dualEquiv.functor = dual

source

theorem bddOrd_dual_comp_forget_to_partOrd :

BddOrd.dual.comp (CategoryTheory.forget₂ BddOrd PartOrd) = (CategoryTheory.forget₂ BddOrd PartOrd).comp PartOrd.dual

source

theorem bddOrd_dual_comp_forget_to_bipointed :

BddOrd.dual.comp (CategoryTheory.forget₂ BddOrd Bipointed) = (CategoryTheory.forget₂ BddOrd Bipointed).comp Bipointed.swap

Documentation

Mathlib.Order.Category.BddOrd

The category of bounded orders #