Category of linear orders #

This defines LinOrd, the category of linear orders with monotone maps.

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structure LinOrd :

Type (u_1 + 1)

The category of linear orders.

carrier : Type u_1
The underlying linearly ordered type.
str : LinearOrder ↑self

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instance LinOrd.instCoeSortType :

CoeSort LinOrd (Type u_1)

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@[reducible, inline]

abbrev LinOrd.of (X : Type u_1) [LinearOrder X] :

LinOrd

Construct a bundled LinOrd from the underlying type and typeclass.

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structure LinOrd.Hom (X Y : LinOrd) :

Type u

The type of morphisms in LinOrd R.

hom' : ↑X →o ↑Y
The underlying OrderHom.

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

x = y ↔ x.hom' = y.hom'

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

x = y

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instance LinOrd.instCategory :

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

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instance LinOrd.instConcreteCategoryOrderHomCarrier :

CategoryTheory.ConcreteCategory LinOrd fun (x1 x2 : LinOrd) => ↑x1 →o ↑x2

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@[reducible, inline]

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

↑X →o ↑Y

Turn a morphism in LinOrd back into a OrderHom.

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@[reducible, inline]

abbrev LinOrd.ofHom {X Y : Type u} [LinearOrder X] [LinearOrder Y] (f : X →o Y) :

of X ⟶ of Y

Typecheck a OrderHom as a morphism in LinOrd.

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

↑X →o ↑Y

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

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

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@[simp]

theorem LinOrd.coe_id {X : LinOrd} :

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

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@[simp]

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

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

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@[simp]

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

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

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

f = g

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

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

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theorem LinOrd.coe_of (X : Type u) [LinearOrder X] :

↑(of X) = X

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@[simp]

theorem LinOrd.hom_id {X : LinOrd} :

Hom.hom (CategoryTheory.CategoryStruct.id X) = OrderHom.id

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theorem LinOrd.id_apply (X : LinOrd) (x : ↑X) :

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

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@[simp]

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

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

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

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

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

f = g

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

f = g ↔ Hom.hom f = Hom.hom g

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@[simp]

theorem LinOrd.hom_ofHom {X Y : Type u} [LinearOrder X] [LinearOrder Y] (f : X →o Y) :

Hom.hom (ofHom f) = f

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@[simp]

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

ofHom (Hom.hom f) = f

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@[simp]

theorem LinOrd.ofHom_id {X : Type u} [LinearOrder X] :

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

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@[simp]

theorem LinOrd.ofHom_comp {X Y Z : Type u} [LinearOrder X] [LinearOrder Y] [LinearOrder Z] (f : X →o Y) (g : Y →o Z) :

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

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theorem LinOrd.ofHom_apply {X Y : Type u} [LinearOrder X] [LinearOrder Y] (f : X →o Y) (x : X) :

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

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

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

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

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

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instance LinOrd.instInhabited :

Inhabited LinOrd

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instance LinOrd.hasForgetToLat :

CategoryTheory.HasForget₂ LinOrd Lat

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

α ≅ β

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

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@[simp]

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

(mk e).inv = ofHom ↑e.symm

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@[simp]

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

(mk e).hom = ofHom ↑e

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def LinOrd.dual :

CategoryTheory.Functor LinOrd LinOrd

OrderDual as a functor.

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@[simp]

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

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

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def LinOrd.dualEquiv :

LinOrd ≌ LinOrd

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

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@[simp]

theorem LinOrd.dualEquiv_inverse :

dualEquiv.inverse = dual

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@[simp]

theorem LinOrd.dualEquiv_functor :

dualEquiv.functor = dual

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theorem linOrd_dual_comp_forget_to_Lat :

LinOrd.dual.comp (CategoryTheory.forget₂ LinOrd Lat) = (CategoryTheory.forget₂ LinOrd Lat).comp Lat.dual

Documentation

Mathlib.Order.Category.LinOrd

Category of linear orders #