Category of preorders

This defines Preord, the category of preorders with monotone maps.

structure Preord : Type (u_1 + 1)

The category of preorders.

• carrier : Type u_1

  The underlying preordered type.

• str : Preorder ↑self

instance Preord.instCoeSortType : CoeSort Preord (Type u)

@[reducible, inline]

abbrev Preord.of (X : Type u) [Preorder X] : Preord

Construct a bundled Preord from the underlying type and typeclass.

structure Preord.Hom (X Y : Preord) : Type u

The type of morphisms in Preord R.

• hom' : ↑X →o ↑Y

  The underlying OrderHom.

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

theorem Preord.Hom.ext_iff {X Y : Preord} {x y : X.Hom Y} : x = y ↔ x.hom' = y.hom'

instance Preord.instCategory : CategoryTheory.Category.{u, u + 1} Preord

instance Preord.instConcreteCategoryOrderHomCarrier : CategoryTheory.ConcreteCategory Preord fun (x1 x2 : Preord) => ↑x1 →o ↑x2

@[reducible, inline]

abbrev Preord.Hom.hom {X Y : Preord} (f : X.Hom Y) : ↑X →o ↑Y

Turn a morphism in Preord back into a OrderHom.

@[reducible, inline]

abbrev Preord.ofHom {X Y : Type u} [Preorder X] [Preorder Y] (f : X →o Y) : of X ⟶ of Y

Typecheck a OrderHom as a morphism in Preord.

def Preord.Hom.Simps.hom (X Y : Preord) (f : X.Hom Y) : ↑X →o ↑Y

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

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

@[simp]

theorem Preord.coe_id {X : Preord} : ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) = id

@[simp]

theorem Preord.coe_comp {X Y Z : Preord} {f : X ⟶ Y} {g : Y ⟶ Z} : ⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.comp f g)) = ⇑(CategoryTheory.ConcreteCategory.hom g) ∘ ⇑(CategoryTheory.ConcreteCategory.hom f)

@[simp]

theorem Preord.forget_map {X Y : Preord} (f : X ⟶ Y) : (CategoryTheory.forget Preord).map f = ⇑(CategoryTheory.ConcreteCategory.hom f)

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

theorem Preord.ext_iff {X Y : Preord} {f g : X ⟶ Y} : f = g ↔ ∀ (x : ↑X), (CategoryTheory.ConcreteCategory.hom f) x = (CategoryTheory.ConcreteCategory.hom g) x

theorem Preord.coe_of (X : Type u) [Preorder X] : ↑(of X) = X

@[simp]

theorem Preord.hom_id {X : Preord} : Hom.hom (CategoryTheory.CategoryStruct.id X) = OrderHom.id

theorem Preord.id_apply (X : Preord) (x : ↑X) : (CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) x = x

@[simp]

theorem Preord.hom_comp {X Y Z : Preord} (f : X ⟶ Y) (g : Y ⟶ Z) : Hom.hom (CategoryTheory.CategoryStruct.comp f g) = (Hom.hom g).comp (Hom.hom f)

theorem Preord.comp_apply {X Y Z : Preord} (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)

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

theorem Preord.hom_ext_iff {X Y : Preord} {f g : X ⟶ Y} : f = g ↔ Hom.hom f = Hom.hom g

@[simp]

theorem Preord.hom_ofHom {X Y : Type u} [Preorder X] [Preorder Y] (f : X →o Y) : Hom.hom (ofHom f) = f

@[simp]

theorem Preord.ofHom_hom {X Y : Preord} (f : X ⟶ Y) : ofHom (Hom.hom f) = f

@[simp]

theorem Preord.ofHom_id {X : Type u} [Preorder X] : ofHom OrderHom.id = CategoryTheory.CategoryStruct.id (of X)

@[simp]

theorem Preord.ofHom_comp {X Y Z : Type u} [Preorder X] [Preorder Y] [Preorder Z] (f : X →o Y) (g : Y →o Z) : ofHom (g.comp f) = CategoryTheory.CategoryStruct.comp (ofHom f) (ofHom g)

theorem Preord.ofHom_apply {X Y : Type u} [Preorder X] [Preorder Y] (f : X →o Y) (x : X) : (CategoryTheory.ConcreteCategory.hom (ofHom f)) x = f x

theorem Preord.inv_hom_apply {X Y : Preord} (e : X ≅ Y) (x : ↑X) : (CategoryTheory.ConcreteCategory.hom e.inv) ((CategoryTheory.ConcreteCategory.hom e.hom) x) = x

theorem Preord.hom_inv_apply {X Y : Preord} (e : X ≅ Y) (s : ↑Y) : (CategoryTheory.ConcreteCategory.hom e.hom) ((CategoryTheory.ConcreteCategory.hom e.inv) s) = s

instance Preord.instInhabited : Inhabited Preord

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

Constructs an equivalence between preorders from an order isomorphism between them.

@[simp]

theorem Preord.Iso.mk_inv {α β : Preord} (e : ↑α ≃o ↑β) : (mk e).inv = ofHom ↑e.symm

@[simp]

theorem Preord.Iso.mk_hom {α β : Preord} (e : ↑α ≃o ↑β) : (mk e).hom = ofHom ↑e

def Preord.dual : CategoryTheory.Functor Preord Preord

OrderDual as a functor.

@[simp]

theorem Preord.dual_map {X✝ Y✝ : Preord} (f : X✝ ⟶ Y✝) : dual.map f = ofHom (OrderHom.dual (Hom.hom f))

def Preord.dualEquiv : Preord ≌ Preord

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

@[simp]

theorem Preord.dualEquiv_inverse : dualEquiv.inverse = dual

@[simp]

theorem Preord.dualEquiv_functor : dualEquiv.functor = dual

def preordToCat : CategoryTheory.Functor Preord CategoryTheory.Cat

The embedding of Preord into Cat.

@[simp]

theorem preordToCat_obj (X : Preord) : preordToCat.obj X = CategoryTheory.Cat.of ↑X

@[simp]

theorem preordToCat_map {X✝ Y✝ : Preord} (f : X✝ ⟶ Y✝) : preordToCat.map f = ⋯.functor

instance instFaithfulPreordCatPreordToCat : preordToCat.Faithful

instance instFullPreordCatPreordToCat : preordToCat.Full