Esakia morphisms

This file defines pseudo-epimorphisms and Esakia morphisms.

We use the FunLike design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.

Types of morphisms

• PseudoEpimorphism: Pseudo-epimorphisms. Maps f such that f a ≤ b implies the existence of a' such that a ≤ a' and f a' = b.
• EsakiaHom: Esakia morphisms. Continuous pseudo-epimorphisms.

Typeclasses

• PseudoEpimorphismClass
• EsakiaHomClass

References

• Wikipedia, Esakia space

structure PseudoEpimorphism (α : Type u_6) (β : Type u_7) [Preorder α] [Preorder β] extends OrderHom : Type (max u_6 u_7)

• toFun : α → β
• monotone' : Monotone s.toFun
• exists_map_eq_of_map_le' : ∀ ⦃a : α⦄ ⦃b : β⦄, OrderHom.toFun s.toOrderHom a ≤ b → ∃ c, a ≤ c ∧ OrderHom.toFun s.toOrderHom c = b

The type of pseudo-epimorphisms, aka p-morphisms, aka bounded maps, from α to β.

structure EsakiaHom (α : Type u_6) (β : Type u_7) [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] extends ContinuousOrderHom : Type (max u_6 u_7)

• toFun : α → β
• monotone' : Monotone s.toFun
• continuous_toFun : Continuous s.toFun
• exists_map_eq_of_map_le' : ∀ ⦃a : α⦄ ⦃b : β⦄, OrderHom.toFun s.toOrderHom a ≤ b → ∃ c, a ≤ c ∧ OrderHom.toFun s.toOrderHom c = b

The type of Esakia morphisms, aka continuous pseudo-epimorphisms, from α to β.

class PseudoEpimorphismClass (F : Type u_6) (α : outParam (Type u_7)) (β : outParam (Type u_8)) [Preorder α] [Preorder β] extends RelHomClass : Type (max (max u_6 u_7) u_8)

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_rel : ∀ (f : F) {a b : α}, a ≤ b → ↑f a ≤ ↑f b
• exists_map_eq_of_map_le : ∀ (f : F) ⦃a : α⦄ ⦃b : β⦄, ↑f a ≤ b → ∃ c, a ≤ c ∧ ↑f c = b

PseudoEpimorphismClass F α β states that F is a type of ⊔-preserving morphisms.

You should extend this class when you extend PseudoEpimorphism.

class EsakiaHomClass (F : Type u_6) (α : outParam (Type u_7)) (β : outParam (Type u_8)) [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] extends ContinuousOrderHomClass : Type (max (max u_6 u_7) u_8)

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_continuous : ∀ (f : F), Continuous ↑f
• map_monotone : ∀ (f : F), Monotone ↑f
• exists_map_eq_of_map_le : ∀ (f : F) ⦃a : α⦄ ⦃b : β⦄, ↑f a ≤ b → ∃ c, a ≤ c ∧ ↑f c = b

EsakiaHomClass F α β states that F is a type of lattice morphisms.

You should extend this class when you extend EsakiaHom.

instance PseudoEpimorphismClass.toTopHomClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [PartialOrder α] [OrderTop α] [Preorder β] [OrderTop β] [PseudoEpimorphismClass F α β] : TopHomClass F α β

instance OrderIsoClass.toPseudoEpimorphismClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [OrderIsoClass F α β] : PseudoEpimorphismClass F α β

instance EsakiaHomClass.toPseudoEpimorphismClass {F : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [EsakiaHomClass F α β] : PseudoEpimorphismClass F α β

instance instCoeTCPseudoEpimorphism {F : Type u_1} {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [PseudoEpimorphismClass F α β] : CoeTC F (PseudoEpimorphism α β)

instance instCoeTCEsakiaHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [EsakiaHomClass F α β] : CoeTC F (EsakiaHom α β)

Pseudo-epimorphisms

instance PseudoEpimorphism.instPseudoEpimorphismClassPseudoEpimorphism {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] : PseudoEpimorphismClass (PseudoEpimorphism α β) α β

instance PseudoEpimorphism.instCoeFunPseudoEpimorphismForAll {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] : CoeFun (PseudoEpimorphism α β) fun x => α → β

Helper instance for when there's too many metavariables to apply FunLike.hasCoeToFun directly.

@[simp]

theorem PseudoEpimorphism.toOrderHom_eq_coe {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : PseudoEpimorphism α β) : ↑f.toOrderHom = ↑f

theorem PseudoEpimorphism.toFun_eq_coe {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] {f : PseudoEpimorphism α β} : f.toFun = ↑f

theorem PseudoEpimorphism.ext {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] {f : PseudoEpimorphism α β} {g : PseudoEpimorphism α β} (h : ∀ (a : α), ↑f a = ↑g a) : f = g

def PseudoEpimorphism.copy {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = ↑f) : PseudoEpimorphism α β

Copy of a PseudoEpimorphism with a new toFun equal to the old one. Useful to fix definitional equalities.

@[simp]

theorem PseudoEpimorphism.coe_copy {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = ↑f) : ↑(PseudoEpimorphism.copy f f' h) = f'

theorem PseudoEpimorphism.copy_eq {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = ↑f) : PseudoEpimorphism.copy f f' h = f

def PseudoEpimorphism.id (α : Type u_2) [Preorder α] : PseudoEpimorphism α α

id as a PseudoEpimorphism.

instance PseudoEpimorphism.instInhabitedPseudoEpimorphism (α : Type u_2) [Preorder α] : Inhabited (PseudoEpimorphism α α)

@[simp]

theorem PseudoEpimorphism.coe_id (α : Type u_2) [Preorder α] : ↑(PseudoEpimorphism.id α) = id

@[simp]

theorem PseudoEpimorphism.coe_id_orderHom (α : Type u_2) [Preorder α] : ↑(PseudoEpimorphism.id α) = OrderHom.id

@[simp]

theorem PseudoEpimorphism.id_apply {α : Type u_2} [Preorder α] (a : α) : ↑(PseudoEpimorphism.id α) a = a

def PseudoEpimorphism.comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) : PseudoEpimorphism α γ

Composition of PseudoEpimorphisms as a PseudoEpimorphism.

@[simp]

theorem PseudoEpimorphism.coe_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) : ↑(PseudoEpimorphism.comp g f) = ↑g ∘ ↑f

@[simp]

theorem PseudoEpimorphism.coe_comp_orderHom {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) : ↑(PseudoEpimorphism.comp g f) = OrderHom.comp ↑g ↑f

@[simp]

theorem PseudoEpimorphism.comp_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) (a : α) : ↑(PseudoEpimorphism.comp g f) a = ↑g (↑f a)

@[simp]

theorem PseudoEpimorphism.comp_assoc {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [Preorder α] [Preorder β] [Preorder γ] [Preorder δ] (h : PseudoEpimorphism γ δ) (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) : PseudoEpimorphism.comp (PseudoEpimorphism.comp h g) f = PseudoEpimorphism.comp h (PseudoEpimorphism.comp g f)

@[simp]

theorem PseudoEpimorphism.comp_id {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : PseudoEpimorphism α β) : PseudoEpimorphism.comp f (PseudoEpimorphism.id α) = f

@[simp]

theorem PseudoEpimorphism.id_comp {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : PseudoEpimorphism α β) : PseudoEpimorphism.comp (PseudoEpimorphism.id β) f = f

@[simp]

theorem PseudoEpimorphism.cancel_right {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] {g₁ : PseudoEpimorphism β γ} {g₂ : PseudoEpimorphism β γ} {f : PseudoEpimorphism α β} (hf : Function.Surjective ↑f) : PseudoEpimorphism.comp g₁ f = PseudoEpimorphism.comp g₂ f ↔ g₁ = g₂

@[simp]

theorem PseudoEpimorphism.cancel_left {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] {g : PseudoEpimorphism β γ} {f₁ : PseudoEpimorphism α β} {f₂ : PseudoEpimorphism α β} (hg : Function.Injective ↑g) : PseudoEpimorphism.comp g f₁ = PseudoEpimorphism.comp g f₂ ↔ f₁ = f₂

Esakia morphisms

def EsakiaHom.toPseudoEpimorphism {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : EsakiaHom α β) : PseudoEpimorphism α β

instance EsakiaHom.instEsakiaHomClassEsakiaHom {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] : EsakiaHomClass (EsakiaHom α β) α β

instance EsakiaHom.instCoeFunEsakiaHomForAll {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] : CoeFun (EsakiaHom α β) fun x => α → β

Helper instance for when there's too many metavariables to apply FunLike.hasCoeToFun directly.

@[simp]

theorem EsakiaHom.toContinuousOrderHom_coe {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] {f : EsakiaHom α β} : ↑f.toContinuousOrderHom = ↑f

theorem EsakiaHom.toFun_eq_coe {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] {f : EsakiaHom α β} : f.toFun = ↑f

theorem EsakiaHom.ext {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] {f : EsakiaHom α β} {g : EsakiaHom α β} (h : ∀ (a : α), ↑f a = ↑g a) : f = g

def EsakiaHom.copy {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : EsakiaHom α β) (f' : α → β) (h : f' = ↑f) : EsakiaHom α β

Copy of an EsakiaHom with a new toFun equal to the old one. Useful to fix definitional equalities.

@[simp]

theorem EsakiaHom.coe_copy {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : EsakiaHom α β) (f' : α → β) (h : f' = ↑f) : ↑(EsakiaHom.copy f f' h) = f'

theorem EsakiaHom.copy_eq {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : EsakiaHom α β) (f' : α → β) (h : f' = ↑f) : EsakiaHom.copy f f' h = f

def EsakiaHom.id (α : Type u_2) [TopologicalSpace α] [Preorder α] : EsakiaHom α α

id as an EsakiaHom.

instance EsakiaHom.instInhabitedEsakiaHom (α : Type u_2) [TopologicalSpace α] [Preorder α] : Inhabited (EsakiaHom α α)

@[simp]

theorem EsakiaHom.coe_id (α : Type u_2) [TopologicalSpace α] [Preorder α] : ↑(EsakiaHom.id α) = id

@[simp]

theorem EsakiaHom.coe_id_pseudoEpimorphism (α : Type u_2) [TopologicalSpace α] [Preorder α] : { toOrderHom := ↑(EsakiaHom.id α), exists_map_eq_of_map_le' := (_ : ∀ ⦃a b : α⦄, ↑(EsakiaHom.id α) a ≤ b → ∃ c, a ≤ c ∧ ↑(EsakiaHom.id α) c = b) } = PseudoEpimorphism.id α

@[simp]

theorem EsakiaHom.id_apply {α : Type u_2} [TopologicalSpace α] [Preorder α] (a : α) : ↑(EsakiaHom.id α) a = a

@[simp]

theorem EsakiaHom.coe_id_continuousOrderHom {α : Type u_2} [TopologicalSpace α] [Preorder α] : ↑(EsakiaHom.id α) = ContinuousOrderHom.id α

def EsakiaHom.comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] (g : EsakiaHom β γ) (f : EsakiaHom α β) : EsakiaHom α γ

Composition of EsakiaHoms as an EsakiaHom.

@[simp]

theorem EsakiaHom.coe_comp_continuousOrderHom {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] (g : EsakiaHom β γ) (f : EsakiaHom α β) : ↑(EsakiaHom.comp g f) = ContinuousOrderHom.comp ↑g ↑f

@[simp]

theorem EsakiaHom.coe_comp_pseudoEpimorphism {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] (g : EsakiaHom β γ) (f : EsakiaHom α β) : { toOrderHom := ↑(EsakiaHom.comp g f), exists_map_eq_of_map_le' := (_ : ∀ ⦃a : α⦄ ⦃b : γ⦄, ↑(EsakiaHom.comp g f) a ≤ b → ∃ c, a ≤ c ∧ ↑(EsakiaHom.comp g f) c = b) } = PseudoEpimorphism.comp { toOrderHom := ↑g, exists_map_eq_of_map_le' := (_ : ∀ ⦃a : β⦄ ⦃b : γ⦄, ↑g a ≤ b → ∃ c, a ≤ c ∧ ↑g c = b) } { toOrderHom := ↑f, exists_map_eq_of_map_le' := (_ : ∀ ⦃a : α⦄ ⦃b : β⦄, ↑f a ≤ b → ∃ c, a ≤ c ∧ ↑f c = b) }

@[simp]

theorem EsakiaHom.coe_comp {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] (g : EsakiaHom β γ) (f : EsakiaHom α β) : ↑(EsakiaHom.comp g f) = ↑g ∘ ↑f

@[simp]

theorem EsakiaHom.comp_apply {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] (g : EsakiaHom β γ) (f : EsakiaHom α β) (a : α) : ↑(EsakiaHom.comp g f) a = ↑g (↑f a)

@[simp]

theorem EsakiaHom.comp_assoc {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] [TopologicalSpace δ] [Preorder δ] (h : EsakiaHom γ δ) (g : EsakiaHom β γ) (f : EsakiaHom α β) : EsakiaHom.comp (EsakiaHom.comp h g) f = EsakiaHom.comp h (EsakiaHom.comp g f)

@[simp]

theorem EsakiaHom.comp_id {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : EsakiaHom α β) : EsakiaHom.comp f (EsakiaHom.id α) = f

@[simp]

theorem EsakiaHom.id_comp {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : EsakiaHom α β) : EsakiaHom.comp (EsakiaHom.id β) f = f

@[simp]

theorem EsakiaHom.cancel_right {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] {g₁ : EsakiaHom β γ} {g₂ : EsakiaHom β γ} {f : EsakiaHom α β} (hf : Function.Surjective ↑f) : EsakiaHom.comp g₁ f = EsakiaHom.comp g₂ f ↔ g₁ = g₂

@[simp]

theorem EsakiaHom.cancel_left {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] {g : EsakiaHom β γ} {f₁ : EsakiaHom α β} {f₂ : EsakiaHom α β} (hg : Function.Injective ↑g) : EsakiaHom.comp g f₁ = EsakiaHom.comp g f₂ ↔ f₁ = f₂