Documentation

Mathlib.Order.Category.Frm

The category of frames #

This file defines Frm, the category of frames.

References #

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structure Frm :
Type (u_1 + 1)

The category of frames.

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instance Frm.instCoeSortType :
CoeSort Frm (Type u_1)
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@[reducible, inline]
abbrev Frm.of (X : Type u_1) [Order.Frame X] :
Frm

Construct a bundled Frm from the underlying type and typeclass.

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

The type of morphisms in Frm R.

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theorem Frm.Hom.ext_iff {X Y : Frm} {x y : X.Hom Y} :
x = y ↔ x.hom' = y.hom'
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theorem Frm.Hom.ext {X Y : Frm} {x y : X.Hom Y} (hom' : x.hom' = y.hom') :
x = y
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instance Frm.instCategory :
CategoryTheory.Category.{u, u + 1} Frm
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instance Frm.instConcreteCategoryFrameHomCarrier :
CategoryTheory.ConcreteCategory Frm fun (x1 x2 : Frm) => FrameHom ↑x1 ↑x2
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@[reducible, inline]
abbrev Frm.Hom.hom {X Y : Frm} (f : X.Hom Y) :
FrameHom ↑X ↑Y

Turn a morphism in Frm back into a FrameHom.

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@[reducible, inline]
abbrev Frm.ofHom {X Y : Type u} [Order.Frame X] [Order.Frame Y] (f : FrameHom X Y) :
of X ⟶ of Y

Typecheck a FrameHom as a morphism in Frm.

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def Frm.Hom.Simps.hom (X Y : Frm) (f : X.Hom Y) :
FrameHom ↑X ↑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 Frm.coe_id {X : Frm} :
⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) = id
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@[simp]
theorem Frm.coe_comp {X Y Z : Frm} {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 Frm.forget_map {X Y : Frm} (f : X ⟶ Y) :
(CategoryTheory.forget Frm).map f = ⇑(CategoryTheory.ConcreteCategory.hom f)
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theorem Frm.ext {X Y : Frm} {f g : X ⟶ Y} (w : ∀ (x : ↑X), (CategoryTheory.ConcreteCategory.hom f) x = (CategoryTheory.ConcreteCategory.hom g) x) :
f = g
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theorem Frm.ext_iff {X Y : Frm} {f g : X ⟶ Y} :
f = g ↔ ∀ (x : ↑X), (CategoryTheory.ConcreteCategory.hom f) x = (CategoryTheory.ConcreteCategory.hom g) x
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theorem Frm.coe_of (X : Type u) [Order.Frame X] :
↑(of X) = X
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@[simp]
theorem Frm.hom_id {X : Frm} :
Hom.hom (CategoryTheory.CategoryStruct.id X) = FrameHom.id ↑X
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theorem Frm.id_apply (X : Frm) (x : ↑X) :
(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) x = x
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@[simp]
theorem Frm.hom_comp {X Y Z : Frm} (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 Frm.comp_apply {X Y Z : Frm} (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 Frm.hom_ext {X Y : Frm} {f g : X ⟶ Y} (hf : Hom.hom f = Hom.hom g) :
f = g
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theorem Frm.hom_ext_iff {X Y : Frm} {f g : X ⟶ Y} :
f = g ↔ Hom.hom f = Hom.hom g
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@[simp]
theorem Frm.hom_ofHom {X Y : Type u} [Order.Frame X] [Order.Frame Y] (f : FrameHom X Y) :
Hom.hom (ofHom f) = f
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@[simp]
theorem Frm.ofHom_hom {X Y : Frm} (f : X ⟶ Y) :
ofHom (Hom.hom f) = f
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@[simp]
theorem Frm.ofHom_id {X : Type u} [Order.Frame X] :
ofHom (FrameHom.id X) = CategoryTheory.CategoryStruct.id (of X)
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@[simp]
theorem Frm.ofHom_comp {X Y Z : Type u} [Order.Frame X] [Order.Frame Y] [Order.Frame Z] (f : FrameHom X Y) (g : FrameHom Y Z) :
ofHom (g.comp f) = CategoryTheory.CategoryStruct.comp (ofHom f) (ofHom g)
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theorem Frm.ofHom_apply {X Y : Type u} [Order.Frame X] [Order.Frame Y] (f : FrameHom X Y) (x : X) :
(CategoryTheory.ConcreteCategory.hom (ofHom f)) x = f x
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theorem Frm.inv_hom_apply {X Y : Frm} (e : X ≅ Y) (x : ↑X) :
(CategoryTheory.ConcreteCategory.hom e.inv) ((CategoryTheory.ConcreteCategory.hom e.hom) x) = x
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theorem Frm.hom_inv_apply {X Y : Frm} (e : X ≅ Y) (s : ↑Y) :
(CategoryTheory.ConcreteCategory.hom e.hom) ((CategoryTheory.ConcreteCategory.hom e.inv) s) = s
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instance Frm.instInhabited :
Inhabited Frm
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instance Frm.hasForgetToLat :
CategoryTheory.HasForget₂ Frm Lat
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def Frm.Iso.mk {α β : Frm} (e : ↑α ≃o ↑β) :
α ≅ β

Constructs an isomorphism of frames from an order isomorphism between them.

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@[simp]
theorem Frm.Iso.mk_hom {α β : Frm} (e : ↑α ≃o ↑β) :
(mk e).hom = ofHom { toFun := ⇑e, map_inf' := ⋯, map_top' := ⋯, map_sSup' := ⋯ }
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@[simp]
theorem Frm.Iso.mk_inv {α β : Frm} (e : ↑α ≃o ↑β) :
(mk e).inv = ofHom { toFun := ⇑e.symm, map_inf' := ⋯, map_top' := ⋯, map_sSup' := ⋯ }