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Mathlib.Order.Category.Lat

The category of lattices #

This defines Lat, the category of lattices.

Note that Lat doesn't correspond to the literature definition of [Lat] (https://ncatlab.org/nlab/show/Lat) as we don't require bottom or top elements. Instead, Lat corresponds to BddLat.

TODO #

The free functor from Lat to BddLat is X → WithTop (WithBot X).

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structure Lat :
Type (u_1 + 1)

The category of lattices.

  • carrier : Type u_1

    The underlying lattices.

  • str : Lattice ↑self
Instances For
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instance Lat.instCoeSortType :
CoeSort Lat (Type u_1)
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@[reducible, inline]
abbrev Lat.of (X : Type u_1) [Lattice X] :
Lat

Construct a bundled Lat from the underlying type and typeclass.

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

The type of morphisms in Lat R.

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theorem Lat.Hom.ext {X Y : Lat} {x y : X.Hom Y} (hom' : x.hom' = y.hom') :
x = y
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instance Lat.instCategory :
CategoryTheory.Category.{u, u + 1} Lat
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instance Lat.instConcreteCategoryLatticeHomCarrier :
CategoryTheory.ConcreteCategory Lat fun (x1 x2 : Lat) => LatticeHom ↑x1 ↑x2
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  • One or more equations did not get rendered due to their size.
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@[reducible, inline]
abbrev Lat.Hom.hom {X Y : Lat} (f : X.Hom Y) :
LatticeHom ↑X ↑Y

Turn a morphism in Lat back into a LatticeHom.

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@[reducible, inline]
abbrev Lat.ofHom {X Y : Type u} [Lattice X] [Lattice Y] (f : LatticeHom X Y) :
of X ⟶ of Y

Typecheck a LatticeHom as a morphism in Lat.

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def Lat.Hom.Simps.hom (X Y : Lat) (f : X.Hom Y) :
LatticeHom ↑X ↑Y

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

Equations

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

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@[simp]
theorem Lat.coe_id {X : Lat} :
⇑(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) = id
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@[simp]
theorem Lat.coe_comp {X Y Z : Lat} {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 Lat.forget_map {X Y : Lat} (f : X ⟶ Y) :
(CategoryTheory.forget Lat).map f = ⇑(CategoryTheory.ConcreteCategory.hom f)
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theorem Lat.ext {X Y : Lat} {f g : X ⟶ Y} (w : ∀ (x : ↑X), (CategoryTheory.ConcreteCategory.hom f) x = (CategoryTheory.ConcreteCategory.hom g) x) :
f = g
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theorem Lat.coe_of (X : Type u) [Lattice X] :
↑(of X) = X
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@[simp]
theorem Lat.hom_id {X : Lat} :
Hom.hom (CategoryTheory.CategoryStruct.id X) = LatticeHom.id ↑X
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theorem Lat.id_apply (X : Lat) (x : ↑X) :
(CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) x = x
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@[simp]
theorem Lat.hom_comp {X Y Z : Lat} (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 Lat.comp_apply {X Y Z : Lat} (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 Lat.hom_ext {X Y : Lat} {f g : X ⟶ Y} (hf : Hom.hom f = Hom.hom g) :
f = g
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@[simp]
theorem Lat.hom_ofHom {X Y : Type u} [Lattice X] [Lattice Y] (f : LatticeHom X Y) :
Hom.hom (ofHom f) = f
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@[simp]
theorem Lat.ofHom_hom {X Y : Lat} (f : X ⟶ Y) :
ofHom (Hom.hom f) = f
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@[simp]
theorem Lat.ofHom_id {X : Type u} [Lattice X] :
ofHom (LatticeHom.id X) = CategoryTheory.CategoryStruct.id (of X)
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@[simp]
theorem Lat.ofHom_comp {X Y Z : Type u} [Lattice X] [Lattice Y] [Lattice Z] (f : LatticeHom X Y) (g : LatticeHom Y Z) :
ofHom (g.comp f) = CategoryTheory.CategoryStruct.comp (ofHom f) (ofHom g)
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theorem Lat.ofHom_apply {X Y : Type u} [Lattice X] [Lattice Y] (f : LatticeHom X Y) (x : X) :
(CategoryTheory.ConcreteCategory.hom (ofHom f)) x = f x
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theorem Lat.inv_hom_apply {X Y : Lat} (e : X ≅ Y) (x : ↑X) :
(CategoryTheory.ConcreteCategory.hom e.inv) ((CategoryTheory.ConcreteCategory.hom e.hom) x) = x
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theorem Lat.hom_inv_apply {X Y : Lat} (e : X ≅ Y) (s : ↑Y) :
(CategoryTheory.ConcreteCategory.hom e.hom) ((CategoryTheory.ConcreteCategory.hom e.inv) s) = s
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instance Lat.hasForgetToPartOrd :
CategoryTheory.HasForget₂ Lat PartOrd
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  • One or more equations did not get rendered due to their size.
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def Lat.Iso.mk {α β : Lat} (e : ↑α ≃o ↑β) :
α ≅ β

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

Equations
  • Lat.Iso.mk e = { hom := Lat.ofHom { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ }, inv := Lat.ofHom { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ }
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@[simp]
theorem Lat.Iso.mk_hom {α β : Lat} (e : ↑α ≃o ↑β) :
(mk e).hom = ofHom { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ }
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@[simp]
theorem Lat.Iso.mk_inv {α β : Lat} (e : ↑α ≃o ↑β) :
(mk e).inv = ofHom { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ }
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def Lat.dual :
CategoryTheory.Functor Lat Lat

OrderDual as a functor.

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  • One or more equations did not get rendered due to their size.
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@[simp]
theorem Lat.dual_map {X✝ Y✝ : Lat} (f : X✝ ⟶ Y✝) :
dual.map f = ofHom (LatticeHom.dual (Hom.hom f))
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def Lat.dualEquiv :
Lat ≌ Lat

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

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  • One or more equations did not get rendered due to their size.
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@[simp]
theorem Lat.dualEquiv_functor :
dualEquiv.functor = dual
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@[simp]
theorem Lat.dualEquiv_inverse :
dualEquiv.inverse = dual
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theorem Lat_dual_comp_forget_to_partOrd :
Lat.dual.comp (CategoryTheory.forget₂ Lat PartOrd) = (CategoryTheory.forget₂ Lat PartOrd).comp PartOrd.dual