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Mathlib.Topology.Category.Locale

The category of locales #

This file defines Locale, the category of locales. This is the opposite of the category of frames.

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def Locale :
Type (u_1 + 1)

The category of locales.

Instances For
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    @[implicit_reducible]
    instance instLargeCategoryLocale :
    CategoryTheory.LargeCategory Locale
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    @[implicit_reducible]
    instance Locale.instCoeSortType :
    CoeSort Locale (Type u_1)
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    @[implicit_reducible]
    instance Locale.instFrameCarrierUnopFrm (X : Locale) :
    Order.Frame ↑(Opposite.unop X)
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    def Locale.of (α : Type u_1) [Order.Frame α] :
    Locale

    Construct a bundled Locale from a Frame.

    Instances For
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      @[simp]
      theorem Locale.coe_of (α : Type u_1) [Order.Frame α] :
      ↑(Opposite.unop (of α)) = α
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      @[implicit_reducible]
      instance Locale.instInhabited :
      Inhabited Locale
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      def topToLocale :
      CategoryTheory.Functor TopCat Locale

      The forgetful functor from Top to Locale which forgets that the space has "enough points".

      Instances For
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        @[simp]
        theorem topToLocale_map {X✝ Y✝ : TopCat} (f : X✝ ⟶ Y✝) :
        topToLocale.map f = (Frm.ofHom (TopologicalSpace.Opens.comap (TopCat.Hom.hom f))).op
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        @[simp]
        theorem topToLocale_obj (X : TopCat) :
        topToLocale.obj X = Opposite.op (topCatOpToFrm.obj (Opposite.op X))
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        instance CompHausToLocale.faithful :
        (compHausToTop.comp topToLocale).Faithful