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Mathlib.Order.Category.BddDistLat

The category of bounded distributive lattices #

This defines BddDistLat, the category of bounded distributive lattices.

Note that this category is sometimes called DistLat when being a lattice is understood to entail having a bottom and a top element.

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structure BddDistLat :
Type (u_1 + 1)

The category of bounded distributive lattices with bounded lattice morphisms.

  • toDistLat : DistLat

    The underlying distrib lattice of a bounded distributive lattice.

  • isBoundedOrder : BoundedOrder ↑self.toDistLat
Instances For
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    instance BddDistLat.instCoeSortType :
    CoeSort BddDistLat (Type u_1)
    Equations
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    instance BddDistLat.instDistribLatticeαToDistLat (X : BddDistLat) :
    DistribLattice ↑X.toDistLat
    Equations
    • X.instDistribLatticeαToDistLat = X.toDistLat.str
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    def BddDistLat.of (α : Type u_1) [DistribLattice α] [BoundedOrder α] :
    BddDistLat

    Construct a bundled BddDistLat from a BoundedOrder DistribLattice.

    Equations
    Instances For
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      @[simp]
      theorem BddDistLat.coe_of (α : Type u_1) [DistribLattice α] [BoundedOrder α] :
      ↑(BddDistLat.of α).toDistLat = α
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      instance BddDistLat.instInhabited :
      Inhabited BddDistLat
      Equations
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      def BddDistLat.toBddLat (X : BddDistLat) :
      BddLat

      Turn a BddDistLat into a BddLat by forgetting it is distributive.

      Equations
      Instances For
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        @[simp]
        theorem BddDistLat.coe_toBddLat (X : BddDistLat) :
        ↑X.toBddLat.toLat = ↑X.toDistLat
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        instance BddDistLat.instLargeCategory :
        CategoryTheory.LargeCategory BddDistLat
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        instance BddDistLat.instConcreteCategory :
        CategoryTheory.ConcreteCategory BddDistLat
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        instance BddDistLat.hasForgetToDistLat :
        CategoryTheory.HasForget₂ BddDistLat DistLat
        Equations
        • One or more equations did not get rendered due to their size.
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        instance BddDistLat.hasForgetToBddLat :
        CategoryTheory.HasForget₂ BddDistLat BddLat
        Equations
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        theorem BddDistLat.forget_bddLat_lat_eq_forget_distLat_lat :
        (CategoryTheory.forget₂ BddDistLat BddLat).comp (CategoryTheory.forget₂ BddLat Lat) = (CategoryTheory.forget₂ BddDistLat DistLat).comp (CategoryTheory.forget₂ DistLat Lat)
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        @[simp]
        theorem BddDistLat.Iso.mk_hom_toLatticeHom_toSupHom_toFun {α : BddDistLat} {β : BddDistLat} (e : ↑α.toDistLat ≃o ↑β.toDistLat) (a : ↑α.toDistLat) :
        (BddDistLat.Iso.mk e).hom.toSupHom a = e a
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        @[simp]
        theorem BddDistLat.Iso.mk_inv_toLatticeHom_toSupHom_toFun {α : BddDistLat} {β : BddDistLat} (e : ↑α.toDistLat ≃o ↑β.toDistLat) (a : ↑β.toDistLat) :
        (BddDistLat.Iso.mk e).inv.toSupHom a = e.symm a
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        def BddDistLat.Iso.mk {α : BddDistLat} {β : BddDistLat} (e : ↑α.toDistLat ≃o ↑β.toDistLat) :
        α ≅ β

        Constructs an equivalence between bounded distributive lattices from an order isomorphism between them.

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        • One or more equations did not get rendered due to their size.
        Instances For
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          @[simp]
          theorem BddDistLat.dual_map {X : BddDistLat} {Y : BddDistLat} (a : BoundedLatticeHom ↑X.toBddLat.toLat ↑Y.toBddLat.toLat) :
          BddDistLat.dual.map a = BoundedLatticeHom.dual a
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          @[simp]
          theorem BddDistLat.dual_obj (X : BddDistLat) :
          BddDistLat.dual.obj X = BddDistLat.of (↑X.toDistLat)ᵒᵈ
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          def BddDistLat.dual :
          CategoryTheory.Functor BddDistLat BddDistLat

          OrderDual as a functor.

          Equations
          • One or more equations did not get rendered due to their size.
          Instances For
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            @[simp]
            theorem BddDistLat.dualEquiv_inverse :
            BddDistLat.dualEquiv.inverse = BddDistLat.dual
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            @[simp]
            theorem BddDistLat.dualEquiv_functor :
            BddDistLat.dualEquiv.functor = BddDistLat.dual
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            def BddDistLat.dualEquiv :
            BddDistLat ≌ BddDistLat

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

            Equations
            • One or more equations did not get rendered due to their size.
            Instances For
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              theorem bddDistLat_dual_comp_forget_to_distLat :
              BddDistLat.dual.comp (CategoryTheory.forget₂ BddDistLat DistLat) = (CategoryTheory.forget₂ BddDistLat DistLat).comp DistLat.dual