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Mathlib.CategoryTheory.Limits.Lattice

Limits in lattice categories are given by infimums and supremums. #

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def CategoryTheory.Limits.CompleteLattice.finiteLimitCone {α : Type u} {J : Type w} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] [SemilatticeInf α] [OrderTop α] (F : CategoryTheory.Functor J α) :
CategoryTheory.Limits.LimitCone F

The limit cone over any functor from a finite diagram into a SemilatticeInf with OrderTop.

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    def CategoryTheory.Limits.CompleteLattice.finiteColimitCocone {α : Type u} {J : Type w} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] [SemilatticeSup α] [OrderBot α] (F : CategoryTheory.Functor J α) :
    CategoryTheory.Limits.ColimitCocone F

    The colimit cocone over any functor from a finite diagram into a SemilatticeSup with OrderBot.

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      @[instance 100]
      instance CategoryTheory.Limits.CompleteLattice.hasFiniteLimits_of_semilatticeInf_orderTop {α : Type u} [SemilatticeInf α] [OrderTop α] :
      CategoryTheory.Limits.HasFiniteLimits α
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      @[instance 100]
      instance CategoryTheory.Limits.CompleteLattice.hasFiniteColimits_of_semilatticeSup_orderBot {α : Type u} [SemilatticeSup α] [OrderBot α] :
      CategoryTheory.Limits.HasFiniteColimits α
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      theorem CategoryTheory.Limits.CompleteLattice.finite_limit_eq_finset_univ_inf {α : Type u} {J : Type w} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] [SemilatticeInf α] [OrderTop α] (F : CategoryTheory.Functor J α) :
      CategoryTheory.Limits.limit F = Finset.univ.inf F.obj

      The limit of a functor from a finite diagram into a SemilatticeInf with OrderTop is the infimum of the objects in the image.

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      theorem CategoryTheory.Limits.CompleteLattice.finite_colimit_eq_finset_univ_sup {α : Type u} {J : Type w} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] [SemilatticeSup α] [OrderBot α] (F : CategoryTheory.Functor J α) :
      CategoryTheory.Limits.colimit F = Finset.univ.sup F.obj

      The colimit of a functor from a finite diagram into a SemilatticeSup with OrderBot is the supremum of the objects in the image.

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      theorem CategoryTheory.Limits.CompleteLattice.finite_product_eq_finset_inf {α : Type u} [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι] (f : ι → α) :
      ∏ᶜ f = Fintype.elems.inf f

      A finite product in the category of a SemilatticeInf with OrderTop is the same as the infimum.

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      theorem CategoryTheory.Limits.CompleteLattice.finite_coproduct_eq_finset_sup {α : Type u} [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι] (f : ι → α) :
      ∐ f = Fintype.elems.sup f

      A finite coproduct in the category of a SemilatticeSup with OrderBot is the same as the supremum.

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      @[instance 100]
      instance CategoryTheory.Limits.CompleteLattice.instHasBinaryProductsOfOrderTop {α : Type u} [SemilatticeInf α] [OrderTop α] :
      CategoryTheory.Limits.HasBinaryProducts α
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      @[simp]
      theorem CategoryTheory.Limits.CompleteLattice.prod_eq_inf {α : Type u} [SemilatticeInf α] [OrderTop α] (x y : α) :
      (x ⨯ y) = x ⊓ y

      The binary product in the category of a SemilatticeInf with OrderTop is the same as the infimum.

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      @[instance 100]
      instance CategoryTheory.Limits.CompleteLattice.instHasBinaryCoproductsOfOrderBot {α : Type u} [SemilatticeSup α] [OrderBot α] :
      CategoryTheory.Limits.HasBinaryCoproducts α
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      @[simp]
      theorem CategoryTheory.Limits.CompleteLattice.coprod_eq_sup {α : Type u} [SemilatticeSup α] [OrderBot α] (x y : α) :
      (x ⨿ y) = x ⊔ y

      The binary coproduct in the category of a SemilatticeSup with OrderBot is the same as the supremum.

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      @[simp]
      theorem CategoryTheory.Limits.CompleteLattice.pullback_eq_inf {α : Type u} [SemilatticeInf α] [OrderTop α] {x y z : α} (f : x ⟶ z) (g : y ⟶ z) :
      CategoryTheory.Limits.pullback f g = x ⊓ y

      The pullback in the category of a SemilatticeInf with OrderTop is the same as the infimum over the objects.

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      @[simp]
      theorem CategoryTheory.Limits.CompleteLattice.pushout_eq_sup {α : Type u} [SemilatticeSup α] [OrderBot α] (x y z : α) (f : z ⟶ x) (g : z ⟶ y) :
      CategoryTheory.Limits.pushout f g = x ⊔ y

      The pushout in the category of a SemilatticeSup with OrderBot is the same as the supremum over the objects.

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      def CategoryTheory.Limits.CompleteLattice.limitCone {α : Type u} [CompleteLattice α] {J : Type u} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J α) :
      CategoryTheory.Limits.LimitCone F

      The limit cone over any functor into a complete lattice.

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        def CategoryTheory.Limits.CompleteLattice.colimitCocone {α : Type u} [CompleteLattice α] {J : Type u} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J α) :
        CategoryTheory.Limits.ColimitCocone F

        The colimit cocone over any functor into a complete lattice.

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          @[instance 100]
          instance CategoryTheory.Limits.CompleteLattice.hasLimits_of_completeLattice {α : Type u} [CompleteLattice α] :
          CategoryTheory.Limits.HasLimits α
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          @[instance 100]
          instance CategoryTheory.Limits.CompleteLattice.hasColimits_of_completeLattice {α : Type u} [CompleteLattice α] :
          CategoryTheory.Limits.HasColimits α
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          theorem CategoryTheory.Limits.CompleteLattice.limit_eq_iInf {α : Type u} [CompleteLattice α] {J : Type u} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J α) :
          CategoryTheory.Limits.limit F = iInf F.obj

          The limit of a functor into a complete lattice is the infimum of the objects in the image.

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          theorem CategoryTheory.Limits.CompleteLattice.colimit_eq_iSup {α : Type u} [CompleteLattice α] {J : Type u} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J α) :
          CategoryTheory.Limits.colimit F = iSup F.obj

          The colimit of a functor into a complete lattice is the supremum of the objects in the image.