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 CategoryTheory.Limits.CompleteLattice.hasFiniteLimits_of_semilatticeInf_orderTop {α : Type u} [SemilatticeInf α] [OrderTop α] :

CategoryTheory.Limits.HasFiniteLimits α

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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.inf Finset.univ 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.sup Finset.univ 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) :

∏ f = Finset.inf Fintype.elems 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 = Finset.sup Fintype.elems f

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

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instance CategoryTheory.Limits.CompleteLattice.instHasBinaryProductsSmallCategoryToPreorderToPartialOrder {α : Type u} [SemilatticeInf α] [OrderTop α] :

CategoryTheory.Limits.HasBinaryProducts α

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theorem CategoryTheory.Limits.CompleteLattice.prod_eq_inf {α : Type u} [SemilatticeInf α] [OrderTop α] (x : α) (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 CategoryTheory.Limits.CompleteLattice.instHasBinaryCoproductsSmallCategoryToPreorderToPartialOrder {α : Type u} [SemilatticeSup α] [OrderBot α] :

CategoryTheory.Limits.HasBinaryCoproducts α

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theorem CategoryTheory.Limits.CompleteLattice.coprod_eq_sup {α : Type u} [SemilatticeSup α] [OrderBot α] (x : α) (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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theorem CategoryTheory.Limits.CompleteLattice.pullback_eq_inf {α : Type u} [SemilatticeInf α] [OrderTop α] {x : α} {y : α} {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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theorem CategoryTheory.Limits.CompleteLattice.pushout_eq_sup {α : Type u} [SemilatticeSup α] [OrderBot α] (x : α) (y : α) (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 CategoryTheory.Limits.CompleteLattice.hasLimits_of_completeLattice {α : Type u} [CompleteLattice α] :

CategoryTheory.Limits.HasLimits α

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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.