Documentation

Mathlib.CategoryTheory.Limits.Lattice

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

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

Equations
  • One or more equations did not get rendered due to their size.

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

Equations
  • One or more equations did not get rendered due to their size.

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

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

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

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

@[simp]
theorem CategoryTheory.Limits.CompleteLattice.prod_eq_inf {α : Type u} [SemilatticeInf α] [OrderTop α] (x y : α) :
(x y) = xy

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

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

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

@[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) :
pullback f g = xy

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

@[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) :
pushout f g = xy

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

The limit cone over any functor into a complete lattice.

Equations
  • One or more equations did not get rendered due to their size.

The colimit cocone over any functor into a complete lattice.

Equations
  • One or more equations did not get rendered due to their size.

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

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