File: Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean

full_name: AlgebraicGeometry.UniversallyClosed
code: /-- A morphism of schemes `f : X ⟶ Y` is universally closed if the base change `X ×[Y] Y' ⟶ Y'`
along any morphism `Y' ⟶ Y` is (topologically) a closed map.
-/
@[mk_iff]
class UniversallyClosed (f : X ⟶ Y) : Prop where
  out : universally (topologically @IsClosedMap) f
goal: Prop

full_name: AlgebraicGeometry.UniversallyClosed.out
code: /-- A morphism of schemes `f : X ⟶ Y` is universally closed if the base change `X ×[Y] Y' ⟶ Y'`
along any morphism `Y' ⟶ Y` is (topologically) a closed map.
-/
@[mk_iff]
class UniversallyClosed (f : X ⟶ Y) : Prop where
  out : universally (topologically @IsClosedMap) f
goal: (AlgebraicGeometry.topologically @IsClosedMap).universally f

full_name: AlgebraicGeometry.universallyClosed_iff
code: /-- A morphism of schemes `f : X ⟶ Y` is universally closed if the base change `X ×[Y] Y' ⟶ Y'`
along any morphism `Y' ⟶ Y` is (topologically) a closed map.
-/
@[mk_iff]
class UniversallyClosed (f : X ⟶ Y) : Prop where
  out : universally (topologically @IsClosedMap) f
goal: AlgebraicGeometry.UniversallyClosed f ↔ (AlgebraicGeometry.topologically @IsClosedMap).universally f

full_name: AlgebraicGeometry.universallyClosed_eq
code: theorem universallyClosed_eq : @UniversallyClosed = universally (topologically @IsClosedMap) 
goal: @AlgebraicGeometry.UniversallyClosed = (AlgebraicGeometry.topologically @IsClosedMap).universally

full_name: AlgebraicGeometry.universallyClosed_respectsIso
code: theorem universallyClosed_respectsIso : RespectsIso @UniversallyClosed 
goal: CategoryTheory.MorphismProperty.RespectsIso @AlgebraicGeometry.UniversallyClosed

full_name: AlgebraicGeometry.universallyClosed_stableUnderBaseChange
code: theorem universallyClosed_stableUnderBaseChange : StableUnderBaseChange @UniversallyClosed 
goal: CategoryTheory.MorphismProperty.StableUnderBaseChange @AlgebraicGeometry.UniversallyClosed

full_name: AlgebraicGeometry.isClosedMap_isStableUnderComposition
code: instance isClosedMap_isStableUnderComposition :
    IsStableUnderComposition (topologically @IsClosedMap) 
goal: (AlgebraicGeometry.topologically @IsClosedMap).IsStableUnderComposition

full_name: AlgebraicGeometry.universallyClosed_isStableUnderComposition
code: instance universallyClosed_isStableUnderComposition :
    IsStableUnderComposition @UniversallyClosed 
goal: CategoryTheory.MorphismProperty.IsStableUnderComposition @AlgebraicGeometry.UniversallyClosed

full_name: AlgebraicGeometry.universallyClosedTypeComp
code: instance universallyClosedTypeComp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
    [hf : UniversallyClosed f] [hg : UniversallyClosed g] : UniversallyClosed (f ≫ g) 
goal: AlgebraicGeometry.UniversallyClosed (f ≫ g)

full_name: AlgebraicGeometry.universallyClosed_fst
code: instance universallyClosed_fst {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hg : UniversallyClosed g] :
    UniversallyClosed (pullback.fst f g) 
goal: AlgebraicGeometry.UniversallyClosed (CategoryTheory.Limits.pullback.fst f g)

full_name: AlgebraicGeometry.universallyClosed_snd
code: instance universallyClosed_snd {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hf : UniversallyClosed f] :
    UniversallyClosed (pullback.snd f g) 
goal: AlgebraicGeometry.UniversallyClosed (CategoryTheory.Limits.pullback.snd f g)

full_name: AlgebraicGeometry.universallyClosed_isLocalAtTarget
code: instance universallyClosed_isLocalAtTarget : IsLocalAtTarget @UniversallyClosed 
goal: AlgebraicGeometry.IsLocalAtTarget @AlgebraicGeometry.UniversallyClosed