Theorems · Theorem · algebraic geometry

AlgebraicGeometry.SheafedSpace.IsOpenImmersion.of_stalk_iso

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimits C]
  [inst_2 : CategoryTheory.Limits.HasColimits C] {FC : C → C → Type u_1} {CC : C → Type v}
  [inst_3 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] [instCC : CategoryTheory.ConcreteCategory C FC]
  [(CategoryTheory.forget C).ReflectsIsomorphisms] [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget C)]
  [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget C)] {X Y : AlgebraicGeometry.SheafedSpace C}
  (f : X ⟶ Y),
  Topology.IsOpenEmbedding ⇑(CategoryTheory.ConcreteCategory.hom f.hom.base) →
    ∀
      [H :
        ∀ (x : ↑↑X.toPresheafedSpace), CategoryTheory.IsIso (AlgebraicGeometry.PresheafedSpace.Hom.stalkMap f.hom x)],
      AlgebraicGeometry.SheafedSpace.IsOpenImmersion f

Suppose X Y : SheafedSpace C, where C is a concrete category, whose forgetful functor reflects isomorphisms, preserves limits and filtered colimits. Then a morphism X ⟶ Y that is a topological open embedding is an open immersion iff every stalk map is an iso.

Defined in

Mathlib.Geometry.RingedSpace.OpenImmersion

Cited by

2 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasLimitsCategoryTheory.Limits.HasColimitsFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Functor.ReflectsIsomorphismsCategoryTheory.Limits.PreservesLimitsCategoryTheory.Limits.PreservesFilteredColimitsCategoryTheory.IsIso