Theorems · Theorem · algebraic geometry

TopCat.Presheaf.EtaleSpace.eventually_nhds

∀ {X : TopCat} {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {CC : C → Type v} {FC : C → C → Type w}
  [inst_1 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] [inst_2 : CategoryTheory.ConcreteCategory C FC]
  [inst_3 : CategoryTheory.Limits.HasColimits C] {F : TopCat.Presheaf C X} (g : F.EtaleSpace)
  {U : TopologicalSpace.Opens ↑X} (h : g.base ∈ U) (s : CategoryTheory.ToType (F.obj (Opposite.op U))),
  (CategoryTheory.ConcreteCategory.hom (F.germ U g.base h)) s = g.germ →
    ∀ᶠ (g' : F.EtaleSpace) in nhds g,
      ∃ (hgU : g'.base ∈ U), g'.germ = (CategoryTheory.ConcreteCategory.hom (F.germ U g'.base hgU)) s

If s is a section of a presheaf F over U with germ at g.base equal to g.germ, then a neighborhood of g consists of germs of s at points x ∈ U.

Defined in

Mathlib.Topology.Sheaves.EtaleSpace

Cited by

2 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.HasColimits