Theorems · Theorem · algebraic geometry

TopCat.Presheaf.EtaleSpace.exists_section_of_tendsto

∀ {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}
  [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget C)] {α : Type u_1} {l : Filter α}
  {g : α → F.EtaleSpace} {g₀ : F.EtaleSpace},
  Filter.Tendsto g l (nhds g₀) →
    ∃ U,
      g₀.base ∈ U ∧
        ∃ f,
          ∀ᶠ (a : α) in l,
            ∃ (ha : (g a).base ∈ U), (g a).germ = (CategoryTheory.ConcreteCategory.hom (F.germ U (g a).base ha)) f

Defined in

Mathlib.Topology.Sheaves.EtaleSpace

Cited by

0 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.HasColimitsCategoryTheory.Limits.PreservesFilteredColimits