Theorems · Theorem · category theory

TopCat.Presheaf.exists_mem_germ_eq_of_isBasis

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasColimits C] {X : TopCat}
  {FC : C → C → Type u_1} {CC : C → Type v} [inst_2 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)]
  [instCC : CategoryTheory.ConcreteCategory C FC]
  [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget C)] {B : Set (TopologicalSpace.Opens ↑X)},
  TopologicalSpace.Opens.IsBasis B →
    ∀ (F : TopCat.Presheaf C X) (x : ↑X) (t : CategoryTheory.ToType (F.stalk x)),
      ∃ U, ∃ (m : x ∈ U) (_ : U ∈ B), ∃ s, (CategoryTheory.ConcreteCategory.hom (F.germ U x m)) s = t

Defined in

Mathlib.Topology.Sheaves.Stalks

Cited by

1 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasColimitsFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.PreservesFilteredColimits