Theorems · Theorem · category theory

TopCat.Presheaf.exists_germ_eq

∀ {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)] (F : TopCat.Presheaf C X) {x : ↑X}
  (t : CategoryTheory.ToType (F.stalk x)),
  ∃ U, ∃ (m : x ∈ U), ∃ s, (CategoryTheory.ConcreteCategory.hom (F.germ U x m)) s = t

For presheaves valued in a concrete category whose forgetful functor preserves filtered colimits, every element of the stalk is the germ of a section.

Defined in

Mathlib.Topology.Sheaves.Stalks

Cited by

14 results in Mathlib

Foundations

Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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