Theorems · Theorem · category theory

TopCat.Presheaf.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)
  {U V : TopologicalSpace.Opens ↑X} (x : ↑X) (mU : x ∈ U) (mV : x ∈ V)
  (s : CategoryTheory.ToType (F.obj (Opposite.op U))) (t : CategoryTheory.ToType (F.obj (Opposite.op V))),
  (CategoryTheory.ConcreteCategory.hom (F.germ U x mU)) s = (CategoryTheory.ConcreteCategory.hom (F.germ V x mV)) t →
    ∃ W,
      ∃ (_ : x ∈ W),
        ∃ iU iV,
          (CategoryTheory.ConcreteCategory.hom (F.map iU.op)) s = (CategoryTheory.ConcreteCategory.hom (F.map iV.op)) t

If two sections have the same germ at x, then their restrictions to some open neighborhood W of x are equal.

Defined in

Mathlib.Topology.Sheaves.Stalks

Cited by

9 results in Mathlib

Foundations

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

Assumes

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