Theorems · Theorem · category theory

TopCat.Presheaf.section_ext

∀ {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)] [CategoryTheory.Limits.HasLimits C]
  [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget C)] [(CategoryTheory.forget C).ReflectsIsomorphisms]
  (F : TopCat.Sheaf C X) (U : TopologicalSpace.Opens ↑X) (s t : CategoryTheory.ToType (F.obj.obj (Opposite.op U))),
  (∀ (x : ↑X) (hx : x ∈ U),
      (CategoryTheory.ConcreteCategory.hom (F.presheaf.germ U x hx)) s =
        (CategoryTheory.ConcreteCategory.hom (F.presheaf.germ U x hx)) t) →
    s = t

Let F be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms, preserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.

Defined in

Mathlib.Topology.Sheaves.Stalks

Cited by

7 results in Mathlib

Foundations

Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasColimitsFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.PreservesFilteredColimitsCategoryTheory.Limits.HasLimitsCategoryTheory.Limits.PreservesLimitsCategoryTheory.Functor.ReflectsIsomorphisms