Theorems · Theorem · category theory

TopCat.Presheaf.app_injective_iff_stalkFunctor_map_injective

∀ {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} {G : TopCat.Presheaf C X} (f : F.obj ⟶ G),
  (∀ (x : ↑X), Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom ((TopCat.Presheaf.stalkFunctor C x).map f))) ↔
    ∀ (U : TopologicalSpace.Opens ↑X), Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom (f.app (Opposite.op U)))

Defined in

Mathlib.Topology.Sheaves.Stalks

Cited by

0 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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