Theorems · Theorem · category theory

TopCat.Presheaf.stalkFunctor_map_injective_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 G : TopCat.Presheaf C X} {α : F ⟶ G},
      (∀ U ∈ B, Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom (α.app (Opposite.op U)))) →
        ∀ (x : ↑X), Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom ((TopCat.Presheaf.stalkFunctor C x).map α))

Defined in

Mathlib.Topology.Sheaves.Stalks

Cited by

1 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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