Theorems · Theorem · category theory

TopCat.Presheaf.locally_surjective_iff_surjective_on_stalks

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {FC : C → C → Type u_1} {CC : C → Type v}
  [inst_1 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] [inst_2 : CategoryTheory.ConcreteCategory C FC] {X : TopCat}
  {ℱ 𝒢 : TopCat.Presheaf C X} [inst_3 : CategoryTheory.Limits.HasColimits C]
  [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget C)] (T : ℱ ⟶ 𝒢),
  TopCat.Presheaf.IsLocallySurjective T ↔
    ∀ (x : ↑X), Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom ((TopCat.Presheaf.stalkFunctor C x).map T))

An equivalent condition for a map of presheaves to be locally surjective is for all the induced maps on stalks to be surjective.

Defined in

Mathlib.Topology.Sheaves.LocallySurjective

Cited by

0 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.HasColimitsCategoryTheory.Limits.PreservesFilteredColimits