Theorems · Theorem · category theory

TopCat.Sheaf.isZero_iff_stalkFunctor_obj_isZero

∀ {C : Type v} [inst : CategoryTheory.Category.{u, v} C] [inst_1 : CategoryTheory.Limits.HasColimits C]
  [CategoryTheory.Limits.HasLimits C] {FC : C → C → Type u_1} {CC : C → Type u}
  [inst_3 : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)] [instCC : CategoryTheory.ConcreteCategory C FC]
  [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget C)]
  [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget C)] [CategoryTheory.Abelian C] {X : TopCat}
  (F : TopCat.Sheaf C X),
  CategoryTheory.Limits.IsZero F ↔
    ∀ (x : ↑X), CategoryTheory.Limits.IsZero (((TopCat.Sheaf.forget C X).comp (TopCat.Presheaf.stalkFunctor C x)).obj F)

A sheaf is zero if and only if its stalks are all zero.

Defined in

Mathlib.Topology.Sheaves.Abelian

Cited by

1 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasColimitsCategoryTheory.Limits.HasLimitsFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.PreservesFilteredColimitsCategoryTheory.Limits.PreservesLimitsCategoryTheory.Abelian