Theorems · Theorem · category theory

CategoryTheory.Precoverage.Generates.isSheaf_of_forall

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {K : CategoryTheory.Precoverage C}
  {J : CategoryTheory.GrothendieckTopology C},
  K.Generates J →
    ∀ (F : CategoryTheory.Functor Cᵒᵖ (Type w)),
      (∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, CategoryTheory.Presieve.IsSheafFor F R) → CategoryTheory.Presieve.IsSheaf J F

If K generates J, then any presheaf Cᵒᵖ ⥤ Type w that satisfies the sheaf condition for all K-coverings, is a J-sheaf.

Defined in

Mathlib.CategoryTheory.Sites.Precoverage.Generates

Cited by

1 results in Mathlib

Foundations

Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.Category