Theorems · Theorem · category theory

CategoryTheory.Precoverage.Generates.isSheaf_type_iff

∀ {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)},
      CategoryTheory.Presieve.IsSheaf J F ↔ ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, CategoryTheory.Presieve.IsSheafFor F R

If K generates J, then any presheaf is a sheaf if and only if it is a sheaf for all K-covers.

Defined in

Mathlib.CategoryTheory.Sites.Precoverage.Generates

Cited by

1 results in Mathlib

Foundations

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

Assumes

CategoryTheory.Category