Theorems · Theorem · category theory

CategoryTheory.Presheaf.isSheaf_iff_isLimit_pretopology

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {A : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} A]
  (P : CategoryTheory.Functor Cᵒᵖ A) [inst_2 : CategoryTheory.Limits.HasPullbacks C] (K : CategoryTheory.Pretopology C),
  CategoryTheory.Presheaf.IsSheaf K.toGrothendieck P ↔
    ∀ ⦃X : C⦄,
      ∀ R ∈ K.coverings X,
        Nonempty (CategoryTheory.Limits.IsLimit (P.mapCone (CategoryTheory.Sieve.generate R).arrows.cocone.op))

A presheaf P is a sheaf for the Grothendieck topology generated by a pretopology K iff for every covering presieve R of K, the natural cone associated to P and Sieve.generate R is a limit cone.

Defined in

Mathlib.CategoryTheory.Sites.Sheaf

Cited by

0 results in Mathlib

Foundations

Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasPullbacks