Theorems · Theorem · category theory

CategoryTheory.Presieve.isSheafFor_arrows_iff_pullbacks

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] (P : CategoryTheory.Functor Cᵒᵖ (Type w)) {B : C}
  {I : Type u_1} {X : I → C} (π : (i : I) → X i ⟶ B)
  [inst_1 : (CategoryTheory.Presieve.ofArrows X π).HasPairwisePullbacks],
  CategoryTheory.Presieve.IsSheafFor P (CategoryTheory.Presieve.ofArrows X π) ↔
    ∀ (x : (i : I) → P.obj (Opposite.op (X i))),
      CategoryTheory.Presieve.Arrows.PullbackCompatible P π x →
        ∃! t, ∀ (i : I), (CategoryTheory.ConcreteCategory.hom (P.map (π i).op)) t = x i

Defined in

Mathlib.CategoryTheory.Sites.IsSheafFor

Cited by

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Foundations

Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Presieve.HasPairwisePullbacks