Theorems · Theorem · category theory

CategoryTheory.Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {P : CategoryTheory.Functor Cᵒᵖ (Type w)} {X : C}
  {R : CategoryTheory.Presieve X},
  (CategoryTheory.Presieve.IsSeparatedFor P R ∧
      ∀ (x : CategoryTheory.Presieve.FamilyOfElements P R), x.Compatible → ∃ t, x.IsAmalgamation t) ↔
    CategoryTheory.Presieve.IsSheafFor P R

P is a sheaf for R iff it is separated for R and there exists an amalgamation.

Defined in

Mathlib.CategoryTheory.Sites.IsSheafFor

Cited by

7 results in Mathlib

Foundations

Depth 16 from the axioms · uses propext, Quot.sound

Assumes

CategoryTheory.Category