Theorems · Theorem · category theory

CategoryTheory.Presheaf.isLimit_iff_isSheafFor_presieve

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {A : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} A]
  (P : CategoryTheory.Functor Cᵒᵖ A) {X : C} (R : CategoryTheory.Presieve X),
  Nonempty (CategoryTheory.Limits.IsLimit (P.mapCone (CategoryTheory.Sieve.generate R).arrows.cocone.op)) ↔
    ∀ (E : Aᵒᵖ), CategoryTheory.Presieve.IsSheafFor (P.comp (CategoryTheory.coyoneda.obj E)) R

Given presieve R and presheaf P : Cᵒᵖ ⥤ A, the natural cone associated to P and the sieve Sieve.generate R generated by R is a limit cone iff Hom (E, P -) is a sheaf of types for the presieve R and all E : A.

Defined in

Mathlib.CategoryTheory.Sites.Sheaf

Cited by

1 results in Mathlib

Foundations

Depth 36 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Category