Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.toSheafification

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    (J : CategoryTheory.GrothendieckTopology C) →
      (D : Type w) →
        [inst_1 : CategoryTheory.Category.{w', w} D] →
          [inst_2 :
              ∀ (P : CategoryTheory.Functor Cᵒᵖ D) (X : C) (S : J.Cover X),
                CategoryTheory.Limits.HasMultiequalizer (S.index P)] →
            [inst_3 : ∀ (X : C), CategoryTheory.Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] →
              CategoryTheory.Functor.id (CategoryTheory.Functor Cᵒᵖ D) ⟶ J.sheafification D

The canonical map from P to its sheafification, as a natural transformation. Note: We only show this is a sheaf under additional hypotheses on D.

Defined in

Mathlib.CategoryTheory.Sites.ConcreteSheafification

Cited by

1 results in Mathlib

Foundations

Depth 55 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasMultiequalizerCategoryTheory.Limits.HasColimitsOfShape