Theorems · Definition · category theory

CategoryTheory.plusPlusIsoSheafify

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    (J : CategoryTheory.GrothendieckTopology C) →
      (D : Type w) →
        [inst_1 : CategoryTheory.Category.{t, 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] →
              {FD : D → D → Type u_1} →
                {CD : D → Type t} →
                  [inst_4 : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)] →
                    [inst_5 : CategoryTheory.ConcreteCategory D FD] →
                      [inst_6 :
                          ∀ (X : C),
                            CategoryTheory.Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ (CategoryTheory.forget D)] →
                        [inst_7 : (CategoryTheory.forget D).ReflectsIsomorphisms] →
                          [inst_8 :
                              ∀ {X : C} (S : J.Cover X),
                                CategoryTheory.Limits.PreservesLimitsOfShape
                                  (CategoryTheory.Limits.WalkingMulticospan S.shape) (CategoryTheory.forget D)] →
                            (P : CategoryTheory.Functor Cᵒᵖ D) → J.sheafify P ≅ CategoryTheory.sheafify J P

plusPlus is isomorphic to sheafify.

Defined in

Mathlib.CategoryTheory.Sites.LeftExact

Cited by

2 results in Mathlib

Foundations

Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasMultiequalizerCategoryTheory.Limits.HasColimitsOfShapeFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Functor.ReflectsIsomorphismsCategoryTheory.Limits.PreservesLimitsOfShape