Theorems · Theorem · category theory

CategoryTheory.GrothendieckTopology.Plus.isSheaf_of_sep

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} {D : Type w}
  [inst_1 : CategoryTheory.Category.{w', w} D] {FD : D → D → Type u_1} {CD : D → Type t}
  [inst_2 : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)] [instCC : CategoryTheory.ConcreteCategory D FD]
  [∀ {X : C} (S : J.Cover X),
      CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Limits.WalkingMulticospan S.shape)
        (CategoryTheory.forget D)]
  [inst_4 : ∀ (X : C), CategoryTheory.Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D]
  [inst_5 :
    ∀ (P : CategoryTheory.Functor Cᵒᵖ D) (X : C) (S : J.Cover X), CategoryTheory.Limits.HasMultiequalizer (S.index P)]
  [∀ (X : C), CategoryTheory.Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ (CategoryTheory.forget D)]
  [(CategoryTheory.forget D).ReflectsIsomorphisms] (P : CategoryTheory.Functor Cᵒᵖ D),
  (∀ (X : C) (S : J.Cover X) (x y : CategoryTheory.ToType (P.obj (Opposite.op X))),
      (∀ (I : S.Arrow),
          (CategoryTheory.ConcreteCategory.hom (P.map I.f.op)) x =
            (CategoryTheory.ConcreteCategory.hom (P.map I.f.op)) y) →
        x = y) →
    CategoryTheory.Presheaf.IsSheaf J (J.plusObj P)

If P is separated, then P⁺ is a sheaf.

Defined in

Mathlib.CategoryTheory.Sites.ConcreteSheafification

Cited by

1 results in Mathlib

Foundations

Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.PreservesLimitsOfShapeCategoryTheory.Limits.HasColimitsOfShapeCategoryTheory.Limits.HasMultiequalizerCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Functor.ReflectsIsomorphisms