Theorems · Theorem · category theory

CategoryTheory.GrothendieckTopology.sheafifyCompIso_inv_eq_sheafifyLift

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) {D : Type u_1}
  [inst_1 : CategoryTheory.Category.{v_1, u_1} D] {E : Type u_2} [inst_2 : CategoryTheory.Category.{v_2, u_2} E]
  (F : CategoryTheory.Functor D E)
  [inst_3 :
    ∀ (J : CategoryTheory.Limits.MulticospanShape),
      CategoryTheory.Limits.HasLimitsOfShape (CategoryTheory.Limits.WalkingMulticospan J) D]
  [inst_4 :
    ∀ (J : CategoryTheory.Limits.MulticospanShape),
      CategoryTheory.Limits.HasLimitsOfShape (CategoryTheory.Limits.WalkingMulticospan J) E]
  [inst_5 : ∀ (X : C), CategoryTheory.Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D]
  [inst_6 : ∀ (X : C), CategoryTheory.Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ E]
  [inst_7 : ∀ (X : C), CategoryTheory.Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
  [inst_8 :
    ∀ (X : C) (W : J.Cover X) (P : CategoryTheory.Functor Cᵒᵖ D),
      CategoryTheory.Limits.PreservesLimit (W.index P).multicospan F]
  (P : CategoryTheory.Functor Cᵒᵖ D) {FD : D → D → Type u_3} {CD : D → Type u_4}
  [inst_9 : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)] [inst_10 : CategoryTheory.ConcreteCategory D FD]
  [inst_11 :
    CategoryTheory.Limits.PreservesLimitsOfSize.{max v u, max v u, v_1, u_4, u_1, u_4 + 1} (CategoryTheory.forget D)]
  [inst_12 : ∀ (X : C), CategoryTheory.Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ (CategoryTheory.forget D)]
  [inst_13 : (CategoryTheory.forget D).ReflectsIsomorphisms],
  (J.sheafifyCompIso F P).inv = J.sheafifyLift (CategoryTheory.Functor.whiskerRight (J.toSheafify P) F) ⋯

Defined in

Mathlib.CategoryTheory.Sites.CompatibleSheafification

Cited by

1 results in Mathlib

Foundations

Depth 59 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasLimitsOfShapeCategoryTheory.Limits.HasLimitsOfShapeCategoryTheory.Limits.HasColimitsOfShapeCategoryTheory.Limits.HasColimitsOfShapeCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.PreservesLimitFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.PreservesLimitsOfSizeCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Functor.ReflectsIsomorphisms