Theorems · Theorem · category theory

CategoryTheory.toSheafify_plusPlusIsoSheafify_hom_assoc

∀ {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) {Z : CategoryTheory.Functor Cᵒᵖ D} (h : CategoryTheory.sheafify J P ⟶ Z),
  CategoryTheory.CategoryStruct.comp (J.toSheafify P)
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.plusPlusIsoSheafify J D P).hom h) =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.toSheafify J P) h

Defined in

Mathlib.CategoryTheory.Sites.LeftExact

Cited by

0 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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