Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.Plus.meqOfSep

{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] →
                  [inst_3 :
                      ∀ {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)] →
                          (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) →
                              (X : C) →
                                (S : J.Cover X) →
                                  (s : CategoryTheory.Meq (J.plusObj P) S) →
                                    (T : (I : S.Arrow) → J.Cover I.Y) →
                                      (t : (I : S.Arrow) → CategoryTheory.Meq P (T I)) →
                                        (∀ (I : S.Arrow), ↑s I = CategoryTheory.GrothendieckTopology.Plus.mk (t I)) →
                                          CategoryTheory.Meq P (S.bind T)

An auxiliary definition to be used in the proof of exists_of_sep below. Given a compatible family of local sections for P⁺, and representatives of said sections, construct a compatible family of local sections of P over the combination of the covers associated to the representatives. The separatedness condition is used to prove compatibility among these local sections of P.

Defined in

Mathlib.CategoryTheory.Sites.ConcreteSheafification

Cited by

1 results in Mathlib

Foundations

Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryFunLikeCategoryTheory.ConcreteCategoryCategoryTheory.Limits.PreservesLimitsOfShapeCategoryTheory.Limits.HasColimitsOfShapeCategoryTheory.Limits.HasMultiequalizerCategoryTheory.Limits.PreservesColimitsOfShape