Theorems · Definition · category theory

CategoryTheory.plusPlusAdjunction

{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 :
                        ∀ (P : CategoryTheory.Functor Cᵒᵖ D) (X : C) (S : J.Cover X),
                          CategoryTheory.Limits.HasMultiequalizer (S.index P)] →
                      [inst_5 : ∀ (X : C), CategoryTheory.Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] →
                        [inst_6 :
                            ∀ (X : C),
                              CategoryTheory.Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ (CategoryTheory.forget D)] →
                          [inst_7 : (CategoryTheory.forget D).ReflectsIsomorphisms] →
                            CategoryTheory.plusPlusSheaf J D ⊣ CategoryTheory.sheafToPresheaf J D

The sheafification functor is left adjoint to the forgetful functor.

Defined in

Mathlib.CategoryTheory.Sites.ConcreteSheafification

Cited by

2 results in Mathlib

Foundations

Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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