Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.sheafifyCompIso

{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] →
                        [∀ (X : C), CategoryTheory.Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ F] →
                          [∀ (X : C) (W : J.Cover X) (P : CategoryTheory.Functor Cᵒᵖ D),
                                CategoryTheory.Limits.PreservesLimit (W.index P).multicospan F] →
                            (P : CategoryTheory.Functor Cᵒᵖ D) → (J.sheafify P).comp F ≅ J.sheafify (P.comp F)

The isomorphism between the sheafification of P composed with F and the sheafification of P ⋙ F. Use the lemmas whisker_right_to_sheafify_sheafify_comp_iso_hom, to_sheafify_comp_sheafify_comp_iso_inv and sheafify_comp_iso_inv_eq_sheafify_lift to reduce the components of this isomorphism to a state that can be handled using the universal property of sheafification.

Defined in

Mathlib.CategoryTheory.Sites.CompatibleSheafification

Cited by

11 results in Mathlib

Foundations

Depth 53 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.PreservesLimit