Theorems · Definition · category theory

CategoryTheory.presheafToSheafCompComposeAndSheafifyIso

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    (J : CategoryTheory.GrothendieckTopology C) →
      {A : Type u_1} →
        {B : Type u_2} →
          [inst_1 : CategoryTheory.Category.{v_1, u_1} A] →
            [inst_2 : CategoryTheory.Category.{v_2, u_2} B] →
              (F : CategoryTheory.Functor A B) →
                [inst_3 : CategoryTheory.HasWeakSheafify J B] →
                  [inst_4 : CategoryTheory.HasWeakSheafify J A] →
                    [J.PreservesSheafification F] →
                      (CategoryTheory.presheafToSheaf J A).comp (CategoryTheory.Sheaf.composeAndSheafify J F) ≅
                        ((CategoryTheory.Functor.whiskeringRight Cᵒᵖ A B).obj F).comp
                          (CategoryTheory.presheafToSheaf J B)

The canonical isomorphism between presheafToSheaf J A ⋙ Sheaf.composeAndSheafify J F and (whiskeringRight Cᵒᵖ A B).obj F ⋙ presheafToSheaf J B when F : A ⥤ B preserves sheafification.

Defined in

Mathlib.CategoryTheory.Sites.PreservesSheafification

Cited by

2 results in Mathlib

Foundations

Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.HasWeakSheafifyCategoryTheory.HasWeakSheafifyCategoryTheory.GrothendieckTopology.PreservesSheafification