Theorems · Definition · category theory

CategoryTheory.Functor.IsDenseSubsite.sheafifyAdjunctionOfIsEquivalence

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    {D : Type u_2} →
      [inst_1 : CategoryTheory.Category.{v_2, u_2} D] →
        (J : CategoryTheory.GrothendieckTopology C) →
          (K : CategoryTheory.GrothendieckTopology D) →
            (G : CategoryTheory.Functor C D) →
              (A : Type u_4) →
                [inst_2 : CategoryTheory.Category.{v_4, u_4} A] →
                  [inst_3 : CategoryTheory.Functor.IsDenseSubsite J K G] →
                    [inst_4 : (G.sheafPushforwardContinuous A J K).IsEquivalence] →
                      [inst_5 : CategoryTheory.HasWeakSheafify J A] →
                        CategoryTheory.Functor.IsDenseSubsite.sheafifyOfIsEquivalence J K G A ⊣
                          CategoryTheory.sheafToPresheaf K A

Assuming that (C, J) is a dense subsite of (D, K) (via a functor G : C ⥤ D) and sheafPushforwardContinuous G A J K is an equivalence of categories, and that HasWeakSheafify J A holds, then this adjunction shows the existence of a left adjoint to sheafToPresheaf K A.

Defined in

Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Cited by

2 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsDenseSubsiteCategoryTheory.Functor.IsEquivalenceCategoryTheory.HasWeakSheafify