Theorems · Definition · category theory

CategoryTheory.Functor.sheafAdjunctionCocontinuous

{C : Type u_1} →
  {D : Type u_2} →
    [inst : CategoryTheory.Category.{v_1, u_1} C] →
      [inst_1 : CategoryTheory.Category.{v_2, u_2} D] →
        (G : CategoryTheory.Functor C D) →
          (A : Type w) →
            [inst_2 : CategoryTheory.Category.{w', w} A] →
              (J : CategoryTheory.GrothendieckTopology C) →
                (K : CategoryTheory.GrothendieckTopology D) →
                  [inst_3 : G.IsCocontinuous J K] →
                    [inst_4 : ∀ (F : CategoryTheory.Functor Cᵒᵖ A), G.op.HasPointwiseRightKanExtension F] →
                      [inst_5 : G.IsContinuous J K] →
                        G.sheafPushforwardContinuous A J K ⊣ G.sheafPushforwardCocontinuous A J K

Given a functor between sites that is continuous and cocontinuous, the pushforward for the continuous functor G is left adjoint to the pushforward for the cocontinuous functor G.

Defined in

Mathlib.CategoryTheory.Sites.CoverLifting

Cited by

7 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsCocontinuousCategoryTheory.Functor.HasPointwiseRightKanExtensionCategoryTheory.Functor.IsContinuous