Theorems · Definition · category theory

SheafOfModules.pullbackPushforwardAdjunction

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {D : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        {J : CategoryTheory.GrothendieckTopology C} →
          {K : CategoryTheory.GrothendieckTopology D} →
            {F : CategoryTheory.Functor C D} →
              {S : CategoryTheory.Sheaf J RingCat} →
                {R : CategoryTheory.Sheaf K RingCat} →
                  [inst_2 : F.IsContinuous J K] →
                    (φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R) →
                      [inst_3 : (SheafOfModules.pushforward φ).IsRightAdjoint] →
                        SheafOfModules.pullback φ ⊣ SheafOfModules.pushforward φ

Given a continuous functor between sites F, and a morphism of sheaves of rings S ⟶ (F.sheafPushforwardContinuous RingCat.{u} J K).obj R, this is the adjunction between the corresponding pullback and pushforward functors on the categories of sheaves of modules.

Defined in

Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackContinuous

Cited by

7 results in Mathlib

Foundations

Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsContinuousCategoryTheory.Functor.IsRightAdjoint