Theorems · Definition · category theory

SheafOfModules.pullbackIso

{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 : (PresheafOfModules.pushforward φ.hom).IsRightAdjoint] →
                        [inst_4 : CategoryTheory.HasWeakSheafify K AddCommGrpCat] →
                          [inst_5 : K.WEqualsLocallyBijective AddCommGrpCat] →
                            SheafOfModules.pullback φ ≅
                              (SheafOfModules.forget S).comp
                                ((PresheafOfModules.pullback φ.hom).comp
                                  (PresheafOfModules.sheafification (CategoryTheory.CategoryStruct.id R.obj)))

The pullback functor on sheaves of modules can be described as a composition of the forget functor to presheaves, the pullback on presheaves of modules, and the sheafification functor.

Defined in

Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackContinuous

Cited by

0 results in Mathlib

Foundations

Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsContinuousCategoryTheory.Functor.IsRightAdjointCategoryTheory.HasWeakSheafifyCategoryTheory.GrothendieckTopology.WEqualsLocallyBijective