Theorems · Definition · category theory

PresheafOfModules.sheafificationAdjunction

{C : Type u'} →
  [inst : CategoryTheory.Category.{v', u'} C] →
    {J : CategoryTheory.GrothendieckTopology C} →
      {R₀ : CategoryTheory.Functor Cᵒᵖ RingCat} →
        {R : CategoryTheory.Sheaf J RingCat} →
          (α : R₀ ⟶ R.obj) →
            [inst_1 : CategoryTheory.Presheaf.IsLocallyInjective J α] →
              [inst_2 : CategoryTheory.Presheaf.IsLocallySurjective J α] →
                [inst_3 : J.WEqualsLocallyBijective AddCommGrpCat] →
                  [inst_4 : CategoryTheory.HasWeakSheafify J AddCommGrpCat] →
                    PresheafOfModules.sheafification α ⊣
                      (SheafOfModules.forget R).comp (PresheafOfModules.restrictScalars α)

Given a locally bijective morphism α : R₀ ⟶ R.val where R₀ is a presheaf of rings and R a sheaf of rings, this is the adjunction sheafification.{v} α ⊣ SheafOfModules.forget R ⋙ restrictScalars α.

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification

Cited by

3 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Presheaf.IsLocallyInjectiveCategoryTheory.Presheaf.IsLocallySurjectiveCategoryTheory.GrothendieckTopology.WEqualsLocallyBijectiveCategoryTheory.HasWeakSheafify