Theorems · Definition · category theory

PresheafOfModules.sheafification

{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) →
            [CategoryTheory.Presheaf.IsLocallyInjective J α] →
              [CategoryTheory.Presheaf.IsLocallySurjective J α] →
                [J.WEqualsLocallyBijective AddCommGrpCat] →
                  [CategoryTheory.HasWeakSheafify J AddCommGrpCat] →
                    CategoryTheory.Functor (PresheafOfModules R₀) (SheafOfModules R)

Given a locally bijective morphism α : R₀ ⟶ R.val where R₀ is a presheaf of rings and R a sheaf of rings (i.e. R identifies to the sheafification of R₀), this is the associated sheaf of modules functor PresheafOfModules.{v} R₀ ⥤ SheafOfModules.{v} R.

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification

Cited by

9 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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