Theorems · Theorem · category theory

PresheafOfModules.sheafification_map

∀ {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]
  {X Y : PresheafOfModules R₀} (f : X ⟶ Y),
  (PresheafOfModules.sheafification α).map f =
    PresheafOfModules.sheafifyMap α (CategoryTheory.toSheafify J X.presheaf) (CategoryTheory.toSheafify J Y.presheaf) f
      ((CategoryTheory.presheafToSheaf J AddCommGrpCat).map ((PresheafOfModules.toPresheaf R₀).map f)) ⋯

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification

Cited by

0 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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