Theorems · Theorem · category theory

PresheafOfModules.toPresheaf_map_sheafificationHomEquiv_def

∀ {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]
  {P : PresheafOfModules R₀} {F : SheafOfModules R} (f : (PresheafOfModules.sheafification α).obj P ⟶ F),
  (PresheafOfModules.toPresheaf R₀).map ((PresheafOfModules.sheafificationHomEquiv α) f) =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.toSheafify J P.presheaf)
      ((PresheafOfModules.toPresheaf R.obj).map f.val)

Defined in

Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification

Cited by

1 results in Mathlib

Foundations

Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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