Theorems · Theorem · category theory

CategoryTheory.ObjectProperty.SerreClassLocalization.essImage_whiskeringLeft

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C] {D : Type u'}
  [inst_2 : CategoryTheory.Category.{v', u'} D] (L : CategoryTheory.Functor C D) (P : CategoryTheory.ObjectProperty C)
  [inst_3 : P.IsSerreClass] (E : Type u'') [inst_4 : CategoryTheory.Category.{v'', u''} E]
  [inst_5 : CategoryTheory.Abelian E] [inst_6 : L.IsLocalization P.isoModSerre] [inst_7 : CategoryTheory.Preadditive D]
  [inst_8 : L.Additive],
  (CategoryTheory.ObjectProperty.SerreClassLocalization.whiskeringLeft L P E).essImage = fun G =>
    P.isoModSerre.IsInvertedBy G.obj

Let L : C ⥤ D be a localization functor with respect to a Serre class P in the abelian category C. If G : C ⥤ₑ E is an exact functor to an abelian category, it "factors" through D (i.e. it is in the essential image of whiskeringLeft L P E : (D ⥤ₑ E) ⥤ C ⥤ₑ E) iff G inverts the class of morphisms P.isoModSerre.

Defined in

Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Cited by

0 results in Mathlib

Foundations

Depth 124 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.CategoryCategoryTheory.ObjectProperty.IsSerreClassCategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.Functor.IsLocalizationCategoryTheory.PreadditiveCategoryTheory.Functor.Additive