Theorems · Definition · category theory

CategoryTheory.ObjectProperty.SerreClassLocalization.abelian

{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] →
                [L.IsLocalization P.isoModSerre] →
                  [inst_5 : CategoryTheory.Preadditive D] → [L.Additive] → CategoryTheory.Abelian D

If L : C ⥤ D is a localization functor with respect to a Serre class P in the abelian category C, then D is an abelian category. Note that we assume that D has already been equipped with a preadditive structure, and that L is additive. Otherwise, see the results in the file Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean which applies because P.isoModSerre has a calculus of left and right fractions.

Defined in

Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Cited by

6 results in Mathlib

Foundations

Depth 118 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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