In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud.

Definition

Let be a Grothendieck category. A full subcategory is called reflective, if the inclusion functor has a left adjoint. If this left adjoint of also preserves kernels, then is called a Giraud subcategory.

Properties

Let be Giraud in the Grothendieck category and the inclusion functor.

  • is again a Grothendieck category.
  • An object in is injective if and only if is injective in .
  • The left adjoint of is exact.
  • Let be a localizing subcategory of and be the associated quotient category. The section functor is fully faithful and induces an equivalence between and the Giraud subcategory given by the -closed objects in .

See also

References

  • Bo Stenström; 1975; Rings of quotients. Springer.
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