Giraud subcategory

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 Localizing subcategory

References

Bo Stenström; 1975; Rings of quotients. Springer.