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 of an [abelian category|quotient category]. The section functor is full and [faithful functors|fully faithful] and induces an equivalence between and the Giraud subcategory given by the -closed objects in.