In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud.
Let A be a Grothendieck category. A full subcategory B is called reflective, if the inclusion functor i : B → A has a left adjoint. If this left adjoint of i also preserves kernels, then B is called a Giraud subcategory.
Let B be Giraud in the Grothendieck category A and i : B → A the inclusion functor.
B is again a Grothendieck category.An object X in B is injective if and only if i ( X ) is injective in A .The left adjoint a : A → B of i is exact.Let C be a localizing subcategory of A and A / C be the associated quotient category. The section functor S : A / C → A is fully faithful and induces an equivalence between A / C and the Giraud subcategory B given by the C -closed objects in A .