Giraud subcategory

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

Definition
Let $$\mathcal{A}$$ be a Grothendieck category. A full subcategory $$\mathcal{B}$$ is called reflective, if the inclusion functor $$i\colon\mathcal{B}\rightarrow\mathcal{A}$$ has a left adjoint. If this left adjoint of $$i$$ also preserves kernels, then $$\mathcal{B}$$ is called a Giraud subcategory.

Properties
Let $$\mathcal{B}$$ be Giraud in the Grothendieck category $$\mathcal{A}$$ and $$i\colon\mathcal{B}\rightarrow\mathcal{A}$$ the inclusion functor.
 * $$\mathcal{B}$$ is again a Grothendieck category.
 * An object $$X$$ in $$\mathcal{B}$$ is injective if and only if $$i(X)$$ is injective in $$\mathcal{A}$$.
 * The left adjoint $$a\colon\mathcal{A}\rightarrow\mathcal{B}$$ of $$i$$ is exact.
 * Let $$\mathcal{C}$$ be a localizing subcategory of $$\mathcal{A}$$ and $$\mathcal{A}/\mathcal{C}$$ be the associated quotient category. The section functor $$S\colon\mathcal{A}/\mathcal{C}\rightarrow\mathcal{A}$$ is fully faithful and induces an equivalence between $$\mathcal{A}/\mathcal{C}$$ and the Giraud subcategory $$\mathcal{B}$$ given by the $$\mathcal{C}$$-closed objects in $$\mathcal{A}$$.