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 ${\mathcal {A}}$.

See also

• Localizing subcategory

References

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