Giraud subcategory

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

Definition

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

Properties

Let ${\displaystyle {\mathcal {B}}}$ be Giraud in the Grothendieck category ${\displaystyle {\mathcal {A}}}$ and ${\displaystyle i\colon {\mathcal {B}}\rightarrow {\mathcal {A}}}$ the inclusion functor.

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

See also

• Localizing subcategory

References

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