Giraud subcategory

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

Definition

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

Properties

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

See also

* Localizing subcategory

References

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

Category theory
Homological algebra