Giraud subcategory

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

Definition

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

Properties

Let $ℬ$ be Giraud in the Grothendieck category $𝒜$ and $i:ℬ\to 𝒜$ 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 $a:𝒜\to ℬ$ of $i$ is exact.
• Let $𝒞$ be a localizing subcategory of $𝒜$ and $𝒜/𝒞$ be the associated quotient category. The section functor $S:𝒜/𝒞\to 𝒜$ is fully faithful and induces an equivalence between $𝒜/𝒞$ and the Giraud subcategory $ℬ$ given by the $𝒞$-closed objects in $𝒜$.

See also

• Localizing subcategory

References

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