Giraud subcategory

Short description: Mathematical subcategories of Grothendieck categories

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.