Giraud subcategory

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

kernels, then $\backslash mathcal\{B\}$ is called a Giraud subcategory.

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

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

• Localizing subcategory

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

Category:Category theory

Category:Homological algebra