Localizing subcategory

In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category. Localizing subcategories are certain Serre subcategories. They are strongly linked to the notion of a quotient category.

Let ${\displaystyle {\mathcal {A}}}$ be an abelian category. A non-empty full subcategory ${\displaystyle {\mathcal {C}}}$ is called a Serre subcategory (or also a dense subcategory), if for every short exact sequence ${\displaystyle 0\rightarrow A'\rightarrow A\rightarrow A''\rightarrow 0}$ in ${\displaystyle {\mathcal {A}}}$ the object ${\displaystyle A}$ is in ${\displaystyle {\mathcal {C}}}$ if and only if the objects ${\displaystyle A'}$ and ${\displaystyle A''}$ belong to ${\displaystyle {\mathcal {C}}}$. In words: ${\displaystyle {\mathcal {C}}}$ is closed under subobjects, quotient objects and extensions.

...
Wikipedia