Let $\mathcal{A}$ be an abelian category. I knew two definitions of Serre subcategory, and I'd like to show they are equivalent. For the first definition, one can refer to https://stacks.math.columbia.edu/tag/02MN
- Serre subcategory of $\mathcal{A}$ is a nonempty full subcategory $\mathcal{C}$ of $\mathcal{A}$ such that given an exact sequence $$ A \to B \to C $$ with $A,C \in \mathrm{Ob} (\mathcal{C})$, then also $B \in \mathrm{Ob}(\mathcal{C})$.
For the second definition, one can refer to https://en.wikipedia.org/wiki/Localizing_subcategory
2.A nonempty full subcategory $\mathcal{C}$ of $\mathcal{A}$ is called a Serre subcategory (or also a dense subcategory), if for every short exact sequence $$ 0 \to A' \to A \to A'' \to 0 $$ in $\mathcal{A}$, the object $A$ is in $\mathcal{C}$ if and only if the objects $A'$ and $A''$ belong to $\mathcal{C}$. In words: $\mathcal{C}$ is closed under subobjects, quotient objects and extensions.
It is easy to prove 2 by 1, since we can consider $0\to A' \to A$ and $A \to A'' \to 0$. But how to prove 1 by 2?
Note that $$ 0 \to A'/kerf \to A \to Img \to 0 $$ , in which the first and third nonzero terms are in $\mathcal{C}$ since $\mathcal{C}$ closed under subobjects, quotient objects.