Definition 12.10.1. Let $\mathcal{A}$ be an abelian category.

A

*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 $A, C \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, then also $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

A

*weak Serre subcategory*of $\mathcal{A}$ is a nonempty full subcategory $\mathcal{C}$ of $\mathcal{A}$ such that given an exact sequence\[ A_0 \to A_1 \to A_2 \to A_3 \to A_4 \]with $A_0, A_1, A_3, A_4$ in $\mathcal{C}$, then also $A_2$ in $\mathcal{C}$.

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