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

1. 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 $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

2. 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 $\mathcal{C}$.

Comment #3951 by Shane on

Is it possible to put a warning in (1) that the sequence is not a short exact sequence? That trips me up every time I look at this.

Comment #3957 by on

Maybe your comment here can serve as a warning. The trouble with putting warnings is where does one stop? In every lemma or definition there is some absolutely essential thing that should not be overlooked (and in well written lemmas all the assumptions are necessary). If we try to emphasize certain of these conditions over others, then when are we going to stop?

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