Definition 12.5.7. Let $\mathcal{A}$ be an additive category. We say a sequence of morphisms

$\ldots \to x \to y \to z \to \ldots$

in $\mathcal{A}$ is a complex if the composition of any two (drawn) arrows is zero. If $\mathcal{A}$ is abelian then we say a sequence as above is exact at $y$ if $\mathop{\mathrm{Im}}(x \to y) = \mathop{\mathrm{Ker}}(y \to z)$. We say it is exact if it is exact at every object. A short exact sequence is an exact complex of the form

$0 \to A \to B \to C \to 0.$

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