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

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

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