The Stacks project

Definition 12.5.1. A category $\mathcal{A}$ is abelian if it is additive, if all kernels and cokernels exist, and if the natural map $\mathop{\mathrm{Coim}}(f) \to \mathop{\mathrm{Im}}(f)$ is an isomorphism for all morphisms $f$ of $\mathcal{A}$.

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View Definition 12.5.1 as pdf