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 $\mathcal{A}$.

Comments (1)

Comment #9917 by Autistic Retard on

Should the "classical" equivalence : is abelian iff additive and every mono is a kernel + dual   be added.

The formulation as such is often given, eventhough it's trash as the true meaning of such a proposition is that a mono is the kernel of it's cokernel, which really legitimise the existence of such property.

There are also:

  • 9 comment(s) on Section 12.5: Abelian categories

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View Definition 12.5.1 as pdf