Definition 12.5.1 (0109)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 12: Homological Algebra
Section 12.5: Abelian categories
Definition 12.5.1 (cite)

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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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Comment #9917 by Autistic Retard on January 04, 2025 at 11:38

Should the "classical" equivalence : $A$ is abelian iff $A$ 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:

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