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In a course of homological algebra the teacher defined an abelian category as a pre-abelian category in which image and coimages are isomorphic. Explicitly, in a pre-abelian category, given a morphism $f:A\to B $, it always splits as $$A\twoheadrightarrow \text{Coim} f \rightarrow \text{Im} f\hookrightarrow B;$$ if this induced morphism from the coimage to the image is an isomorphism for every arrow $f$, then the category is abelian. We didn't prove that this definition is equivalent to the usual one, and I'm not finding anything online, so maybe I didn't understand correctly. Is this actually a definition of abelian category? The proof is not really necessary, since in the course we'll only use this particular definition.

Arnaud D.
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Dr. Scotti
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    Yes, an Abelian category is a pre-Abelian category where all the induced maps ${\rm Coim},f\to{\rm Im},f$ are isomorphisms. – Berci Sep 13 '21 at 09:42

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