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I'm trying to understand Abelian Categories and I have a problem with the AB4 axiom which states that monomorphisms are kernels and epimorphisms are cokernels. Indeed, on nLab (https://ncatlab.org/nlab/show/abelian+category#MacLane), it is said that this axiom is equivalent to *: "The canonical morphism $Coker(Ker f) \rightarrow Ker(Coker f)$ is an isomorphism". I proved that * implies AB4 but not the converse and the MacLane does not help me.

My idea is to proof that the morphism is a mono and an epi because AB4 implies that mono+epi=iso.

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  • $\begingroup$ Look at the very beginning of the book of Freyd on the subject. $\endgroup$ – Ivan Di Liberti Dec 23 '17 at 14:36
  • $\begingroup$ Can you be more precise? I did not find it the first time I tried in this book $\endgroup$ – Sov Dec 23 '17 at 15:18
  • $\begingroup$ Notation is a bit confusing. If $f : A \to B$ is a morphism, then $ker(f)$ and $coker (f)$ are morphisms. Let us write $ker(f) : KER(f) \to A$ with a "kernel object" $KER(f)$, similarly $coker(f) : B \to COKER(f)$. Then $(\ast)$ reads as: The canonical morphism $COKER(Ker(f)) \to KER(Coker(f))$ is an isomorphism. $\endgroup$ – Paul Frost Sep 5 '18 at 22:11

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