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In a pre-abelian category, every morphism $f: A \to B$ has a canonical decomposition:

$$ A \to coker(kerf) \to ker(cokerf) \to B $$

How do we obtain the middle morphism, the one from $coker(kerf)$ to $ker(cokerf)$?

Thank you.

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We have the commutative diagram

enter image description here

Then, by universal property of the limit $\ker(\mathrm{coker}\, f)$, we have arrows such that

enter image description here

Then, by universal property of the colimit $\mathrm{coker}(\ker f)$, we have an arrow

enter image description here

which is the searched one !

Edit. My diagrams are a bit tall. Let me know if it is uncomfortable to read and I will reduce them.

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  • $\begingroup$ Thank you very much, that was really understandable. $\endgroup$ – Polydarya Dec 9 '13 at 8:08
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Well, $\DeclareMathOperator{\coker}{coker}$ $\coker f \circ f = 0$, so $f = \ker \coker f \circ c$; but $f \circ \ker f = 0$ and $\ker \coker f$ is monic, so $c \circ \ker f = 0$, so $c = d \circ \coker \ker f$ for some $d$, as required.

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