# In an abelian category,every morphism can be written as composition of epi and mono. [duplicate]

Following Weibel's book on homological algebra, he states without proof that every morphism $f\colon A \to B$ can be written as composition of an epimorphism followed by a monomorphism.

After many attempts,I am unable prove this. It is easy to show that

$f\colon A \to B$ factors as $A \twoheadrightarrow Coker(Ker(f)) \to B$ and $A \to Ker(Coker(f))\rightarrowtail B$ .Moreover there is an arrow from $Ker(Coker(f)) \to Coker(Ker(f))$ such that the all the diagrams commute.

Additionaly, in Hilton and Stammbach's book on the same subject, they take this property as an axiom.

## marked as duplicate by Zhen Lin category-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 7 '14 at 9:53

• For completeness, you should state the definition of abelian category Weibel uses. Of course, all you have to do is prove that the canonical $\operatorname{Ker} \operatorname{coker} f \to \operatorname{Coker} \operatorname{ker} f$ is an isomorphism. I am sure this has been asked before on this site somewhere... – Zhen Lin Aug 22 '14 at 11:26