Hi all, 


I was reading a proof that the localization of categories preserves abelianess. Although the author didn't mentioned explicitly, it seems to me that the proof is reduced to the statement that



> 
> An additive category is a balanced category(i.e., every monic epimorphism is an isomorphism).
> 
> 
> 


I know that an abelian category is balanced, but I'm afraid not so an additive category. Any suggestions or reference?


Thanks.



Is an additive category a balanced category?
An additive category need not be balanced. Consider the full subcategory of abelian groups consisting of all torsion-free groups. Then for any $n\neq0$, the map $n:\mathbb{Z}\to\mathbb{Z}$ is monic and epic, but it is not an isomorphism unless $n=\pm1$. The only nontrivial part of this is that it is epic, and this is simply the statement that a map from $\mathbb{Z}$ to a torsion-free group is uniquely determined by its value at $n$.


