Is every balanced pre-abelian category abelian? That is, given an additive category $\mathcal{A}$ in which cokernels and kernels exists, such that every morphism, which is a mono- and an epimorphism, is an isomorphism; does it follow that $\mathcal{A}$ is abelian? Note that it would suffice to prove that the canonical morphism $coim(f) \to im(f)$, where $f$ is an arbitrary morphism, is a mono- and an epimorphism.


Note that the usual examples for non-abelian categories somehow suggest this (filtered modules, topological abelian groups). See also this [related question](https://mathoverflow.net/questions/41353/abelian-categories-vs-additive-categories).


After a google search I have found the following theorem (in "Basic homological algebra" by M. Scott Osborne, Cor. 7.18): If $\mathcal{A}$ is a balanced pre-abelian category with a separating class of projectives and a coseparating class of injectives, then $\mathcal{A}$ is abelian. Ok then this does not seem to be true in general. Does anybody know an example?



Is every balanced pre-abelian category abelian?
Let $B$ be the abelian category of 3-term sequences of vector spaces and linear maps $V^{(1)}\to V^{(2)}\to V^{(3)}$ (the composition can be nonzero). There are 6 indecomposable objects in this category; denote them by $E\_1$, $E\_2$, $E\_3$, $E\_{12}$, $E\_{23}$, and $E\_{123}$. Here $dim E^{(i)}\_J=1$ for $i\in J$ and $0$ otherwise. Let $A\subset B$ be the full additive subcategory whose objects are the direct sums of all the indecomposables except $E\_{12}$ and $f$ be the morphism $E\_3\to E\_{123}$.


Then one has $Coker\_B f=E\_{12}$, $Coker\_A f=E\_1$ and $Im\_A f=E\_{23}$, while $Coim\_A f=E\_3$ and therefore $Coker\_A(Coim\_A f\to Im\_A f)=E\_2\ne0$. It is easy to see that any morphism in $A$ has a kernel and cokernel. Besides, any morphism in $A$ with zero kernel and cokernel has also zero kernel and cokernel in $B$, hence is an isomorphism.


Further discussion can be found in my paper <http://arxiv.org/abs/1006.4343> , Example A.5(7) (pages 59-60 in version 2).


