Wikipedia says that a semi-abelian category is a pre-abelian category in which for each morphism ${\displaystyle f}$ the induced morphism ${\displaystyle {\overline {f}}:\operatorname {coim} f\rightarrow \operatorname {im} f}$ is always a monomorphism and an epimorphism. But examples in my brain which is pre-abelian but not abelian are like Banach space and Hausdorff abelian topological groups, which are all semi-abelian. So can anyone give me an example which is pre-abelian but not semi-abelian?
Thank you!
Let $\mathcal{C}$ be the (abelian) category of diagrams $U\stackrel{\alpha}{\to}V\stackrel{\beta}{\to}W$ of vector spaces over a fixed field $k$.
Let $\mathcal{D}$ be the full subcategory consisting of those diagrams where $V=\text{im}\alpha+\ker\beta$.
$\mathcal{D}$ is closed under taking quotients in $\mathcal{C}$, and so has cokernels, which are the same as in $\mathcal{C}$.
Every object $U\stackrel{\alpha}{\to}V\stackrel{\beta}{\to}W$ of $\mathcal{C}$ has a unique subobject $U\to(\text{im}\alpha+\ker\beta)\to W$ that is maximal subject to being in $\mathcal{D}$. So $\mathcal{D}$ has kernels: a kernel in $\mathcal{D}$ is the maximal $\mathcal{D}$-subobject of the kernel in $\mathcal{C}$.
So $\mathcal{D}$ is pre-abelian.
Now consider the obvious nonzero map from $k\stackrel{\sim}{\to}k\stackrel{\sim}{\to}k$ to $k\to0\to0$. The coimage in $\mathcal{D}$ of this map is $k\stackrel{\sim}{\to}k\to0$, and the image is $k\to0\to0$, and the natural map from the coimage to the image is not monic.
So $\mathcal{D}$ is not semi-abelian.