In the following, we work with additive categories.

We say that a category is weakly idempotent complete if all the epimorphisms which admit a section have kernel. It's equivalent to the dual statement: all the monomorphisms which admit a retract have cokernel.

A stronger notion is the notion of idempotent complete. A category is idempotent complete if each morphism in $\{f:A \rightarrow A \;|\; f^2=f\}$ has a kernel or equivalently a cokernel. As suggested by the name of these notions, an idempotent complete category is weakly idempotent.

We know that an abelian category is idempotent complete.

We have the following examples :

• We consider the category $K_{vect} ^{ ^{\ge n}}$ of vector spaces over a field $K$ with no non-nul vector space of smaller dimension than $n$. Then $K_{vect} ^{\ge n}$ is not weakly idempotent complete since the trivial projection $K^{n+1} \twoheadrightarrow K^n$ has a section but no kernel.

• We consider the category $K_{vect} ^ {^{\equiv 0 [2]}}$ of vector spaces over a field $K$ with pair dimension or infinite dimension. Then $K_{vect} ^ {^{\equiv 0 [2]}}$ is weakly idempotent complete. But it's not idempotent complete since the projector $K^{2} \stackrel{\begin{pmatrix}Id & 0 \\ 0 & 0\end{pmatrix}}{\rightarrow} K^2$ has no kernel or cokernel.

So this is the question :

Is someone knows an example of an idempotent complete category which is not abelian?

(here we don't need to have Grothendieck's axioms)

Thanks you, Timothée