In an additive category, given a morphism $f : A\to B$ let $\operatorname{im} f = \ker(\operatorname{coker}(f))$ and $\operatorname{coim} f = \operatorname{coker}(\ker(f))$, if they exist. If they do, then there is a unique morphism $\tilde f$ such that $f = \operatorname{im} f \circ \tilde f \circ \operatorname{coim} f$.

An additive category is "*semi-Abelian*", if every morphism a has a kernel and cokernel and the $\tilde f$ is mono and epi for all $f$.

I put 'semi-Abelian' in quotation marks because this definition conflicts with the on the nLab, but this definition is given in "Derived Functors in Functional Analysis" by J. Wengenroth, which was originally introduced by Palamodov in "Homological Methods in the Theory of Locally Convex Functions" (see J. Wengenroths comment).

> Are "semi-Abelian" categories regular? If not do they have regular
> epi-mono factorizations?

"Regular" should be interpreted in the weakest sense possible in terms of limits, i.e. "pullbacks along strong epis exists".

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This might be "obvious", but it isn't to me. 
A̶l̶s̶o̶,̶ ̶I̶'̶m̶ ̶a̶l̶s̶o̶ ̶i̶n̶t̶e̶r̶e̶s̶t̶e̶d̶ ̶i̶n̶ ̶t̶h̶e̶ ̶q̶u̶e̶s̶t̶i̶o̶n̶ ̶w̶i̶t̶h̶ ̶"̶r̶e̶g̶u̶l̶a̶r̶"̶ ̶r̶e̶p̶l̶a̶c̶e̶d̶ ̶b̶y̶ ̶"̶p̶r̶o̶t̶o̶m̶o̶d̶u̶l̶a̶r̶"̶,̶ ̶b̶u̶t̶ ̶I̶'̶m̶ ̶g̶o̶i̶n̶g̶ ̶t̶o̶ ̶p̶o̶s̶t̶p̶o̶n̶e̶ ̶t̶h̶a̶t̶ ̶q̶u̶e̶s̶t̶i̶o̶n̶ ̶f̶o̶r̶ ̶n̶o̶w̶.̶  

EDIT: Nevermind this latter question: in a left exact pointed category, we have "additive = protomodular + all subobjects are normal" ([Bourceux-Bourn][1] Thm. 3.2.16). At least this covers all the cases where the category is left exact.


  [1]: https://ncatlab.org/nlab/show/Borceux-Bourn