Semi-abelian category

In mathematics, specifically in category theory, a semi-abelian category is a pre-abelian category in which the induced morphism is a bimorphism, i.e., a monomorphism and an epimorphism, for every morphism.
The history of the notion is intertwined with that of a quasi-abelian category, as, for awhile, it was not known whether the two notions are distinct.

Properties

The two properties used in the definition can be characterized by several equivalent conditions.
Every semi-abelian category has a maximal exact structure.
If a semi-abelian category is not quasi-abelian, then the class of all kernel-cokernel pairs does not form an exact structure.

Examples

Every quasiabelian category is semiabelian. In particular, every abelian category is semi-abelian. Non-quasiabelian examples are the following.

The category of bornological spaces is semiabelian.
Let be the quiver

Left and right semi-abelian categories

By dividing the two conditions on the induced map in the definition, one can define left semi-abelian categories by requiring that is a monomorphism for each morphism. Accordingly, right semi-abelian categories are pre-abelian categories such that is an epimorphism for each morphism.
If a category is left semi-abelian and right quasi-abelian, then it is already quasi-abelian. The same holds, if the category is right semi-abelian and left quasi-abelian.