Semi-abelian category

In mathematics, specifically in category theory, a semi-abelian category is a pre-abelian category in which the induced morphism formula_1 is a bimorphism, i.e., a monomorphism and an epimorphism, for every morphism formula_2.
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 quasi-abelian category is semi-abelian. In particular, every abelian category is semi-abelian. Non quasi-abelian examples are the following.
and formula_4 be a field. The category of finitely generated projective modules over the algebra formula_5 is semi-abelian.
History.
The concept of a semi-abelian category was developed in the 1960s. Raikov conjectured that the notion of a quasi-abelian category is equivalent to that of a semi-abelian category. Around 2005 it turned out that the conjecture is false.
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 formula_6 is a monomorphism for each morphism formula_2. Accordingly, right semi-abelian categories are pre-abelian categories such that formula_6 is an epimorphism for each morphism formula_2.
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.