In mathematics, specifically in category theory, a semi-abelian category is a pre-abelian category in which the induced morphism ${\displaystyle {\overline {f)):\operatorname {coim} f\rightarrow \operatorname {im} f}$ is a bimorphism, i.e., a monomorphism and an epimorphism, for every morphism ${\displaystyle f}$.

Properties

The two properties used in the definition can be characterized by several equivalent conditions.^[1]

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 (possibly non-Hausdorff) bornological spaces is semiabelian.^[2][3][4]
• Let ${\displaystyle Q}$ be the quiver

${\displaystyle {\begin{array}{ccc}1&\xrightarrow {} &2&\xleftarrow {} &3\\\downarrow {}&&\downarrow {}&&\downarrow {}\\4&\xrightarrow {} &5&\xleftarrow {} &6\\\end{array))}$

and ${\displaystyle K}$ be a field. The category of finitely generated projective modules over the algebra ${\displaystyle KQ}$ is semiabelian.^[5]

History

The concept of a semiabelian category was developed in the 1960s. Raikov conjectured that the notion of a quasi-abelian category is equivalent to that of a semiabelian category. Around 2005 it turned out that the conjecture is false.^[6]

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 ${\displaystyle {\overline {f))}$ is a monomorphism for each morphism ${\displaystyle f}$. Accordingly, right semi-abelian categories are pre-abelian categories such that ${\displaystyle {\overline {f))}$ is an epimorphism for each morphism ${\displaystyle f}$.^[7]

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.^[8]