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.

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 be a field. The category of finitely generated projective modules over the algebra 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 is a monomorphism for each morphism. Accordingly, right quasi-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.

Citations