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 (see quasi-abelian category#History).
Properties
The two properties used in the definition can be characterized by several equivalent conditions.Kopylov et al., 2012.
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.Bonet et al., 2004/2005.Sieg et al., 2011, Example 4.1.Rump, 2011, p. 44. Let be the quiver
and be a field. The category of finitely generated projective modules over the algebra is semiabelian.Rump, 2008, p. 993.
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 .Rump, 2001.
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.Rump, 2001.
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