Semi-abelian category

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

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.

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.
• Let $Q$ be the quiver

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

and $K$ be a field. The category of finitely generated projective modules over the algebra $KQ$ is semiabelian.

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

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.

References

• José Bonet, J., Susanne Dierolf, The pullback for bornological and ultrabornological spaces. Note Mat. 25(1), 63–67 (2005/2006).
• Yaroslav Kopylov and Sven-Ake Wegner, On the notion of a semi-abelian category in the sense of Palamodov, Appl. Categ. Structures 20 (5) (2012) 531–541.
• Wolfgang Rump, A counterexample to Raikov's conjecture, Bull. London Math. Soc. 40, 985–994 (2008).
• Wolfgang Rump, Almost abelian categories, Cahiers Topologie Géom. Différentielle Catég. 42(3), 163–225 (2001).
• Wolfgang Rump, Analysis of a problem of Raikov with applications to barreled and bornological spaces, J. Pure and Appl. Algebra 215, 44–52 (2011).
• Dennis Sieg and Sven-Ake Wegner, Maximal exact structures on additive categories, Math. Nachr. 284 (2011), 2093–2100.