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

• Bonet, J.; Dierolf, S. (1 January 2006). "The Pullback for bornological and ultrabornological spaces". Note di Matematica. 25 (1): 63–67. doi:10.1285/i15900932v25n1p63.
• Kopylov, Yaroslav; Wegner, Sven-Ake (October 2012). "On the Notion of a Semi-Abelian Category in the Sense of Palamodov". Applied Categorical Structures. 20 (5): 531–541. arXiv:1406.6804. doi:10.1007/s10485-011-9249-0.
• Rump, Wolfgang (December 2008). "A counterexample to Raikov's conjecture". Bulletin of the London Mathematical Society. 40 (6): 985–994. doi:10.1112/blms/bdn080.
• Rump, Wolfgang (2001). "Almost abelian categories" (PDF). Cahiers de Topologie et Géométrie Différentielle Catégoriques. 42 (3): 163–225.
• Rump, Wolfgang (January 2011). "Analysis of a problem of Raikov with applications to barreled and bornological spaces". Journal of Pure and Applied Algebra. 215 (1): 44–52. doi:10.1016/j.jpaa.2010.02.031.
• Sieg, Dennis; Wegner, Sven-Ake (November 2011). "Maximal exact structures on additive categories: Maximal exact structures". Mathematische Nachrichten. 284 (16): 2093–2100. doi:10.1002/mana.200910154.