Semi-abelian category

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 quasi-abelian category is semi-abelian. In particular, every abelian category is semi-abelian. Non quasi-abelian examples are the following.

• The category of (possibly non Hausdorff) bornological spaces is semi-abelian.[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 semi-abelian.[5]

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.[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 quasi-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]

Citations

1. Kopylov et. al., 2012.
2. Bonet et. al., 2004/2005.
3. Sieg et. al., 2011, Example 4.1.
4. Rump, 2011, p. 44.
5. Rump, 2008, p. 993.
6. Rump, 2011, p. 44f.
7. Rump, 2001.
8. Rump, 2001.

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.