Semi-abelian category explained
\overline{f}:\operatorname{coim}f → \operatorname{im}f
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.[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 quasiabelian category is semiabelian. In particular, every abelian category is semi-abelian. Non-quasiabelian examples are the following.
be the
quiverand
be a
field. The category of
finitely generated projective modules over the
algebra
is semiabelian.
[5] 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
.
[6] 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.[7]
Citations
- Kopylov et al., 2012.
- Bonet et al., 2004/2005.
- Sieg et al., 2011, Example 4.1.
- Rump, 2011, p. 44.
- Rump, 2008, p. 993.
- Rump, 2001.
- 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.