• Source: Semi-abelian category

In mathematics, specifically in category theory, a semi-abelian category is a pre-abelian category in which the induced morphism





f
¯


:
coim
⁡
f
→
im
⁡
f


{\displaystyle {\overline {f}}:\operatorname {coim} f\rightarrow \operatorname {im} f}

is a bimorphism, i.e., a monomorphism and an epimorphism, for every morphism



f


{\displaystyle 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


{\displaystyle Q}

be the quiver

and



K


{\displaystyle K}

be a field. The category of finitely generated projective modules over the algebra



K
Q


{\displaystyle 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





f
¯




{\displaystyle {\overline {f}}}

is a monomorphism for each morphism



f


{\displaystyle f}

. Accordingly, right semi-abelian categories are pre-abelian categories such that





f
¯




{\displaystyle {\overline {f}}}

is an epimorphism for each morphism



f


{\displaystyle 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.




Citations






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.

Kata Kunci Pencarian:

• Produk (teori kategori)
• Semi-abelian category
• Pre-abelian category
• Abelian category
• Quasi-abelian category
• Category of groups
• Semi-simplicity
• Abelian group
• Magma (algebra)
• Product (category theory)
• Semidirect product