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Semi abelian category

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

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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 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.
  • Let Q be the quiver
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    and k be a field. The category of finitely generated projective modules over the algebra k Q is semi-abelian.

    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.

    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 ¯ is a monomorphism for each morphism f . Accordingly, right quasi-abelian categories are pre-abelian categories such that 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

    Semi-abelian category Wikipedia