Let be a pre-abelian category. A morphism is a kernel (a cokernel) if there exists a morphism such that is a kernel (cokernel) of . The category is quasi-abelian if for every kernel and every morphism in the pushout diagram
the morphism is again a kernel and, dually, for every cokernel and every morphism in the pullback diagram
the morphism is again a cokernel.
Equivalently, a quasi-abelian category is a pre-abelian category in which the system of all kernel-cokernel pairs forms an exact structure.
Given a pre-abelian category, those kernels, which are stable under arbitrary pushouts, are sometimes called the semi-stable kernels. Dually, cokernels, which are stable under arbitrary pullbacks, are called semi-stable cokernels.[1]
Properties
Let be a morphism in a quasi-abelian category. Then the induced morphism is always a bimorphism, i.e., a monomorphism and an epimorphism. A quasi-abelian category is therefore always semi-abelian.
Examples and non-examples
Every abelian category is quasi-abelian. Typical non-abelian examples arise in functional analysis.[2]
Contrary to the claim by Beilinson,[3] the category of complete separated topological vector spaces with linear topology is not quasi-abelian.[4] On the other hand, the category of (arbitrary or Hausdorff) topological vector spaces with linear topology is quasi-abelian.[4]
History
The concept of quasi-abelian category was developed in the 1960s. The history is involved.[5] This is in particular due to Raikov's conjecture, which stated that the notion of a semi-abelian category is equivalent to that of a quasi-abelian category. Around 2005 it turned out that the conjecture is false.[6]
Left and right quasi-abelian categories
By dividing the two conditions in the definition, one can define left quasi-abelian categories by requiring that cokernels are stable under pullbacks and right quasi-abelian categories by requiring that kernels stable under pushouts.[7]