In mathematics, specifically in category theory, a quasi-abelian category is a pre-abelian category in which the pushout of a kernel along arbitrary morphisms is again a kernel and, dually, the pullback of a cokernel along arbitrary morphisms is again a cokernel.
is a kernel (a cokernel) if there exists a morphism
is again a kernel and, dually, for every 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.
A quasi-abelian category is therefore always semi-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] The concept of quasi-abelian category was developed in the 1960s.
[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.
[6] 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.