Quasi-abelian category

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.
Definition.
Let formula_1 be a pre-abelian category. A morphism formula_2 is a kernel (a cokernel) if there exists a morphism formula_3 such that formula_2 is a kernel (cokernel) of formula_3. The category formula_1 is quasi-abelian if for every kernel formula_7 and every morphism formula_8 in the pushout diagram
the morphism formula_9 is again a kernel and, dually, for every cokernel formula_10 and every morphism formula_11 in the pullback diagram
the morphism formula_12 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.
Properties.
Let formula_2 be a morphism in a quasi-abelian category. Then the induced morphism formula_14 is always a bimorphism, i.e., a monomorphism and an epimorphism. A quasi-abelian category is therefore always semi-abelian.
Examples.
Every "abelian category" is quasi-abelian. Typical non-abelian examples arise in functional analysis.
History.
The concept of quasi-abelian category was developed in the 1960s. The history is involved. 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.
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.