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.