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.
A quasi-abelian category is an exact category.

Definition

Let be a pre-abelian category. A morphism is a kernel if there exists a morphism such that is a kernel 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.

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.

The category of Banach spaces is quasi-abelian.
The category of Fréchet spaces is quasi-abelian.
The category of locally convex spaces is quasi-abelian.

Contrary to the claim by Beilinson, the category of complete separated topological vector spaces with linear topology is not quasi-abelian. On the other hand, the category of topological vector spaces with linear topology is quasi-abelian.

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.