# 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 ${\mathcal {A}}$ be a pre-abelian category. A morphism $f$ is a kernel (a cokernel) if there exists a morphism $g$ such that $f$ is a kernel (cokernel) of $g$. The category ${\mathcal {A}}$ is quasi-abelian if for every kernel $f:X\rightarrow Y$ and every morphism $h:X\rightarrow Z$ in the pushout diagram
${\begin{array}{ccc}X&{\xrightarrow {f}}&Y\\\downarrow _{h}&&\downarrow _{h'}\\Z&{\xrightarrow {f'}}&Q\end{array}}$
the morphism $f'$ is again a kernel and, dually, for every cokernel $g:X\rightarrow Y$ and every morphism $h:Z\rightarrow Y$ in the pullback diagram
${\begin{array}{ccc}P&{\xrightarrow {g'}}&Z\\\downarrow _{h'}&&\downarrow _{h}\\X&{\xrightarrow {g}}&Y\end{array}}$
the morphism $g'$ 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 $f$ be a morphism in a quasi-abelian category. Then the induced morphism ${\overline {f}}:\operatorname {cok} \ker f\to \ker \operatorname {cok} f$ 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 (Hausdorff) 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 (arbitrary or Hausdorff) 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.

## Citations

* ↑ Richman and Walker, 1977.
* ↑ Prosmans, 2000.
* ↑ Beilinson, A (2008). "Remarks on topological algebras". Moscow Mathematical Journal. 8 (1).
* 1 2 Positselski, Leonid (2024). "Exact categories of topological vector spaces with linear topology". Moscow Math. Journal. 24 (2): 219–286.
* ↑ Rump, 2008, p. 986f.
* ↑ Rump, 2011, p. 44f.
* ↑ Rump, 2001.