quasi-abelian category

Let $\backslash mathcal\; A$ 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 $\backslash mathcal\; A$ is quasi-abelian if for every kernel $f:\; X\backslash rightarrow\; Y$ and every morphism $h:\; X\backslash rightarrow\; Z$ in the pushout diagram

{{center|$$

\begin{array}{ccc}

X & \xrightarrow{f} & Y \\

\downarrow_{h} & & \downarrow_{h'}\\

Z & \xrightarrow{f'} & Q

\end{array}

}}

the morphism $f\text{'}$ is again a kernel and, dually, for every cokernel $g:\; X\backslash rightarrow\; Y$ and every morphism $h:\; Z\backslash rightarrow\; Y$ in the pullback diagram

{{center|$$

\begin{array}{ccc}

P & \xrightarrow{g'} & Z \\

\downarrow_{h'} & & \downarrow_{h}\\

X & \xrightarrow{g} & Y

\end{array}

}}

the morphism $g\text{'}$ 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.Richman and Walker, 1977.

Let $f$ be a morphism in a quasi-abelian category. Then the induced morphism $\backslash overline\{f\}\; :\; \backslash operatorname\{cok\}\; \backslash ker\; f\; \backslash to\; \backslash ker\; \backslash operatorname\{cok\}\; f$ is always a bimorphism, i.e., a monomorphism and an epimorphism. A quasi-abelian category is therefore always semi-abelian.

Every abelian category is quasi-abelian. Typical non-abelian examples arise in functional analysis.Prosmans, 2000.

• 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,{{cite journal |last1=Beilinson |first1=A |title=Remarks on topological algebras |journal=Moscow Mathematical Journal |date=2008 |volume=8 |issue=1}} the category of complete separated topological vector spaces with linear topology is not quasi-abelian.{{cite journal |last1=Positselski |first1=Leonid |title=Exact categories of topological vector spaces with linear topology |journal=Moscow Math. Journal |date=2024 |volume=24 |issue=2 |pages=219–286}} On the other hand, the category of (arbitrary or Hausdorff) topological vector spaces with linear topology is quasi-abelian.

The concept of quasi-abelian category was developed in the 1960s. The history is involved.Rump, 2008, p. 986f. 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.Rump, 2011, p. 44f.

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.Rump, 2001.

• Fabienne Prosmans, Derived categories for functional analysis. Publ. Res. Inst. Math. Sci. 36(5–6), 19–83 (2000).
• Fred Richman and Elbert A. Walker, Ext in pre-Abelian categories. Pac. J. Math. 71(2), 521–535 (1977).
• Wolfgang Rump, A counterexample to Raikov's conjecture, Bull. London Math. Soc. 40, 985–994 (2008).
• Wolfgang Rump, Almost abelian categories, Cahiers Topologie Géom. Différentielle Catég. 42(3), 163–225 (2001).
• Wolfgang Rump, Analysis of a problem of Raikov with applications to barreled and bornological spaces, J. Pure and Appl. Algebra 215, 44–52 (2011).
• Jean Pierre Schneiders, Quasi-abelian categories and sheaves, Mém. Soc. Math. Fr. Nouv. Sér. 76 (1999).

Category:Additive categories