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.^{[citation needed]}

Let ${\displaystyle {\mathcal {A}}}$ be a pre-abelian category. A morphism ${\displaystyle f}$ is a kernel (a cokernel) if there exists a morphism ${\displaystyle g}$ such that ${\displaystyle f}$ is a kernel (cokernel) of ${\displaystyle g}$. The category ${\displaystyle {\mathcal {A}}}$ is quasi-abelian if for every kernel ${\displaystyle f:X\rightarrow Y}$ and every morphism ${\displaystyle h:X\rightarrow Z}$ in the pushout diagram

${\displaystyle {\begin{array}{ccc}X&{\xrightarrow {f}}&Y\\\downarrow _{h}&&\downarrow _{h'}\\Z&{\xrightarrow {f'}}&Q\end{array}}}$

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

${\displaystyle {\begin{array}{ccc}P&{\xrightarrow {g'}}&Z\\\downarrow _{h'}&&\downarrow _{h}\\X&{\xrightarrow {g}}&Y\end{array}}}$

the morphism ${\displaystyle g'}$ 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.^[1]

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

Every abelian category is quasi-abelian. Typical non-abelian examples arise in functional analysis.^[2]

• 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,^[3] the category of complete separated topological vector spaces with linear topology is not quasi-abelian.^[4] On the other hand, the category of (arbitrary or Hausdorff) topological vector spaces with linear topology is quasi-abelian.^[4]

The concept of quasi-abelian category was developed in the 1960s. The history is involved.^[5] 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.^[6]

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.^[7]

• 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).