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 be a pre-abelian category. A morphism is a kernel (a cokernel) if there exists a morphism such that is a kernel (cokernel) 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.[1]
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
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.
History
The concept of quasi-abelian category was developed in the 1960s. The history is involved.[3] 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.[4]
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.[5]
Citations
- ↑ Richman and Walker, 1977.
- ↑ Prosmans, 2000.
- ↑ Rump, 2008, p. 986f.
- ↑ Rump, 2011, p. 44f.
- ↑ Rump, 2001.
References
- 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).