Supriya Ghosh (Editor)

Quasi abelian category

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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.

Contents

Definition

Let 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 A is quasi-abelian if for every kernel f : X → Y and every morphism h : X → Z in the pushout diagram

X → f Y ↓ h ↓ h ′ Z → f ′ Q

the morphism f ′ is again a kernel and, dually, for every cokernel g : X → Y and every morphism h : Z → Y in the pullback diagram

P → g ′ Z ↓ h ′ ↓ h X → g Y

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 f ¯ : cok ⁡ ker ⁡ f → ker ⁡ cok ⁡ f 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.

  • 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. 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.

    References

    Quasi-abelian category Wikipedia
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