Pseudo-abelian category

In mathematics, specifically in category theory, a pseudo-abelian category is a category that is preadditive and is such that every idempotent has a kernel.[1] Recall that an idempotent morphism $p$ is an endomorphism of an object with the property that $p\circ p=p$ . Elementary considerations show that every idempotent then has a cokernel.[2] The pseudo-abelian condition is stronger than preadditivity, but it is weaker than the requirement that every morphism have a kernel and cokernel, as is true for abelian categories.

Synonyms in the literature for pseudo-abelian include pseudoabelian and Karoubian.

Examples

Any abelian category, in particular the category Ab of abelian groups, is pseudo-abelian. Indeed, in an abelian category, every morphism has a kernel.

The category of associative rngs (not rings!) together with multiplicative morphisms is pseudo-abelian.

A more complicated example is the category of Chow motives. The construction of Chow motives uses the pseudo-abelian completion described below.

Pseudo-abelian completion

The Karoubi envelope construction associates to an arbitrary category $C$ a category $kar(C)$ together with a functor

s:C\rightarrow kar(C)

such that the image $s(p)$ of every idempotent $p$ in $C$ splits in $kar(C)$ . When applied to a preadditive category $C$ , the Karoubi envelope construction yields a pseudo-abelian category $kar(C)$ called the pseudo-abelian completion of $C$ . Moreover, the functor

C\rightarrow kar(C)

is in fact an additive morphism.

To be precise, given a preadditive category $C$ we construct a pseudo-abelian category $kar(C)$ in the following way. The objects of $kar(C)$ are pairs $(X,p)$ where $X$ is an object of $C$ and $p$ is an idempotent of $X$ . The morphisms

f:(X,p)\rightarrow (Y,q)

in $kar(C)$ are those morphisms

f:X\rightarrow Y

such that $f=q\circ f=f\circ p$ in $C$ . The functor

C\rightarrow kar(C)

is given by taking $X$ to $(X,id_{X})$ .

Citations

Artin, 1972, p. 413.
Lars Brünjes, Forms of Fermat equations and their zeta functions, Appendix A

References

Artin, Michael (1972). Alexandre Grothendieck; Jean-Louis Verdier (eds.). Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1 (Lecture notes in mathematics 269) (in French). Berlin; New York: Springer-Verlag. xix+525.

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[1] Artin, 1972, p. 413.

[2] Lars Brünjes, Forms of Fermat equations and their zeta functions, Appendix A