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. 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. 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 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 $\operatorname {Kar} C$ together with a functor

$s:C\to \operatorname {Kar} C$

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

$C\to \operatorname {Kar} C$

is in fact an additive morphism.

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

$f:(X,p)\to (Y,q)$

in $\operatorname {Kar} C$ are those morphisms

$f:X\to Y$

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

$C\to \operatorname {Kar} C$

is given by taking $X$ to $(X,\mathrm {id} _{X})$.

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.