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

$C\rightarrow kar(C)$

is given by taking $X$ to $(X,id_X)$.