# 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

$s:C\rightarrow kar(C)$ $C\rightarrow kar(C)$ is in fact an additive morphism.

$f:(X,p)\rightarrow (Y,q)$ $f:X\rightarrow Y$ $C\rightarrow kar(C)$ 