# pseudo-abelian category / Karoubian category in K-theory

A pseudo-abelian category or Karoubian category $$mathcal{C}$$ is a preaditive
category such that every idempotent morphism
$$i: A to A$$ in $$mathcal{C}$$ has a kernel and consequently a
cokernel.

Moreover the Karoubi or preudo-abelian completion
associates to an arbitrary category preadditive category
$$mathcal{D}$$ a pseudo-abelian category $$Kar(mathcal{D})$$
called the pseudo-abelian completion or Karoubian envelope of $$mathcal{D}$$
Here
is is shortly noted unfortunately without any references that pseudo-abelian completion
is also used in the construction of the category of pure motives, and in K-theory.

I found heaps of meterials how it is used in the
construction of the category of pure motives but almost nothing