The Karoubian envelope is also used in the construction of the category of pure motives,

and in K-theory.

Although there is a lot of online notes/courses available where is precisely explaned how taking Karoubian envelope

is involved in the construction of (pure) motives, there seems to be a serious lack of sources where is explained how

the Karoubian envolope is involved in construtions in K-theory.

(appart from the 'basic' construction of algebraic K-theory K_0 (A) for a ring A as K_0 (P_A), where P_A

is the category of finitely generated A-modules, where P_A can also be recognized as Karoubi completion of the

category F_A of finite generated free A-modules.

Nevertheless this construction of algebraic K_0 might be considered as a 'toy' example.

Is there in the quoted sentence above also referred to certain constructions in K-theory in more general setting (eg for K-groups of exact or Waldhausen-categories)

which make use of the Karoubian completion? ]]>