Pseudo-abelian Category - Pseudo-abelian Completion

Pseudo-abelian Completion

The Karoubi envelope construction associates to an arbitrary category a category together with a functor

such that the image of every idempotent in splits in . When applied to a preadditive category, the Karoubi envelope construction yields a pseudo-abelian category called the pseudo-abelian completion of . Moreover, the functor

is in fact an additive morphism.

To be precise, given a preadditive category we construct a pseudo-abelian category in the following way. The objects of are pairs where is an object of and is an idempotent of . The morphisms

in are those morphisms

such that in . The functor

is given by taking to .