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In mathematics the Karoubi envelope of a category C is a classification of the idempotents of C, by means of an auxiliary category. Taking the Karoubi envelope of a preadditive category gives a pseudo-abelian category, hence the construction is sometimes called the pseudo-abelian completion. It is named for the French mathematician Max Karoubi.
Given a category C, an idempotent of C is an endomorphism
with
An idempotent e: A → A is said to split if there is an object B and morphisms f: A → B,g : B → A such that e = g f and 1B = f g.
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