In mathematics the Karoubi envelope (or Cauchy completion or idempotent completion) 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. The Karoubi envelope of C, sometimes written Split(C), is the category whose objects are pairs of the form (A, e) where A is an object of C and
Composition in Split(C) is as in C, but the identity morphism on
The category C embeds fully and faithfully in Split(C).
The Karoubi envelope of a category C can therefore be considered as the "completion" of C which splits idempotents.
The Karoubi envelope of a category C can equivalently be defined as the full subcategory of
(the presheaves over C) of retracts of representable functors.
An automorphism in Split(C) is of the form
, then f is a partial automorphism (with inverse g).