Karoubi envelope
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 1 = "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 formula_3 is an idempotent of C, and whose morphisms are the triples where formula_5 is a morphism of C satisfying formula_6 (or equivalently formula_7). Composition in Split(C) is as in C, but the identity morphism on formula_8 in Split(C) is formula_9, rather than the identity on formula_10. The category C embeds fully and faithfully in Split(C). In Split(C) every idempotent  splits, and Split(C) is the universal category with this property. 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 formula_11 (the presheaves over C) of retracts of representable functors. The category of presheaves on C is equivalent to the category of presheaves on Split(C). Automorphisms in the Karoubi envelope. An automorphism in Split(C) is of the form formula_12, with inverse formula_13 satisfying: If the first equation is relaxed to just have formula_17, then "f  is a partial automorphism (with inverse g"). A (partial) involution in Split(C) is a self-inverse (partial) automorphism.