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"B" = "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.