In mathematics the Karoubi envelope of a categoryC 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 = gf and 1B = fg. The Karoubi envelope of C, sometimes written Split, is the category whose objects are pairs of the form where A is an object of C and is an idempotent of C, and whose morphisms are the triples where is a morphism of C satisfying . Composition in Split is as in C, but the identity morphism on in Split is, rather than the identity on. The category C embeds fully and faithfully in Split. In Split every idempotent splits, and Split 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 of retracts of representable functors. The category of presheaves on C is equivalent to the category of presheaves on Split.
An automorphism in Split is of the form, with inverse satisfying: If the first equation is relaxed to just have, then f is a partial automorphism. A involution in Split is a self-inverse automorphism.
If C is a triangulated category, the Karoubi envelope Split can be endowed with the structure of a triangulated category such that the canonical functor C → Split becomes a triangulated functor.
The Karoubi envelope is used in the construction of several categories of motives.
The Karoubi envelope construction takes semi-adjunctions to adjunctions. For this reason the Karoubi envelope is used in the study of models of the untyped lambda calculus. The Karoubi envelope of an extensional lambda model is cartesian closed.
The category of vector bundles over any paracompact space is the Karoubi envelope of its full subcategory of trivial bundles. This is in fact a special case of the previous example by the Serre-Swan theorem and conversely this theorem can be proved by first proving both these facts, the observation that the global sections functor is an equivalence between trivial vector bundles over and free modules over and then using the universal property of the Karoubi envelope.
