Concept

Enveloppe de Karoubi

Résumé
In mathematics the Karoubi envelope (or Cauchy completion or idempotent completion) of a C is a classification of the idempotents of C, by means of an auxiliary category. Taking the Karoubi envelope of a gives a , 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 is an idempotent of C, and whose morphisms are the triples where is a morphism of C satisfying (or equivalently ). Composition in Split(C) is as in C, but the identity morphism on in Split(C) is , rather than the identity on . 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 of (the over C) of retracts of representable functors. The category of presheaves on C is equivalent to the category of presheaves on Split(C). An automorphism in Split(C) is of the form , with inverse satisfying: If the first equation is relaxed to just have , then f is a partial automorphism (with inverse g). A (partial) involution in Split(C) is a self-inverse (partial) automorphism. If C has products, then given an isomorphism the mapping , composed with the canonical map of symmetry, is a partial involution. If C is a , the Karoubi envelope Split(C) can be endowed with the structure of a triangulated category such that the canonical functor C → Split(C) becomes a triangulated functor. The Karoubi envelope is used in the construction of several categories of motives.
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