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.
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Publications associées

Chargement

Personnes associées

Chargement

Unités associées

Chargement

Concepts associés

Chargement

Cours associés

Chargement

Séances de cours associées

Chargement

MOOCs associés

Chargement