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Concept# Karoubi envelope

Summary

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.

Official source

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