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 triples of the form
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 full subcategory of (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).
Read more about Karoubi Envelope: Automorphisms in The Karoubi Envelope, Examples
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