- Karoubi envelope
In
mathematics the Karoubi envelope (or Cauchy completion, but that term has other meanings) of a category C is a classification of theidempotent s of C, by means of an auxiliary category. It is named for the French mathematicianMax Karoubi .Given a category C, an idempotent of C is an
endomorphism :
with
:"e"2 = "e".
The Karoubi envelope of C, sometimes written Split(C), is the category whose objects are pairs of the form ("A", "e") where is an idempotent of C, and whose
morphism s 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 thanthe identity on .
The category C embeds fully and faithfully in Split(C). Moreover, in Split(C) every idempotent splits. This means that for every idempotent , there exists a pair of arrows and such that: and .The Karoubi envelope of a category C can therefore be considered as the "completion" of C which "splits idempotents", thus the notation Split(C).
The Karoubi envelope of a category C can equivalently be defined as the
full subcategory of (the presheaves over C) of retracts ofrepresentable functor s.Automorphisms in the Karoubi envelope
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.
Examples
* If C has products, then given an
isomorphism the mapping , composed with the canonical map of symmetry, is a partialinvolution .
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