Editing Karoubi envelope (section)

{{Short description|Category theory}}
In [[mathematics]] the '''Karoubi envelope''' (or '''Cauchy completion''' or '''idempotent completion''') of a [[category (mathematics)|category]] '''C''' is a classification of the [[idempotent]]s of '''C''', by means of an auxiliary category. Taking the Karoubi envelope of a [[preadditive category]] gives a [[pseudo-abelian category]], hence for additive categories, 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]] 

:<math>e: A \rightarrow A</math>

with 

:<math>e\circ e = e</math>.

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 1<sub>''B''</sub> = ''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 <math>e : A \rightarrow A</math> is an idempotent of '''C''', and whose [[morphism]]s are the triples

: <math>(e, f, e^{\prime}): (A, e) \rightarrow (A^{\prime}, e^{\prime})</math>

where <math>f: A  \rightarrow A^{\prime}</math> is a morphism of '''C''' satisfying <math>e^{\prime} \circ f = f = f \circ e</math> (or equivalently <math>f=e'\circ f\circ e</math>).

Composition in '''Split(C)''' is as in '''C''', but the identity morphism 
on <math>(A,e)</math> in '''Split(C)''' is <math>(e,e,e)</math>, rather than
the identity on <math>A</math>.

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 <math>\hat{\mathbf{C}}</math> (the [[presheaf (category theory)|presheaves]] over '''C''') of retracts of [[representable functor]]s. The category of presheaves on '''C''' is equivalent to the category of presheaves on '''Split(C)'''.