In mathematics the Karoubi envelope (or Cauchy completion or idempotent completion) of a category C is a classification of the idempotents 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
e : A → A {\displaystyle e:A\rightarrow A}
with
e ∘ e = e {\displaystyle e\circ e=e} . 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 e : A → A {\displaystyle e:A\rightarrow A} is an idempotent of C, and whose morphisms are the triples
( e , f , e ′ ) : ( A , e ) → ( A ′ , e ′ ) {\displaystyle (e,f,e^{\prime }):(A,e)\rightarrow (A^{\prime },e^{\prime })}
where f : A → A ′ {\displaystyle f:A\rightarrow A^{\prime }} is a morphism of C satisfying e ′ ∘ f = f = f ∘ e {\displaystyle e^{\prime }\circ f=f=f\circ e} (or equivalently f = e ′ ∘ f ∘ e {\displaystyle f=e'\circ f\circ e} ). Composition in Split(C) is as in C, but the identity morphism on ( A , e ) {\displaystyle (A,e)} in Split(C) is ( e , e , e ) {\displaystyle (e,e,e)} , rather than the identity on A {\displaystyle A} . 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 C ^ {\displaystyle {\hat {\mathbf {C} }}} (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).
Automorphisms in the Karoubi envelope An automorphism in Split(C) is of the form ( e , f , e ) : ( A , e ) → ( A , e ) {\displaystyle (e,f,e):(A,e)\rightarrow (A,e)} , with inverse ( e , g , e ) : ( A , e ) → ( A , e ) {\displaystyle (e,g,e):(A,e)\rightarrow (A,e)} satisfying:
g ∘ f = e = f ∘ g {\displaystyle g\circ f=e=f\circ g}
g ∘ f ∘ g = g {\displaystyle g\circ f\circ g=g}
f ∘ g ∘ f = f {\displaystyle f\circ g\circ f=f}
If the first equation is relaxed to just have g ∘ f = f ∘ g {\displaystyle g\circ f=f\circ g} , then f is a partial automorphism (with inverse g). A (partial) involution in Split(C) is a self-inverse (partial) automorphism.
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