Karoubi envelope

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 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

${\displaystyle e:A\rightarrow A}$

with

${\displaystyle e\circ e=e}$.

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 1_B = 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 ${\displaystyle e:A\rightarrow A}$ is an idempotent of C, and whose morphisms are the triples

${\displaystyle (e,f,e^{\prime }):(A,e)\rightarrow (A^{\prime },e^{\prime })}$

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

Composition in Split(C) is as in C, but the identity morphism on ${\displaystyle (A,e)}$ in Split(C) is ${\displaystyle (e,e,e)}$, rather than the identity on ${\displaystyle A}$.

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 ${\displaystyle {\hat {\mathbf {C} }}}$ (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).