Splitting idempotents

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

with

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

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

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