Karoubi envelope

In mathematics the Karoubi envelope (or Cauchy completion, but that term has other meanings) of a category C is a classification of the idempotents of C, by means of an auxiliary category. It is named for the French mathematician Max Karoubi.

Given a category C, an idempotent of C is an endomorphism

$e:A\rightarrow A$

with

e² = e.

Its Karoubi envelope, sometimes written Split(C), is a category with objects pairs of the form (A, e) where $e:A\rightarrow A$ is an idempotent of C, and morphisms triples of the form

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

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

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

The category C embeds fully and faithfully in Split(C). Moreover, in Split(C) every idempotent splits. This means that for every idempotent $f:(A,e)\to (A',e')$, there exists a pair of arrows $g:(A,e)\to (A'',e'')$ and $h:(A'',e'')\to (A',e')$ such that

$f=h\circ g$ and $g\circ h=1$.

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 ${\hat {\mathbf {C} }}$ (the presheaves over C) of retracts of representable functors.

Automorphisms in the Karoubi envelope

An automorphism in Split(C) is of the form $(e,f,e):(A,e)\rightarrow (A,e)$, with inverse $(e,g,e):(A,e)\rightarrow (A,e)$ satisfying:

$g\circ f=e=f\circ g$

$g\circ f\circ g=g$

$f\circ g\circ f=f$

If the first equation is relaxed to just have $g\circ f=f\circ g$, then f is a partial automorphism (with inverse g). A (partial) involution in Split(C) is a self-inverse (partial) automorphism.

Examples

• If C has products, then given an isomorphism $f:A\rightarrow B$ the mapping $f\times f^{-1}:A\times B\rightarrow B\times A$, composed with the canonical map $\gamma :B\times A\rightarrow A\times B$ of symmetry, is a partial involution.