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

$e:\; A\; \backslash rightarrow\; A$

with

e² = e.

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

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

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

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

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

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

Automorphisms in the Karoubi envelope

---

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

$g\; \backslash circ\; f\; =\; e\; =\; f\; \backslash circ\; g$

$g\; \backslash circ\; f\; \backslash circ\; g\; =\; g$

$f\; \backslash circ\; g\; \backslash circ\; f\; =\; f$

If the first equation is relaxed to just have $g\; \backslash circ\; f\; =\; f\; \backslash circ\; g$, 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 $f:\; A\; \backslash rightarrow\; B$ the mapping $f\; \backslash times\; f^\{-1\}:\; A\; \backslash times\; B\; \backslash rightarrow\; B\; \backslash times\; A$, composed with the canonical map $\backslash gamma:B\; \backslash times\; A\; \backslash rightarrow\; A\; \backslash times\; B$ of symmetry, is a partial involution.

Category theory