Karoubi envelope

The Karoubi envelope is a classification of the idempotents of a category. Precisely, given a category C, an idempotent of C is an endomorphism $e:A\rightarrow A$ with e² = e. The Karoubi envelope of C, 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 C-morphism satisfying $e^{\prime }\circ f=f=f\circ e$.

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 "symmetric" map $sym:B\times A\rightarrow A\times B$, is a partial involution.