Karoubi envelope

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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

<math>e: A \rightarrow A</math>

with

e² = e.

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

<math>(e, f, e^{\prime}): (A, e) \rightarrow (A^{\prime}, e^{\prime})</math>

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

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

<math>f=h\circ g</math> and <math>g\circ h=1</math>.

The Karoubi envelope of a category C can therefore be considered as the "completion" of C which splits idempotents, thus the notation Split(C).

Automorphisms in the Karoubi envelope

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

<math>g \circ f = e = f \circ g</math>

<math>g \circ f \circ g = g</math>

<math>f \circ g \circ f = f</math>

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