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

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.

[edit] 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.

[edit] 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.
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