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 \to A

with

e² = e.

The Karoubi envelope of C, sometimes written Split(C), is the category whose objects are pairs of the form (A, e) where $e : A \to A$ is an idempotent of C, and whose morphisms are 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 $(e, f, e):(A, e)\to(A, e)$ , with inverse $(e, g, e):(A, e)\to(A, e)$ 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 $f : A \to B$ the mapping $f \times f - 1 : A \times B \to B \times A$ , composed with the canonical map $γ: B \times A \to A \times B$ of symmetry, is a partial involution.

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Karoubi envelope

Automorphisms in the Karoubi envelope

Examples

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