Karoubi envelope

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 ightarrow A

with

:"e"2 = "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 ightarrow A is an idempotent of C, and whose morphisms are triples of the form

: (e, f, e^{prime}): (A, e) ightarrow (A^{prime}, e^{prime})

where f: A ightarrow A^{prime} is a morphism of C satisfying e^{prime} circ f = f = f circ e (or equivalently f=e'circ fcirc e).

Composition in Split(C) is as in C, but the identity morphism on (A,e) in Split(C) is (e,e,e), rather thanthe 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) o (A',e'), there exists a pair of arrows g:(A,e) o(A",e") and h:(A",e") o(A',e') such that:f=hcirc g and gcirc 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) ightarrow (A, e), with inverse (e, g, e): (A, e) ightarrow (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 ightarrow B the mapping f imes f^{-1}: A imes B ightarrow B imes A, composed with the canonical map gamma:B imes A ightarrow A imes B of symmetry, is a partial involution.


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