Karoubi envelope

In mathematics the Karoubi envelope (or Cauchy completion or idempotent completion) of a category C is a classification of the idempotents of C, by means of an auxiliary category. Taking the Karoubi envelope of a preadditive category gives a pseudo-abelian category, hence the construction is sometimes called the pseudo-abelian completion. It is named for the French mathematician Max Karoubi.

Given a category C, an idempotent of C is an endomorphism

with

.

An idempotent e: AA is said to split if there is an object B and morphisms f: AB, g : BA such that e = g f and 1B = f g.

The Karoubi envelope of C, sometimes written Split(C), is the category whose objects are pairs of the form (A, e) where A is an object of C and is an idempotent of C, and whose morphisms are the triples

where is a morphism of C satisfying (or equivalently ).

Composition in Split(C) is as in C, but the identity morphism on in Split(C) is , rather than the identity on .

The category C embeds fully and faithfully in Split(C). In Split(C) every idempotent splits, and Split(C) is the universal category with this property. The Karoubi envelope of a category C can therefore be considered as the "completion" of C which splits idempotents.

The Karoubi envelope of a category C can equivalently be defined as the full subcategory of (the presheaves over C) of retracts of representable functors. The category of presheaves on C is equivalent to the category of presheaves on Split(C).

Automorphisms in the Karoubi envelope

An automorphism in Split(C) is of the form , with inverse satisfying:

If the first equation is relaxed to just have , then f is a partial automorphism (with inverse g). A (partial) involution in Split(C) is a self-inverse (partial) automorphism.

Examples

References

  1. Balmer & Schlichting 2001
  2. Susumu Hayashi (1985). "Adjunction of Semifunctors: Categorical Structures in Non-extensional Lambda Calculus". Theoretical Computer Science. 41: 95–104. doi:10.1016/0304-3975(85)90062-3.
  3. C.P.J. Koymans (1982). "Models of the lambda calculus". Information and Control. 52: 306–332. doi:10.1016/s0019-9958(82)90796-3.
  4. DS Scott (1980). "Relating theories of the lambda calculus". To HB Curry: Essays in Combinatory Logic.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.