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
An idempotent e: A → A is said to split if there is an object B and morphisms f: A → B, g : B → A 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.
- If C has products, then given an isomorphism the mapping , composed with the canonical map of symmetry, is a partial involution.
- If C is a triangulated category, the Karoubi envelope Split(C) can be endowed with the structure of a triangulated category such that the canonical functor C → Split(C) becomes a triangulated functor.
- The Karoubi envelope is used in the construction of several categories of motives.
- The Karoubi envelope construction takes semi-adjunctions to adjunctions[disambiguation needed]. For this reason the Karoubi envelope is used in the study of models of the untyped lambda calculus. The Karoubi envelope of an extensional lambda model (a monoid, considered as a category) is cartesian closed.
- Balmer & Schlichting 2001
- Susumu Hayashi (1985). "Adjunction of Semifunctors: Categorical Structures in Non-extensional Lambda Calculus". Theoretical Computer Science 41: 95–104.
- C.P.J. Koymans (1982). "Models of the lambda calculus". Information and Control 52: 306–332.
- DS Scott (1980). "Relating theories of the lambda calculus". To HB Curry: Essays in Combinatory Logic.
- Balmer, Paul; Schlichting, Marco (2001), "Idempotent completion of triangulated categories", Journal of Algebra 236 (2): 819–834, ISSN 0021-8693