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
- e: A \rightarrow A
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 triples of the form
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. 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
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