Karoubi envelope

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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

with

e\circ e = e.

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 e : A \rightarrow A is an idempotent of C, and whose morphisms are the triples

(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). 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 \hat{\mathbf{C}} (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[edit]

An automorphism in Split(C) is of the form (e, f, e): (A, e) \rightarrow (A, e), with inverse (e, g, e): (A, e) \rightarrow (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[edit]

  • If C has products, then given an isomorphism f: A \rightarrow B the mapping f \times f^{-1}: A \times B \rightarrow B \times A, composed with the canonical map \gamma:B \times A \rightarrow A \times B 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 CSplit(C) becomes a triangulated functor.[1]
  • The Karoubi envelope is used in the construction of several categories of motives.
  • The Karoubi envelope construction takes semi-adjunctions to adjunctions[disambiguation needed].[2] 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.[3][4]

References[edit]

  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). "To HB Curry: Essays in Combinatory Logic".  |chapter= ignored (help)