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

${\displaystyle e:A\rightarrow A}$

with

${\displaystyle e\circ e=e}$.

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 1_B = 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 ${\displaystyle e:A\rightarrow A}$ is an idempotent of C, and whose morphisms are the triples

${\displaystyle (e,f,e^{\prime }):(A,e)\rightarrow (A^{\prime },e^{\prime })}$

where ${\displaystyle f:A\rightarrow A^{\prime }}$ is a morphism of C satisfying ${\displaystyle e^{\prime }\circ f=f=f\circ e}$ (or equivalently ${\displaystyle f=e'\circ f\circ e}$).

Composition in Split(C) is as in C, but the identity morphism on ${\displaystyle (A,e)}$ in Split(C) is ${\displaystyle (e,e,e)}$, rather than the identity on ${\displaystyle A}$.

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 ${\displaystyle {\hat {\mathbf {C} }}}$ (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

An automorphism in Split(C) is of the form ${\displaystyle (e,f,e):(A,e)\rightarrow (A,e)}$, with inverse ${\displaystyle (e,g,e):(A,e)\rightarrow (A,e)}$ satisfying:

${\displaystyle g\circ f=e=f\circ g}$

${\displaystyle g\circ f\circ g=g}$

${\displaystyle f\circ g\circ f=f}$

If the first equation is relaxed to just have ${\displaystyle g\circ f=f\circ g}$, 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 ${\displaystyle f:A\rightarrow B}$ the mapping ${\displaystyle f\times f^{-1}:A\times B\rightarrow B\times A}$, composed with the canonical map ${\displaystyle \gamma :B\times A\rightarrow A\times B}$ 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 C → Split(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.[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]
• The category of projective modules over any ring is the Karoubi envelope of its full subcategory of free modules.
• The category of vector bundles over any paracompact space is the Karoubi envelope of its full subcategory of trivial bundles. This is in fact a special case of the previous example by the Serre-Swan theorem and conversely this theorem can be proved by first proving both these facts, the observation that the global sections functor is an equivalence between trivial vector bundles over ${\displaystyle X}$ and free modules over ${\displaystyle C(X)}$ and then using the universal property of the Karoubi envelope.

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.

• Balmer, Paul; Schlichting, Marco (2001), "Idempotent completion of triangulated categories" (PDF), Journal of Algebra, 236 (2): 819–834, doi:10.1006/jabr.2000.8529, ISSN 0021-8693