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

$e: A \rightarrow A$

with

$e\circ e = e$.

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 $e : A \rightarrow A$ 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

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

• 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 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 $X$ and projective modules over $C(X)$ 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