nLab
Karoubian category

Contents

Idea

A Karoubian category or pseudo-abelian category (also: pseudoabelian). is a pre-additive category CC such that every idempotent morphism p:AAp: A \to A in CC has a kernel, and hence (one can easily show) also a cokernel.

This is stronger than pre-additivity but weaker than abelianness, which requires that every morphism has a kernel and cokernel.

Definition

Let CC be a category and p:XXp : X \to X an idempotent endomorphism of an object XX. One says that pp admits an image if the functor Ker(id X,p)Ker(id_X, p) is representable, and the representing object is called the image of pp. Here Ker(id X,p)Ker(id_X, p) is the functor C opSet̲C^{op} \to \underline{Set} mapping

YKer(Hom(Y,X)Hom(Y,X));(*) Y \mapsto Ker(Hom(Y, X) \rightrightarrows Hom(Y, X)); \qquad (\ast)

in other words the image of pp is the difference kernel of (id X,p)(id_X, p), when it exists.

Now CC is called Karoubian if every idempotent pp admits an image. Since p:XXp: X \to X is idempotent iff id Xpid_X - p is idempotent, this is the same as saying every idempotent has a kernel.

Properties

General

One can show that for any idempotent pp, Ker(id X,p)Ker(id_X, p) is representable if and only if Coker(id X,p)Coker(id_X, p) is, and that in fact their representing objects are canonically isomorphic.

Recall that one says pp splits if there exists an object YY, and morphisms f:XYf : X \to Y, g:YXg : Y \to X, such that fg=id Yf \circ g = id_Y and gf=pg \circ f = p. Observe that when pp admits an image KK, it splits: by definition there are functorial isomorphisms Φ Y\Phi_Y for all YY between the image of the functor (*)(\ast) and Hom(Y,K)\Hom(Y, K); now take f:XKf : X \to K the morphism corresponding to pp via Φ X\Phi_X, g:KXg : K \to X the morphism corresponding to id KHom(K,K)id_K \in Hom(K, K) via Φ K\Phi_K. Conversely, if pp splits via a pair (f,g)(f, g), then g:YXg: Y \to X is a difference kernel of (id X,p)(id_X, p): we have g=gfg=pgg = g \circ f \circ g = p \circ g, and if h:ZXh: Z \to X satisfies h=ph=gfhh = p \circ h = g \circ f \circ h, then hh clearly factors through gg, and uniquely so since sections gg are monomorphisms.

Karoubi envelope

There is a universal functor from the category of (say, small) preadditive categories to the category of Karoubian categories, the Karoubinization functor; its value on a preadditive category CC is also called the Karoubian envelope or the pseudo-abelian completion of CC.

In more detail, there exists a Karoubian category kar(C)kar(C) associated to any category CC, and a fully faithful functor φ:Ckar(C)\varphi : C \to kar(C), which is universal in the sense that for any Karoubian category CC', the functor

Hom̲(kar(C),C)Hom̲(C,C)\underline{Hom}(kar(C), C') \to \underline{Hom}(C, C')

taking a functor F:kar(C)CF : kar(C) \to C' to the composite FφF \circ \varphi is an equivalence of categories. kar(C)kar(C) is called the Karoubi envelope of CC (aka the Cauchy completion, or the idempotent-splitting completion). It can be realized explicitly by taking as objects pairs (X,p)(X, p), with pp idempotent, and as morphisms (X,p)(Y,q)(X, p) \to (Y, q) the morphisms f:XYf : X \to Y that satisfy f=qfpf = q \circ f \circ p.

Examples

The requirement that, say, a dg-category or a triangulated category be Karoubian is a natural requirement in a number of contexts.

The Karoubian envelope is also used in the construction of the category of pure motives, and in K-theory.

References

Revised on August 20, 2013 18:42:32 by Adeel Ahmad Khan (160.45.40.186)