nLab
Karoubian category

Idea

A Karoubian category or pseudo-abelian category (also: pseudoabelian). is a pre-additive category $C$ such that every idempotent morphism $p:A\to A$ in $C$ 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 $C$ be a category and $p:X\to X$ an idempotent endomorphism of an object $X$. One says that $p$ admits an image if the functor $\mathrm{Ker}({\mathrm{id}}_X,p)$ is representable, and the representing object is called the image of $p$. Here $\mathrm{Ker}({\mathrm{id}}_X,p)$ is the functor $C^{\mathrm{op}}\to \underline{\mathrm{Set}}$ mapping

$$Y\mapsto \mathrm{Ker}(\mathrm{Hom}(Y,X)\rightrightarrows \mathrm{Hom}(Y,X));(*)$$Y \mapsto Ker(Hom(Y, X) \rightrightarrows Hom(Y, X)); \qquad (\ast)

in other words the image of $p$ is the difference kernel of $({\mathrm{id}}_X,p)$, when it exists.

Now $C$ is called Karoubian if every idempotent $p$ admits an image. Since $p:X\to X$ is idempotent iff ${\mathrm{id}}_X-p$ is idempotent, this is the same as saying every idempotent has a kernel.

Properties

General

One can show that for any idempotent $p$, $\mathrm{Ker}({\mathrm{id}}_X,p)$ is representable if and only if $\mathrm{Coker}({\mathrm{id}}_X,p)$ is, and that in fact their representing objects are canonically isomorphic.

Recall that one says $p$ splits if there exists an object $Y$, and morphisms $f:X\to Y$, $g:Y\to X$, such that $f\circ g={\mathrm{id}}_Y$ and $g\circ f=p$. Observe that when $p$ admits an image $K$, it splits: by definition there are functorial isomorphisms ${\Phi }_Y$ for all $Y$ between the image of the functor $(*)$ and $Hom(Y,K)$; now take $f:X\to K$ the morphism corresponding to $p$ via ${\Phi }_X$, $g:K\to X$ the morphism corresponding to ${\mathrm{id}}_K\in \mathrm{Hom}(K,K)$ via ${\Phi }_K$. Conversely, if $p$ splits via a pair $(f,g)$, then $g:Y\to X$ is a difference kernel of $({\mathrm{id}}_X,p)$: we have $g=g\circ f\circ g=p\circ g$, and if $h:Z\to X$ satisfies $h=p\circ h=g\circ f\circ h$, then $h$ clearly factors through $g$, and uniquely so since sections $g$ 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 $C$ is also called the Karoubian envelope or the pseudo-abelian completion of $C$.

More in detail, there exists a Karoubian category $\mathrm{kar}(C)$ associated to any category $C$, and a fully faithful functor $\phi :C\to \mathrm{kar}(C)$, which is universal in the sense that for any Karoubian category $C\prime$, the functor

$$\underline{\mathrm{Hom}}(\mathrm{kar}(C),C\prime )\to \underline{\mathrm{Hom}}(C,C\prime )$$\underline{Hom}(kar(C), C') \to \underline{Hom}(C, C')

taking a functor $F:\mathrm{kar}(C)\to C\prime$ to the composite $F\circ \phi$ is an equivalence of categories. $\mathrm{kar}(C)$ is called the Karoubi envelope of $C$ (aka the Cauchy completion, or the idempotent-splitting completion). It can be realized explicitly by taking as objects pairs $(X,p)$, with $p$ idempotent, and as morphisms $(X,p)\to (Y,q)$ the morphisms $f:X\to Y$ that satisfy $f=q\circ f\circ p$.

Exampls

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

• Alexandre Grothendieck, Jean-Louis Verdier. Exercice 7.5 in Topos, Exposé IV of SGA 4, volume 1.