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 $Ker(id_X, p)$ is representable, and the representing object is called the image of $p$. Here $Ker(id_X, p)$ is the functor $C^{op} \to \underline{Set}$ mapping

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

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

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

Properties

General

One can show that for any idempotent $p$, $Ker(id_X, p)$ is representable if and only if $Coker(id_X, p)$ 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 $f \circ g = id_Y$ 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 $(\ast)$ 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 $id_K \in Hom(K, K)$ via $\Phi_K$. Conversely, if $p$ splits via a pair $(f, g)$, then $g: Y \to X$ is a difference kernel of $(id_X, p)$: 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 $C$.

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

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

taking a functor $F : kar(C) \to C'$ to the composite $F \circ \varphi$ is an equivalence of categories. $kar(C)$ 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 $f = q \circ f \circ p$.

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

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