Pseudo-abelian category

In mathematics, specifically in category theory, a pseudo-abelian category is a category that is preadditive and is such that every idempotent has a kernel.^[1] Recall that an idempotent morphism ${\displaystyle p}$ is an endomorphism of an object with the property that ${\displaystyle p\circ p=p}$. Elementary considerations show that every idempotent then has a cokernel.^[2] The pseudo-abelian condition is stronger than preadditivity, but it is weaker than the requirement that every morphism have a kernel and cokernel, as is true for abelian categories.

Synonyms in the literature for pseudo-abelian include pseudoabelian and Karoubian.

Any abelian category, in particular the category Ab of abelian groups, is pseudo-abelian. Indeed, in an abelian category, every morphism has a kernel.

The category of rngs (not rings!) together with multiplicative morphisms is pseudo-abelian.

A more complicated example is the category of Chow motives. The construction of Chow motives uses the pseudo-abelian completion described below.

The Karoubi envelope construction associates to an arbitrary category ${\displaystyle C}$ a category ${\displaystyle \operatorname {Kar} C}$ together with a functor

${\displaystyle s:C\to \operatorname {Kar} C}$

such that the image ${\displaystyle s(p)}$ of every idempotent ${\displaystyle p}$ in ${\displaystyle C}$ splits in ${\displaystyle \operatorname {Kar} C}$. When applied to a preadditive category ${\displaystyle C}$, the Karoubi envelope construction yields a pseudo-abelian category ${\displaystyle \operatorname {Kar} C}$ called the pseudo-abelian completion of ${\displaystyle C}$. Moreover, the functor

${\displaystyle C\to \operatorname {Kar} C}$

is in fact an additive morphism.

To be precise, given a preadditive category ${\displaystyle C}$ we construct a pseudo-abelian category ${\displaystyle \operatorname {Kar} C}$ in the following way. The objects of ${\displaystyle \operatorname {Kar} C}$ are pairs ${\displaystyle (X,p)}$ where ${\displaystyle X}$ is an object of ${\displaystyle C}$ and ${\displaystyle p}$ is an idempotent of ${\displaystyle X}$. The morphisms

${\displaystyle f:(X,p)\to (Y,q)}$

in ${\displaystyle \operatorname {Kar} C}$ are those morphisms

${\displaystyle f:X\to Y}$

such that ${\displaystyle f=q\circ f=f\circ p}$ in ${\displaystyle C}$. The functor

${\displaystyle C\to \operatorname {Kar} C}$

is given by taking ${\displaystyle X}$ to ${\displaystyle (X,\mathrm {id} _{X})}$.

• Artin, Michael (1972). Alexandre Grothendieck; Jean-Louis Verdier (eds.). Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1 (Lecture notes in mathematics 269) (in French). Berlin; New York: Springer-Verlag. xix+525.