# 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 is an endomorphism of an object with the property that . 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**.

## Examples

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 associative 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.

## Pseudo-abelian completion

The Karoubi envelope construction associates to an arbitrary category a category together with a functor

such that the image of every idempotent in splits in . When applied to a preadditive category , the Karoubi envelope construction yields a pseudo-abelian category called the pseudo-abelian completion of . Moreover, the functor

is in fact an additive morphism.

To be precise, given a preadditive category we construct a pseudo-abelian category in the following way. The objects of are pairs where is an object of and is an idempotent of . The morphisms

in are those morphisms

such that in . The functor

is given by taking to .

## Citations

- Artin, 1972, p. 413.
- Lars Brünjes, Forms of Fermat equations and their zeta functions, Appendix A

## References

- 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*(in French). Berlin; New York: Springer-Verlag. xix+525.**269**)