# 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

## Citations

## References

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