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
. 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. 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 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