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 formula_1 is an endomorphism of an object with the property that formula_2. 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 (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 formula_3 a category formula_4 together with a functor
such that the image formula_6 of every idempotent formula_1 in formula_3 splits in formula_4.
When applied to a preadditive category formula_3, the Karoubi envelope construction yields a pseudo-abelian category formula_4
called the pseudo-abelian completion of formula_3. Moreover, the functor
is in fact an additive morphism.
To be precise, given a preadditive category formula_3 we construct a pseudo-abelian category formula_4 in the following way. The objects of formula_4 are pairs formula_17 where formula_18 is an object of formula_3 and formula_1 is an idempotent of formula_18. The morphisms
in formula_4 are those morphisms
such that formula_25 in formula_3.
The functor
is given by taking formula_18 to formula_29.