[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.
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
, 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