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 p 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 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 C a category together with a functor such that the image s(p) of every idempotent p in C splits in. When applied to a preadditive category C, the Karoubi envelope construction yields a pseudo-abelian category called the pseudo-abelian completion of C. Moreover, the functor is in fact an additive morphism. To be precise, given a preadditive category C we construct a pseudo-abelian category in the following way. The objects of are pairs (X,p) where X is an object of C and p is an idempotent of X. The morphisms in are those morphisms such that in C. The functor is given by taking X to.

Citations