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 ${\displaystyle p}$ is an endomorphism of an object with the property that ${\displaystyle p\circ p=p}$. 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.