Karoubi envelope

Summary

The Karoubi envelope of a category $\mathcal{C}$ is the category obtained by formally adding an image for every idempotent morphism. Its objects are pairs of an object and an idempotent endomorphism of it. It is the smallest category containing $\mathcal{C}$ in which every idempotent splits, and it is also called the idempotent completion. For example, the Karoubi envelope of the category of finitely generated free modules over a ring is the category of finitely generated projective modules.