nLab additive envelope

The additive envelope of an Ab-enriched category $A$ is defined by taking the objects as formal direct sums of objects in $A$, and morphisms as matrices of coefficients, giving an additive category. This is a universal construction.

For instance, a semisimple category is the additive envelope of its full subcategory consisting of simple objects. Also, any additive category is equivalent to its own additive envelope.

By further taking the Karoubi envelope (i.e. formally adding images of idempotent elements), one constructs a Karoubian category called the pseudo-abelian envelope of $A$. In general, the pseudo-abelian envelope of $A$ is not abelian.

The combination of additive envelope and Karoubi envelope gives the Cauchy completion in the case of Ab-enriched categories.