nLab
linear category

A linear category is an additive category that is enriched over Vect.

As $\mathrm{Vect}$ is really ${\mathrm{Vect}}_K$ for some field $K$, a linear category is really $K$-linear (or linear over $K$) for some field $K$. We can also make $K$ a commutative ring without much trouble; then a linear category over the ring of integer?s is the same thing as an additive category.