Additive category

Remarks

Then every finite family of objects has a biproduct, and a morphism $\bigoplus_{j} X_j \to \bigoplus_{i} Y_i$ is the same as a matrix $(f_{ij})$ with $f_{ij} \colon X_j \to Y_i$, composed by matrix multiplication. Being additive is a property of the underlying category: the addition of morphisms is determined by the biproducts, through $f + g = \nabla \circ (f \oplus g) \circ \Delta$, where $\Delta$ is the diagonal and $\nabla$ the codiagonal.

Example: the category of left modules over a ring $R$ is additive. The zero module is a zero object, and the direct sum $M_1 \oplus M_2$ is a biproduct.

Non-example: the ring $\Z$, as a preadditive category with one object $\ast$, is not additive. It has no zero object, since a zero object $Z$ has $\Hom(Z, Z) = \set{0}$, while $\Hom(\ast, \ast) = \Z$.