Preadditive category

A category $\mathcal C$ is preadditive if every morphism set $\operatorname{Hom}_{\mathcal C}(X,Y)$ is an abelian group and composition is bilinear:

(f+g)\circ h=f\circ h+g\circ h, \qquad k\circ(f+g)=k\circ f+k\circ g.

In particular, there is a zero morphism between every ordered pair of objects.

An additive category is a preadditive category with finite biproducts. Preadditivity supplies the sums and signs used in complexes and distinguished triangles.