Preadditive category

Remarks

Equivalently, a preadditive category is a category enriched over the monoidal category of abelian groups with the tensor product. The zero element of $\Hom(X, Y)$ is the zero morphism $0_{XY}$, and bilinearity gives $h \circ 0 = 0$ and $0 \circ k = 0$. Some authors call these categories $\mathbf{Ab}$-categories.

Example: for a ring $R$, the category of left $R$-modules with pointwise addition $(f + g)(x) = f(x) + g(x)$ is preadditive. The sum of homomorphisms is a homomorphism, and $h(f(x) + g(x)) = h(f(x)) + h(g(x))$ because $h$ is additive, which gives the first identity; the second holds by definition of the sum.

Example: a ring $R$ is a preadditive category with one object $\ast$, with $\Hom(\ast, \ast) = R$ and composition the multiplication of $R$.

Non-example: the category of sets is not preadditive, since $\Hom(\set{0}, \emptyset)$ is empty and an abelian group is never empty.