Ab-enriched category
1. Definition
An ab-enriched category is a category enriched over \(\mathrm{Ab}\) as closed symmetric monoidal category
see:
2. Explicite
Let \(\mathcal{C}\) be a locally small category. Then \(\mathcal{C}\) is said to be Ab-enriched, if for objects \(x,y \in \mathrm{Ob}(\mathcal{C})\) the hom-set \(\mathrm{Hom}_{\mathcal{C}}(x,y)\) is an abelian group and that the composition
\begin{align*} \mathrm{Hom}_{\mathcal{C}}(x,y) \times \mathrm{Hom}_{\mathcal{C}}(y,z) \rightarrow \mathrm{Hom}_{\mathcal{C}}(x,z) \end{align*}