Suppose I have an category additive category C (i.e. the hom sets are enriched in abelian groups and there are finite direct sums). Suppose further that C has cokernels. Then I can make C tensored over finitely presentable Abelian groups by the following ad hoc construction:


First define $\mathbb{Z}^n \otimes X := \oplus\_n X$. Now given a finitely presentable abelian group A, choose a presentation, i.e. realize $A$ as the cokernel of $f:\mathbb{Z}^r \to \mathbb{Z}^g$. Define $A \otimes X$ as the cokernel of the induced map:


$\mathbb{Z}^r \otimes X \to \mathbb{Z}^g \otimes X$


My questions: Is there a way to do the same thing which feels more canonical and less ad hoc? Under what conditions will C be tensored over all abelian groups?



Tensored Over Abelian Groups?
Given an object $X$ in an additive category $C$ and an abelian group $A$, define the object $A\otimes X$ in $C$ by the rule $Hom\_C(A\otimes X,\:Y) = Hom\_{Ab}(A,Hom\_C(X,Y))$, where $Ab$ denotes the category of abelian groups. If arbitrary direct sums and cokernels (arbitrary colimits, in other words) exist in an additive category $C$, the tensor product $A\otimes X$ exists in $C$ for any abelian group $A$ and any $X\in C$. It can be constructed just as you describe in your question, except that finite direct sums should be replaced with infinite direct sums.


