TITLE: Is every abelian group a colimit of copies of Z?
QUESTION [31 upvotes]: More precisely, is every abelian group a colimit $\text{colim}_{j \in J} F(j)$ over a diagram $F : J \to \text{Ab}$ where each $F(j)$ is isomorphic to $\mathbb{Z}$?
Note that this does not follow from the statement that every abelian group has a presentation, which is equivalent to the statement that every abelian group is a coequalizer of a pair of maps between free abelian groups, hence every abelian group is an iterated colimit of copies of $\mathbb{Z}$. A single colimit $A = \text{colim}_{j \in J} F(j)$ of copies of $\mathbb{Z}$ is in particular the coequalizer of a pair of maps between free abelian groups, but the maps have a very special form, which works out explicitly to imposing the following constraint:

$A$ must have a presentation by generators and relations in which the only relations say that some generator is a multiple of some other generator. 

Examples of abelian groups admitting a presentation of this form include cyclic groups and localizations of $\mathbb{Z}$, and the class of all such groups is closed under coproducts.
But I see no reason to believe that every abelian group admits a presentation of this form, and in particular I believe that the $p$-adic integers doesn't. Tyler Lawson sketched a proof of this in the homotopy theory chat but it had a gap; the subsequent discussion may have filled the gap but I didn't follow it, and in any case I'd like someone to write up the details. 
Mike Shulman wrote a lovely note about various different senses in which an object or objects of a category can generate it; in the terminology of that note, the question is whether $\mathbb{Z}$ is colimit-dense in $\text{Ab}$. Until a week or so ago, if you had asked me, I would have answered without hesitation that $R$ is colimit-dense in $\text{Mod}(R)$, and I doubt that I was alone in this...

REPLY [17 votes]: Abelian groups of this type are known under the name "simply presented abelian group" and there are various investigations on them. 
Among others, it is known that simply presented abelian $p$-groups that are reduced (that is the divisible supgroup is trvial) are characterized by their Ulm sequence. 
As a consequence, two distinct abelian $p$-groups that are reduced and have the same Ulm sequence cannot be both simply presented. 
An explcit example is the torsion subgroup of   $\prod_n\mathbb{Z}/p^n\mathbb{Z}$ that has the same Ulm sequence as $\oplus_n\mathbb{Z}/p^n\mathbb{Z}$, and the latter is simply presented. 
There is also a homological characterization of such groups; indeed, another name is totally projective $p$-groups. 
I believe some information on this is in Fuchs' classic books on abelian groups. A more recent source would be Loth "Classifications of Abelian Groups and Pontrjagin Duality" (1998)