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$\begingroup$ Thanks! The colimit in the second case is also known as a Prüfer group: en.wikipedia.org/wiki/Pr%C3%BCfer_group. The only thing I'd add to your proof is that the inclusion $Grp \to sSet$ is fully faithful, so that if the colimits on the simplicial set side were isomorphic, they'd have to be isomorphic on the group side, leading to a contradiction as you explained. $\endgroup$
– Todd Trimble ♦
Commented Jun 23, 2019 at 1:42
$\begingroup$ Yes! You are right. Both are fully faithful! $\endgroup$
– Andrea Marino
Commented Jun 23, 2019 at 7:31

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