Let C be the model category of simplicial commutative monoids (with underlying weak equivalences and fibrations), or equivalently the (∞,1)-category PΣ(Top), where T is the Lawvere theory for commutative monoids. In C, as in any pointed (∞,1)-category with finite limits and colimits, we can define adjoint functors ΣC and ΩC as ΣCX = hocolim [• ← X → •] and ΩCX = holim [• → X ← •].
The category CommMon of commutative monoids sits inside C as a full subcategory (as the constant objects, or the objectwise-discrete presheaves). Consider the two functors CommMon → C given by sending M to ΩCΣCM and to the group completion of M, respectively. Is there a natural equivalence between these functors?
(This question is closely related to Chris's question here. A thorough answer to that question would probably yield this immediately.)