Let C be the model category of simplicial commutative monoids (with underlying weak equivalences and fibrations), or equivalently the (∞,1)-category P<sub>Σ</sub>(T<sup>op</sup>), where T is the Lawvere theory for commutative monoids. 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 ΩΣM and to the group completion of M, respectively. Is there a natural equivalence between these functors? (This question is closely related to <a href="http://mathoverflow.net/questions/430/homological-algebra-for-commutative-monoids">Chris's question here</a>. A thorough answer to that question would probably yield this immediately.)