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. In C, as in any pointed (∞,1)-category with finite limits and colimits, we can define adjoint functors Σ<sub>C</sub> and Ω<sub>C</sub> as Σ<sub>C</sub>X = hocolim [• ← X → •] and Ω<sub>C</sub>X = 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 Ω<sub>C</sub>Σ<sub>C</sub>M and to the group completion of M, respectively. Is there a natural equivalence between these functors? (This question is closely related to <a href="https://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.)