Let C be the model category of simplicial commutative monoids (with underlying weak equivalences and fibrations), or equivalently the (∞,1)-category P_{Σ}(T^{op}), 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_{C} and Ω_{C} as Σ_{C}X = hocolim [• ← X → •] and Ω_{C}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_{C}Σ_{C}M 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.)

exactlya model for $BM$, using the fact that finite comproducts of commutative monoids are set theoretic products. And my "cotensor suspension" is really the same as your "pushout suspension", since $C$ is a proper simplicial model category. Or am I missing the point here? $\endgroup$2more comments