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_CX = hocolim [• ← X → •] and Ω_CX = 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_CΣ_CM 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.)

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_CX = hocolim [• ← X → •] and Ω_CX = 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_CΣ_CM 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.)

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. 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 Chris's question here. A thorough answer to that question would probably yield this immediately.)

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_CX = hocolim [• ← X → •] and Ω_CX = 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_CΣ_CM 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.)