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](https://mathoverflow.net/questions/430/homological-algebra-for-commutative-monoids). A thorough answer to that question would probably yield this immediately.)



Is ΩΣ in {simplicial commutative monoids} group completion?
I think an answer is given by the arguments that Segal gives in Section 4 of his paper on "Categories and Cohomology Theories" (aka, the $\Gamma$-space paper), in Topology, v.13. I'll try to sketch the main idea, translated into the context of simplicial commutative monoids. I'll show that if $M$ is a discrete simplicial commutative monoid, then it's group completion is homotopically discrete; according to the comments, this should answer the question.


Given a commutative monoid $M$, we can define a simplicial commuative monoid $M'$ as the nerve of the category whose objects are $(m\_1,m\_2)\in M\times M$, and where morphisms $(m\_1,m\_2)\to (m\_1',m\_2')$ are $m\in M$ such that $m\_im=m\_i'$. We can prolong this to a functor on simplicial commutative monoids.


Let $H=H\_\*|M|=H\_\*(|M|,F)$ (the homology of the geometric realization of $M$, with coefficients in some field $F$), viewed as a commutative ring under the pontryagin product. Then Segal shows that $H\_\*|M'|\approx H[\pi^{-1}]$, where $\pi$ denotes the image of $\pi\_0|M|$ in $H\_0|M]$. His proof amounts to computing the homology spectral sequence for a simplicial space whose realization is $M'$, and whose $E\_2$-term is $\mathrm{Tor}\_i^H(H\otimes H,F)$, and observing that the higher tor-groups vanish.


This means that if $M$ is discrete, then $H\_\*|M'|$ is concentrated in degree $0$. Since $|M'|$ is a grouplike commutative monoid, the Hurewicz theorem should tell us that $|M'|$ is weakly equivalent to a discrete space, namely the group completion of the monoid $M$.


Segal goes on to show that $BM\to BM'$ is a weak equivalence, using the above homology calculation and another spectral sequence. Since $M'$ is weakly equivalent to a group, $\Omega BM\approx \Omega BM'\approx M'$.


