Is ΩΣ in {simplicial commutative monoids} group completion? 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.  A thorough answer to that question would probably yield this immediately.)
 A: 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, $\Omega BM\approx \Omega BM'\approx M'$.
