homological algebra - 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 $(m_1,m_2)in Mtimes M$, and where morphisms $(m_1,m_2)to (m_1',m_2')$ are $min M$ 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 $mathrm{Tor}_i^H(Hotimes H,F)$, 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 $BMto BM'$ is a weak equivalence, using the above homology calculation and another spectral sequence. Since $M'$ is weakly equivalent to a group, $Omega BMapprox Omega BM'approx M'$.

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