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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.)

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This doesn't work at the level of $\pi_0$. If $M$ is such that $\pi_0 M$ is say the natural numbers you want its group completion to satisfy $\pi_0$ is the integers, right? But $\pi_0 \Omega \Sigma$ of any topological monoid equivalent to the naturals should be huge, a free group on infinitely many generators. Or am I misunderstanding your question? – Ryan Budney Jan 31 '10 at 0:01
The underlying space of $\Sigma M$ won't be the suspension of the underlying space of $M$, because the forgetful functor from C to Spaces doesn't commute with colimits. For instance, if I replaced "commutative monoids" with "abelian groups", then C would be the category of nonnegatively-graded chain complexes, $\Sigma$ would be a shift so that $(\Sigma X)_n = X_{n-1}$, $\Omega$ would be a shift in the other direction, and $\Omega \Sigma$ would be the identity functor. – Reid Barton Jan 31 '10 at 0:07
So "$\Sigma M$" is really a model for the classifying space $BM$? So you are really asking: what does $BM$ of a discrete commutative monoid $M$ look like? In particular, does it have non-trivial homotopy groups in dimensions greater than $1$? (That's how I read your question, anyway.) – Charles Rezk Jan 31 '10 at 4:22
That's part of my question, and the other part is to identify $\Sigma_C M$ with $BM$ (is that obvious? I don't see it right away.) I tried to clarify exactly what my notation was intended to mean. – Reid Barton Jan 31 '10 at 4:58
Well, simplicial commutative monoids is cotensored over pointed simplicial sets. If I use this to define $\Sigma_C M$ to mean $(\Delta^1/\partial \Delta^1)\otimes M$, then I think this is exactly a 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? – Charles Rezk Jan 31 '10 at 5:38
up vote 10 down vote accepted

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, $\Omega BM\approx \Omega BM'\approx M'$.

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