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Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial set?

(This is mostly an idle question, but also motivated by the fact that it is a theorem of Milnor that a similar construction with the free group gives a model for the $\Omega \Sigma X$, perhaps believable in view of the fact that $\Omega \Sigma$ is supposed to be the left adjoint from spaces into grouplike $A_\infty $-(i.e., with a coherently associative multiplication law) spaces.)

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I seemed to have asked the same question twice by accident. Sorry about that; I've deleted the duplicate. –  Akhil Mathew May 7 '12 at 0:58
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1 Answer

up vote 9 down vote accepted

Let $R[-]$ be the free $R$-module functor, from sets to $R$-modules, and $T_R$ the free (tensor) $R$-algebra functor, from $R$-modules to $R$-algebras. The free $R$ algebra functor from sets to $R$-algebras is the compoisite $T_RR[-]$.

Given a simplicial set $X$, the homotopy type of $R[X]$ is well understood. It is a product of Eilenberg-MacLane spaces, and its homotopy groups are the homology groups of $X$ with coefficients in $R$,

$$R[X]\simeq \prod_{n\geq 0}K(H_n(X,R),n).$$

Since

$$T_RM=\bigoplus_{m\geq 0}M^{\otimes m}$$

we have

$$T_RR[X]=\bigoplus_{m\geq 0}R[X\times\stackrel{m}\cdots\times X]$$

therefore

$$T_RR[X]\simeq \prod_{n\geq 0}K\left(\bigoplus_{m\geq 0}H_n(X\times\stackrel{m}\cdots\times X,R),n\right).$$

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Thanks! This is very nice. –  Akhil Mathew May 7 '12 at 15:31
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