Take the 2-minute tour ×  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.)  share|improve this
question 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  1 Answer 1  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 co