Let the topological monoid $M$ be the configuration space $C(\mathbb{R}^n;X)=C\_n(X)$ as in the book [The geometry of iterated loop spaces](http://www.google.com.sg/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CBwQFjAAahUKEwj7uO74wITGAhXlIqYKHZa-AJw&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.170.2840%26rep%3Drep1%26type%3Dpdf&ei=Utp3VfvHN-XFmAWW_YLgCQ&usg=AFQjCNEyNd--WqdU8Tb_uDmhRFbKH0rOOg&sig2=XSVEefdxsSoS7qeDklNoeA&bvm=bv.95277229,d.dGY), Theorem 5.2. I want to prove that the map $\alpha\_n$ in Theorem~5.2 there is the same map as the one described in the answer to [this question on math.stackexchange](https://math.stackexchange.com/questions/1313289/canonical-map-of-a-monoid-to-its-classifying-space/1313361#1313361). How to prove?



configuration space and iterated loop space
$C\_n(X)$ is not a monoid in any natural way; I never said it was. And the target $\Omega^n\Sigma^n X$ of $\alpha\_n$ has $n$ different loop products. The question is not meaningful as posed. Nevertheless the proof of Theorem 6.1 (the approximation theorem) answers it by describing $\alpha\_n$ as the fiber map of a map from a quasifibration to "the" path loop fibration $\Omega \Omega^{n-1}\Sigma^n X\to P \Omega^{n-1}\Sigma^n X \to \Omega^{n-1}\Sigma^n X$ (which one being determined by an ordering of the loop coordinates).
From here, there are papers by Fiedorowicz and by Thomason that compare this $1$-fold delooping to the classifying space delooping of monoids given by Moore loop spaces.


