configuration space and iterated loop space

Question

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, 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. How to prove?

Answer 1

$C_n(X)$ 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.