Let $M$ be a monoid up to homotopy. The simplicial set $WM$ is defined by setting $$ WM_n=M^{n+1}=\{(g_0, g_1,\cdots,g_n)\mid g_i\in M\} $$ with faces and degeneracies given by \begin{eqnarray*} d_i(g_0,g_1,\cdots,g_n)&=& \left\{ \begin{array}{lcl} (g_0,g_1,\cdots,g_{i-1},g_ig_{i+1},g_{i+2},\cdots,g_n)& \text{if} & i<n, \\ (g_0,g_1,\cdots,g_{n-1})& \text{if} & i=n; \end{array} \right.\\ s_i(g_0,g_1,\cdots,g_n)&=&(g_0,g_1,\cdots,g_i,e,g_{i+1},\cdots,g_n). \end{eqnarray*} Define the left action of $M$ on $M^{n+1}$ by setting $$ g\cdot (g_0,g_1,\cdots,g_n)=(gg_0,g_1,\cdots,g_n). $$ Then $d_i(g\cdot (g_0,g_1,\cdots,g_n))=g\cdot d_i (g_0,g_1,\cdots,g_n)$ and $s_ig\cdot (g_0,g_1,\cdots,g_n)=g \cdot s_i (g_0,g_1,\cdots,g_n)$, that is, the action $$ M\times WM\to WM $$ is simplicial. By modolo the action of $M$, we obtain the simplicial set $$ \bar WM=WM/M.$$
The standart geometric $n$-simplex is the space $$ \Delta^n=\{(t_0,t_1,\cdots,t_n)\mid t_i\geq 0 and \sum_{i=0}^n t_i=1\}. $$ Define $d^i: \Delta^{n-1}\to \Delta^n$ and $s^i: \Delta^{n+1}\to \Delta^n$ by setting \begin{eqnarray*} d^i(t_0,t_1,\cdots, t_{n-1})&=&(t_0, \cdots, t_{i-1},0,t_i,\cdots, t_{n-1}),\\ s^i(t_0,t_1,\cdots, t_{n+1})&=&(t_0, \cdots, t_{i-1},t_i+t_{i+1},\cdots, t_{n+1}) \end{eqnarray*} for $0\leq i\leq n$. For a simplicial set $X$, the geometric realization $|X|$ is a CW-complex defined by \begin{eqnarray*} |X|&=&\bigsqcup_{x\in X_n,n\geq 0}(\Delta^n,x)/\sim\\ &=&\bigsqcup_{n=0}^\infty \Delta^n\times X_n/\sim \end{eqnarray*} where $\sim$ is generated by $$ (z,d_ix)\sim (d^iz,x) $$ for any $x\in X_n$ and $z\in \Delta^{n-1}$, and $$ (z,s_ix)\sim(s^iz,x) $$ for any $x\in X_n$ and $z\in \Delta^{n+1}$.
For a monoid up to homotopy $M$, we denote the geometric realization of $WM$ by $EM$ and the geometric realization of $\bar WM$ by $BM$. The CW-complex $BM$ is called the classifying space of $M$.
My question:
Conjecture: (1). There is a canonical map $p: EM\to BM$ such that for any $b\in BM$, the preimage $p^{-1}(M)$ is homeomorphic to $M$. (2). Moreover, $EM$ is contractible.
How to prove the conjecture? Any references for the proof? Thanks for giving help! If my construction of the classifying space $BM$ and the "universal map" $EM\to BM$ is wrong, where can I find the correct construction?