Notation 2.4.7.6. Let $W$ be a Coxeter group with identity element $1$. We let $M_0(W)$ denote the free monoid generated by the set $W \setminus \{ 1\} $. We will identify the elements of $M_0(W)$ with finite sequences $\overrightarrow {w} = ( w_1, w_2, \ldots , w_ n )$, where each $w_{i}$ is an element of $W \setminus \{ 1\} $. We will say that $\overrightarrow {v}$ is a refinement of $\overrightarrow {w}$ if there exists a strictly increasing sequence of integers $0 = i_0 < i_1 < \cdots < i_ n = m$ having the property that
for $1 \leq j \leq n$. We write $\overrightarrow {v} \preceq \overrightarrow {w}$ to indicate that $\overrightarrow {v}$ is a refinement of $\overrightarrow {w}$. Then $\preceq $ determines a partial ordering on the set $M_0(W)$. We denote the nerve of this partially ordered set by $M_{\bullet }(W)$. Note that the multiplication on $M_0(W)$ (given by concatenation) endows $M_{\bullet }(W)$ with the structure of a simplicial monoid.