Proposition 2.4.7.12. Let $(W,S)$ be a Coxeter system. Then the map of simplicial sets $u: B_{\bullet }^{\circ }(W) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( BM(W) )$ of Construction 2.4.7.11 exhibits $BM_{\bullet }(W)$ as a path category of the simplicial set $B_{\bullet }^{\circ }(W)$, in the sense of Definition 2.4.4.1.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Fix an integer $m \geq 0$. Then $BM_{m}(W)$ is the delooping of the monoid $M_{m}(W)$ whose elements are tuples
\[ \overrightarrow {w}_0 \preceq \overrightarrow {w}_1 \preceq \overrightarrow {w}_2 \preceq \cdots \preceq \overrightarrow {w}_ m, \]
where each $\overrightarrow {w}_{i} \in M_0(W)$ is a sequence $( w_{i,1}, w_{i,2}, \ldots , w_{i, n_ i} )$ of elements of $W \setminus \{ 1\} $. Moreover, the monoid structure on $M_{m}(W)$ is given by concatenation. From this description, it is easy to see that the monoid $M_{m}(W)$ is freely generated by its indecomposable elements, which are precisely those sequences for which the sequence $\overrightarrow {w}_ m$ has length $1$. In this case, the relation $\overrightarrow {w}_0 \preceq \overrightarrow {w}_ m$ guarantees that $\overrightarrow {w}_0$ is a nondegenerate $n_0$-simplex of the simplicial set $B_{\bullet }^{\circ }(W)$. It follows that the map $u$ induces a bijection from the set $E( B^{\circ }(W), m)$ of Notation 2.4.4.9 to the set of indecomposable elements of the monoid $M_{m}(W)$. The desired result now follows from the criterion of Remark 2.4.4.11. $\square$