Corollary 2.4.7.13. Let $W$ be a finite Coxeter group, and let $B_{\bullet }^{\circ }(W) \subseteq B_{\bullet }(W)$ be the simplicial subset of Notation 2.4.7.10. Then the simplicial path category $\operatorname{Path}[ B^{\circ }(W) ]_{\bullet }$ has a single object $X$, whose endomorphism monoid $\operatorname{Hom}_{ \operatorname{Path}[ B^{\circ }(W) ] }( X, X)_{\bullet }$ is weakly homotopy equivalent to the braid monoid $\mathrm{Br}^{+}(W)$ of Construction 2.4.7.5.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$