# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Construction 2.4.7.11. Let $(W,S)$ be a Coxeter system, let $M_{\bullet }(W)$ be the simplicial monoid of Notation 2.4.7.6, and let $BM_{\bullet }(W)$ denote the simplicial category obtained by delooping $M_{\bullet }(W)$ (Example 2.4.2.3), having a single object $X$ with $\operatorname{Hom}_{BM(W)}(X,X)_{\bullet } = M_{\bullet }(W)$.

Let $\sigma = (w_{n}, \ldots , w_1)$ be a nondegenerate $n$-simplex of the simplicial set $B_{\bullet }^{\circ }(W)$ (Notation 2.4.7.10). Then $\sigma$ determines a simplicial functor $u(\sigma ): \operatorname{Path}[n]_{\bullet } \rightarrow BM_{\bullet }(W)$, which carries each object of $\operatorname{Path}[n]_{\bullet }$ to the unique object $X$ of $BM_{\bullet }(W)$, and each morphism $I = \{ i_0 < \ldots < i_ k \} \in \operatorname{Hom}_{\operatorname{Path}[n]}( i_0, i_ k)$ to the sequence

$( v_1, v_{2}, \ldots , v_ k ) \in M_0(W) \quad \quad v_{j} = w_{ i_{j-1} +1 } w_{i_{j-1}+2} \cdots w_{ i_{j} }.$

Regarding $u(\sigma )$ as an $n$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( BM(W) )$, the construction $\sigma \mapsto u(\sigma )$ extends to a map of simplicial sets $u: B_{\bullet }^{\circ }(W) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( BM(W) )$.