Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Notation 2.4.7.10. Let $(W,S)$ be a Coxeter system and let $B_{\bullet }W$ denote the classifying simplicial set of the group $W$ (Example 1.2.4.3). For each nonnegative integer $n$, let us identify $B_{n}W$ with the collection of all $n$-tuples $(w_ n, w_{n-1}, \ldots , w_1)$ of elements of $W$. Let $B_{n}^{\circ }W$ denote the subset of $B_{n}W$ consisting of those sequences $(w_ n, w_{n-1}, \ldots , w_1)$ satisfying the identity

\[ \ell ( w_1 w_{2} \cdots w_ n ) = \ell (w_{1} ) + \ell (w_{2}) + \cdots + \ell ( w_ n ). \]

It is easy to see that the collection of subsets $B_{n}^{\circ }W \subseteq B_ n W$ are stable under the face and degeneracy operators of $B_{\bullet } W$, and therefore determine a simplicial subset $B_{\bullet }^{\circ } W \subseteq B_{\bullet } W$.