Kerodon

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Theorem 2.4.7.9 (Deligne). Let $(W,S)$ be a Coxeter system for which the underlying Coxeter group $W$ is finite, and let $\mathrm{Br}^{+}(W)$ denote the braid monoid of Construction 2.4.7.5. Then:

$(a)$

There is an isomorphism of monoids $f: \pi _0( M_{\bullet }(W) ) \rightarrow \mathrm{Br}^{+}(W)$ which is uniquely determined by the following property: if $\overrightarrow {s} = (s_1, s_2, \ldots , s_ m) \in M_0(W)$ is a sequence of elements of $S$, then $f$ carries the connected component of $\overrightarrow {s}$ to the product $s_1 s_2 \cdots s_ m \in \mathrm{Br}^{+}(W)$.

$(b)$

Each connected component of $M_{\bullet }(W)$ is weakly contractible (Definition 3.2.6.1).

In other words, the isomorphism $f$ determines a weak homotopy equivalence of simplicial monoids $M_{\bullet }(W) \rightarrow \mathrm{Br}^{+}(W)$.

Proof. This is a special case of Théorème 2.4 of [MR1437497]. $\square$