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2.4.7 Example: Braid Monoids

In general, the path category $\operatorname{Path}[S]_{\bullet }$ associated to a simplicial set $S_{\bullet }$ is a fairly complicated object. In this section, we describe one situation in which it admits a particularly concrete description, which arises in the theory of Coxeter groups. Let us begin by reviewing some terminology.

Definition 2.4.7.1. A Coxeter system is a pair $(W,S)$, where $W$ is a group and $S \subseteq W$ is a subset with the following properties:

  • Each element of $S$ has order $2$.

  • For each $s,t \in S$, let $m_{s,t} \in \operatorname{\mathbf{Z}}_{ > 0} \cup \{ \infty \} $ denote the order of the product $st$ in the group $W$. Then the inclusion $S \hookrightarrow W$ exhibits $W$ as the quotient of the free group generated by $S$ by the relations $(st)^{m_{s,t}} = 1$ (indexed by those pairs $(s,t)$ with $m_{s,t} < \infty $).

Remark 2.4.7.2. We will use the term Coxeter group to refer to a group $W$ together with a choice of subset $S \subseteq W$ for which the pair $(W,S)$ is a Coxeter system. Beware that the subset $S$ is not determined by the structure of $W$ as an abstract group: for example, if $(W,S)$ is a Coxeter system, then so is $(W, wSw^{-1})$ for each $w \in W$. In other words, a Coxeter group is not merely a group, but a group equipped with some additional structure (namely, the structure of a Coxeter system $(W,S)$).

Notation 2.4.7.3 (Lengths). Let $(W,S)$ be a Coxeter system. Then the group $W$ is generated by $S$: that is, every element of $W$ can be written as a product of elements of $S$. For each $w \in W$, we let $\ell (w)$ denote the smallest nonnegative integer $n$ for which $w$ factors as a product $s_1 s_2 \cdots s_ n$, where each $s_ i$ belongs to $S$. We will refer to $\ell (w)$ as the length of $w$.

Remark 2.4.7.4. Let $(W,S)$ be a Coxeter system. Then the length function $\ell : W \rightarrow \operatorname{\mathbf{Z}}_{\geq 0}$ has the following properties:

  • An element $w \in W$ satisfies $\ell (w) = 0$ if and only if $w = 1$ is the identity element of $W$.

  • An element $w \in W$ satisfies $\ell (w) = 1$ if and only if $w$ belongs to $S$.

  • For every pair of elements $w,w' \in W$, we have $\ell ( w w' ) \leq \ell (w) + \ell (w')$. Moreover, we also have $\ell ( w w' ) \equiv \ell (w) + \ell (w') \pmod{2}$.

Construction 2.4.7.5 (The Braid Group). Let $(W,S)$ be a Coxeter system. We let $\mathrm{B}(W)$ denote the quotient of the free group generated by $S$ by the relations $(st)^{m_{s,t}} = 1$, where $s$ and $t$ range over distinct elements of $S$ satisfying $m_{s,t} < \infty $; here $m_{s,t}$ denotes the order of the product $st$ in the group $W$. We will refer to $\mathrm{B}(W)$ as the braid group of the Coxeter system $(W,S)$. By construction, the braid group $\mathrm{Br}(W)$ is equipped with a surjective group homomorphism $\mathrm{Br}(W) \twoheadrightarrow W$, which exhibits $W$ as the quotient of $\mathrm{Br}(W)$ by the relations $s^2 = 1$ for $s \in S$.

Let $\mathrm{Br}^{+}(W)$ denote the submonoid of $\mathrm{Br}(W)$ generated by the elements of $S$. We will refer to $\mathrm{Br}^{+}(W)$ as the braid monoid of the Coxeter system $(W,S)$.

In [MR1437497], Deligne gave a convenient simplicial presentation for the braid monoid $\mathrm{Br}^{+}(W)$ in the case where the Coxeter group $W$ is finite. To formulate it, we need a bit more terminology.

Notation 2.4.7.6. Let $W$ be a Coxeter group with identity element $1$. We let $M_0(W)$ denote the free monoid generated by the set $W \setminus \{ 1\} $. We will identify the elements of $M(W)$ with finite sequences $\vec{w} = ( w_1, w_2, \ldots , w_ n )$, where each $w_{i}$ is an element of $W \setminus \{ 1\} $.

Let $\vec{v} = ( v_1, v_2, \ldots , v_ m )$ and $\vec{w} = ( w_1, \ldots , w_ n )$ be elements of $M_0(W)$. We will say that $\vec{w}$ is a refinement of $\vec{v}$ if there exists a strictly increasing function $\varphi : [m] \rightarrow [n]$ satisfying $\varphi (0) = 0$, $\varphi (m) = n$, and

\[ v_{i} = w_{ \varphi (i-1) + 1 } w_{ \varphi (i-1) + 2} \cdots w_{ \varphi (i) } \]
\[ \ell ( v_ i ) = \ell ( w_{ \varphi (i-1) + 1 } ) + \ell ( w_{ \varphi (i-1) + 2 } ) + \cdots + \ell ( w_{ \varphi (i) } ) \]

for $1 \leq i \leq m$. We write $\vec{v} \preceq \vec{w}$ to indicate that $\vec{w}$ is a refinement of $\vec{v}$. Then $\preceq $ determines a partial ordering on the set $M_0(W)$. We denote the nerve of this partially ordered set by $M_{\bullet }(W)$. Note that the multiplication on $M_0(W)$ (given by concatenation) endows $M_{\bullet }(W)$ with the structure of a simplicial monoid.

Exercise 2.4.7.7. Let $W$ be a Coxeter group, and let $\vec{u} = ( u_1, u_2, \ldots , u_ m )$ and $\vec{w} = ( w_1, \ldots , w_ n )$ be elements of $M(W)$. Show that, if $\vec{w}$ is a refinement of $\vec{u}$, then there is a unique sequence of integers $0 = j_0 < j_1 < \cdots < j_ m = n$ satisfying the condition specified in Notation 2.4.7.6.

Remark 2.4.7.8. Let $(W,S)$ be a Coxeter system. Then an element $\vec{w} = (w_1, w_2, \ldots , w_ m )$ of $M_0(W)$ is maximal (with respect to the refinement ordering $\preceq $) if and only if each $w_ i$ belongs to $S$. Moreover, every element $\vec{w} \in M_0(W)$ admits a refinement $\vec{s} = (s_1, s_2, \ldots , s_ n )$ which is maximal in $M_0(W)$ (given by choosing a decomposition of each $w_ i$ as a product of elements of $S$). In particular, every connected component of the simplicial set $M_{\bullet }(W)$ contains a vertex $\vec{s} = (s_1, \ldots , s_ n)$, where each $s_{i}$ belongs to $S$.

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 $\vec{s} = (s_1, s_2, \ldots , s_ n) \in M_0(W)$ is a sequence of elements of $S$, then $f$ carries the connected component of $\vec{s}$ to the product $s_1 s_2 \cdots s_ n \in \mathrm{Br}^{+}(W)$.

$(b)$

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

In particular, $f$ 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]. $\square$

We now reformulate the definition of the simplicial monoid $M_{\bullet }(W)$ using the theory of simplicial path categories.

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 $B_{\bullet }^{\circ } W \subseteq B_{\bullet } W$.

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 2.4.2.2), 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 $u: B_{\bullet }^{\circ }(W) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( BM(W) )$.

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.

Proof. Fix an integer $m \geq 0$. Then $BM_{m}(W)$ is the delooping of the monoid $M_{m}(W)$ whose elements are tuples

\[ \vec{w}_0 \preceq \vec{w}_1 \preceq \vec{w}_2 \preceq \cdots \preceq \vec{w}_ m, \]

where each $\vec{w}_{i} \in M_0(W)$ 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 $\vec{w}_0$ has length $1$. In this case, the relation $\vec{w}_0 \preceq \vec{w}_ m$ guarantees that $\vec{w}_ m$ is a nondegenerate $n_ m$-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. $\square$

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.