Universal wall groupoid on relative chambers

PDF p. 15: Definition 4.3 and following paragraph

Let $(W,S)$ be a Coxeter system, let $J$ be a subset of its diagram's vertex set, and let $\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{m}(J)$ be the algebraic chamber-label set. Thus its elements are pairs $(x,I)$ with $x\in W$, $W_{J}x=x{W}_I$, and $x$ of minimal length in $x{W}_I$.

The universal wall groupoid $\widetilde{\mathcal{B}\mathcal{H}}(J)$ has object set $\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{m}(J)$. For objects $A=(x,I)$ and $B=(y,K)$, its hom-set $\mathrm{Hom}(A,B)$ is the singleton whose arrow is labelled $x^{-1}y\in W$. Arrows are tagged by their source and target, even when their labels in $W$ coincide. Composition is read from left to right: the arrow $A\to B$ labelled $x^{-1}y$ followed by the arrow $B\to C=(z,L)$ labelled $y^{-1}z$ is the arrow $A\to C$ labelled $(x^{-1}y)(y^{-1}z)=x^{-1}z$.

For the geometric interpretation, assume additionally that $(W,S)$ has finite rank. Identify the labels with the relative chambers by the chamber-labelling bijection, and call two labels adjacent when their corresponding chambers share a wall. For adjacent labels $A=(x,I)$ and $B=(y,K)$, let $a\mathrm{\notin }I$ be the wall index supplied by the adjacent-wall proposition. That proposition supplies finiteness of the standard parabolic on the crossing component, as required to define the simple wall-crossing element $v_{a,I}$. The unique arrow $A\to B$ has label $v_{a,I}$. No finiteness of $W$, $W_I$ or $W_{I\cup \{a\}}$ is assumed.

Algebraic chamber-label set

Connected small groupoids and vertex groups

Set-sized groupoids, functors and composition conversion

Chambers, walls and adjacency

Labelling theorem for relative chambers

Simple wall-crossing elements and maps

Adjacent chambers and inverse wall-crossing elements

The chamber-label definition supplies a nonempty set of objects and a group $W$. Taking the arrows to be the tagged triples $(A,B,x^{-1}y)$ gives a set, with the stated singleton hom-sets. Group associativity makes the prescribed composition associative. The arrow $(A,A,1)$ is an identity, and the arrow $B\to A$ labelled $y^{-1}x$ is the two-sided inverse of the arrow $A\to B$. Hence the category and groupoid axioms in Set-sized groupoids, functors and composition conversion hold. There is an arrow between every ordered pair of objects, so the nonempty groupoid is connected in the sense of Connected small groupoids and vertex groups.

In the finite-rank geometric setting, Labelling theorem for relative chambers identifies the algebraic labels with the relative chambers, and Chambers, walls and adjacency defines adjacency by a shared wall. For adjacent labels, Adjacent chambers and inverse wall-crossing elements supplies the wall index $a\mathrm{\notin }I$, finiteness of its crossing component's standard parabolic, and ${\omega }_{a,I}(x,I)=(y,K)$. The finiteness condition permits the definition in Simple wall-crossing elements and maps, whose formula gives first coordinate $x{v}_{a,I}$ for ${\omega }_{a,I}(x,I)$. Hence $y=x{v}_{a,I}$. Multiplication by $x^{-1}$ on the left gives $x^{-1}y=v_{a,I}$, as claimed.