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Natural functors to the universal wall groupoid

PDF p. 16: Remark 4.5

Statement

Let $(W, S)$ be a finite-rank Coxeter system and fix a subset $J$ of its diagram vertices. Let $D^{+} (J)$ be the positive category of the relative chamber arrangement and let $D (J)$ be its groupoid completion. Its chambers have labels $(x, I)$ , where $W_{J} x = x W_{I}$ and $x$ has minimum length in $x W_{I}$ . Use the chamber-labelling bijection to identify these chambers with their algebraic labels. Let $\tilde{B H} (J)$ be the groupoid with one arrow from each label $(x, I)$ to each label $(y, K)$ , labelled $x^{- 1} y$ .

There are full functors $D^{+} (J) \to \tilde{B H} (J)$ and $D (J) \to \tilde{B H} (J)$ , both identity on objects. The first sends a chamber path $p : (x, I) \to (y, K)$ to its wall word $v_{p} = x^{- 1} y$ ; the second extends the first and sends formal inverses to inverses. Every based loop in either source maps to the identity. If the arrangement has two adjacent chambers, neither functor is faithful. Here full means surjective on every hom-set, and faithful means injective on every hom-set.

Statement references

Positive arrangement category and its groupoid completion

Universal wall groupoid on relative chambers

Algebraic chamber-label set

Labelling theorem for relative chambers

Chamber paths, concatenation and wall words

The universal wall groupoid is generated by adjacent crossings

Path presentations, congruences and formal inverses

Set-sized groupoids, functors and composition conversion

Facts & Assumptions

[F1]

The positive category is the quotient of the directed chamber-path category by the congruence generated by pairs of geodesic paths with common endpoints. Each adjacent pair gives one original arrow in each direction; geodesic means having minimal length among paths with the same endpoints. The arrangement groupoid is its groupoid completion. Positive arrangement category and its groupoid completion

[F2]

The universal wall groupoid has one arrow between any ordered pair of chamber labels, labelled $x^{- 1} y$ , with composition given by multiplication of labels. It is a groupoid. Universal wall groupoid on relative chambers

[F3]

A chamber path has a wall word given by the ordered product of its crossing elements, and for every pair of chambers there is a positive path whose wall word labels the unique universal-groupoid arrow between them. Chamber paths, concatenation and wall words The universal wall groupoid is generated by adjacent crossings

[F4]

A functor from a presented path category into a groupoid descends when it respects the relations; a functor from the resulting category into a groupoid extends uniquely to its groupoid completion. Path presentations, congruences and formal inverses

[F5]

A functor preserves identities and composition, and every arrow of a groupoid has a two-sided inverse. Set-sized groupoids, functors and composition conversion

[F6]

For the finite-rank Coxeter system and arbitrary $J$ fixed here, the chamber-labelling theorem gives a bijection between the relative chambers and the algebraic labels $(x, I)$ . We identify the object sets through this bijection. Labelling theorem for relative chambers

Proof

technique · direct

Given: The finite-rank Coxeter system, subset $J$ , and categories of the Statement.

Goal: Construct the functors and establish their fullness, loop images, and the stated criterion for nonfaithfulness.

1.1F1F2F3F4F6

Construct the positive functor. Use the bijection in [F6] to identify each chamber with its unique label, and send each directed chamber edge to the unique universal-groupoid arrow with the same endpoints. Composition gives a path from $(x, I)$ to $(y, K)$ the arrow labelled $x^{- 1} y$ , by [F2]. Any two paths with the same endpoints therefore have the same image, so in particular the geodesic relations hold. The assignment descends to a functor $F^{+}$ by [F1] and [F4]. For a path $p$ , each crossing element is the label of its edge, and their product is $v_{p}$ ; hence $v_{p} = x^{- 1} y$ by [F2] and [F3].

2.1F2F3F4F5step 1.1

Extend and prove fullness. Since the target is a groupoid, [F4] extends $F^{+}$ uniquely to a functor $F$ from $D (J)$ , sending formal inverses to inverses. By [F3], every ordered pair of chambers is joined by a positive path, and its image is the sole arrow in the corresponding target hom-set. Thus $F^{+}$ is full; because $F$ extends it, $F$ is full too. A based loop has the same source and target, so its image is the sole target endomorphism, namely the identity. This applies to loops in either source.

3.1F1F5step 2.1

Distinguish a positive loop. Suppose chambers $A$ and $B$ are adjacent. Let $e : A \to B$ and $f : B \to A$ be their two original directed edges. Path length is constant on each generating geodesic relation because two geodesics with the same endpoints have the same minimal length; it is also preserved under concatenation. It therefore descends to a functor from $D^{+} (J)$ to the one-object category whose arrows are the integers under addition, sending every edge to $1$ . This is a groupoid: the inverse of an integer is its negative. The loop $e f$ has length $2$ , whereas the identity at $A$ has length $0$ , so they are distinct in $D^{+} (J)$ . Their images under $F^{+}$ are both the identity by 2.1.

4.1F4F5step 2.1step 3.1∎

Distinguish the loop after completion. By [F4], the length functor of 3.1 extends to $D (J)$ , since the integer target is a groupoid. The image of the canonical image of $e f$ is still $2$ , while that of the identity is $0$ ; hence these arrows remain distinct in the completion. By 2.1 their images under $F$ coincide. Consequently neither functor is faithful whenever adjacent chambers exist.

Source notes

Remark 4.5 attributes the full functors and the general failure of faithfulness to the article. The adjacent-chamber criterion above makes a concrete sufficient condition explicit; it does not assert failure in the degenerate case with no adjacent chambers.

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF page 16, Remark 4.5