The empty-object ribbon component and its braid relations

Let $(A,S)$ be a finite-rank spherical Artin system, so its associated Coxeter group $W$ is finite, and let $A^{+}$ be its positive monoid. For $I\subseteq S$, let $A_I^{+}$ be the standard parabolic submonoid generated by $I$ and let ${\mathrm{\Delta }}_I$ be its standard Garside element, the positive lift of the longest element of $W_I$; in particular ${\mathrm{\Delta }}_{\mathrm{\varnothing }}=1$ and ${\mathrm{\Delta }}_{\{a\}}=a$.

Use the ribbon category with objects $I\subseteq S$ and arrows $(I,g,J)$ satisfying $g^{-1}Ig=J$, with $g\in A^{+}$ for positive arrows and $g\in A$ for groupoid arrows. Equality requires equal endpoints and equal labels; composition is $(I,g,J)(J,h,K)=(I,gh,K)$. For $a\in S\setminus I$, the reduced generator has label $\nu (I,a)={\mathrm{\Delta }}_I^{-1}{\mathrm{\Delta }}_{I\cup \{a\}}$ and target $\nu (I,a{)}^{-1}I\nu (I,a)$. The reduced positive category and reduced groupoid are the subcategory and subgroupoid generated by these arrows.

At $I=\mathrm{\varnothing }$, every reduced generator is the loop labelled $a$. This object is an isolated component: its reduced positive component is the one-object monoid $A^{+}$, and its reduced groupoid component is the one-object group $A$. For distinct $a,b\in S$, the lcm for left divisibility has exactly its two alternating braid paths of length $m_{ab}$, and these braid equalities present this component as a groupoid.

Independent spherical Artin Garside input

Set-sized groupoids, functors and composition conversion

The two-generator braid element is the lcm

Godelle, Definitions 3.7–3.8, defines the positive ribbon category and ribbon groupoid using conjugation of standard positive parabolics. In the monoidal description before Definition 3.12, a positive arrow $I\to J$ is a label $g\in A^{+}$ with $g^{-1}Ig=J$. Groupoid labels lie in $A$ with the same endpoint condition; the natural-transformation construction gives label equality at fixed endpoints and multiplication for composition.

For the spherical structure of F1, Example 3.14 supplies $\nu (I,a)={\mathrm{\Delta }}_I^{-1}{\mathrm{\Delta }}_{I\cup \{a\}}$ for $a\mathrm{\notin }I$. The reduced-generator notation is from §4.2; the paragraph before Lemma 4.17 defines the reduced subcategory and subgroupoid generated by these arrows. These construction inputs are imported under the reference hypotheses; their general proofs are not reconstructed.