The reduced ribbon subgroupoid uses exactly the exterior atoms

Let $(A,S)$ be a spherical Artin system, with positive monoid $A^{+}$ and standard parabolics $A_I^{+}$ and $A_I$ for every $I\subseteq S$. Use the ribbon groupoid $\mathcal{R}(A)$ with all these $A_I$ as objects, labelled by $I$, and arrows $(I,g,J)$ for $g\in A$ satisfying $g^{-1}{A}_I^{+}g=A_J^{+}$. Composition is $(I,g,J)(J,h,K)=(I,gh,K)$, the identity is $(I,1,I)$, and the inverse is $(J,g^{-1},I)$. Positive arrows have $g\in A^{+}$.

For $K\subseteq S$, let $w_K$ be the longest element of $W_K=⟨K⟩$ in the associated Coxeter system. With $(e_s{)}_{s\in S}$ its canonical simple-root basis, use the opposition permutation ${\iota }_K:K\to K$ characterized by $w_K{e}_s=-e_{{\iota }_K(s)}$ for $s\in K$.

For the spherical Artin nu-structure, Godelle's subgroupoid ${\mathcal{R}}^{\nu }(A)$ is exactly the subgroupoid on this full object set generated by all arrows $r(I,a)={\mathrm{\Delta }}_I^{-1}{\mathrm{\Delta }}_{I\cup \{a\}}:I\to {\iota }_{I\cup \{a\}}(I)$ for $a\mathrm{\notin }I$, together with their inverses and every object identity. Thus it is generated by the exterior elementary ribbons, without the internal atoms ${\mathrm{\Delta }}_{I_s}$ in the generating family. “Generated” permits finite composable products and inverses; it does not assert freeness or a list of relations.

We import Godelle's ribbon definitions in their classical spherical Artin specialization (first reference). Definition 2.7 and Definitions 3.7–3.8 give all standard parabolic objects and ribbon arrows whose conjugation preserves the indicated positive parabolic monoids; the paragraph opening Section 3.3 gives the positive-arrow criterion. In the one-object monoid setting, the natural transformation is represented by $g$, and its associated isomorphism is $x\mapsto g^{-1}xg$. Thus the concrete tagged-arrow description in the Statement is this ambient ribbon groupoid, rather than the larger groupoid of arbitrary subgroup conjugators.

We import the defining conventions in Section 4.2 (second reference): an elementary atom is denoted $\tau (I,s)$ when it lies in $A_I^{+}$ and $\nu (I,s)$ otherwise. The paragraph immediately before Lemma 4.17 defines ${\mathcal{R}}^{\nu }(A)$ as the subgroupoid generated by the latter arrows. Corollary 4.18 uses the same generating convention and specifies the same vertex set as the full ribbon groupoid. No presentation theorem or lcm assertion is needed here.

We import the spherical nu-structure and its exterior formula from Godelle's Example 3.14 (third reference, printed/PDF p. 15). For every $I\subseteq S$ and $a\in S\setminus I$, that example gives ${\nu }_I(a)={\mathrm{\Delta }}_I^{-1}{\mathrm{\Delta }}_{I\cup \{a\}}$. This supplies the equality with $r(I,a)$ directly; the endpoint lemma supplies its endpoint and exclusion from $A_I^{+}$. We do not reconstruct the proof attributed there to [15].