FIGURE 6. The braided monoidal structure for the category $\mathcal{G}_d$
that $\oplus_r D_{d,1}^\infty$ has a naturally induced $d$-rigid structure and we can identify it with $D_{d,r}^\infty$, which will be of use to us later.
We can now define the category $\mathcal{G}d$ to be the monoidal category with objects $\oplus_r D{d,1}^\infty$, $r \ge 0$, $\oplus$ as the operation, and $D$ as the $0$ object. So far it is the same as defining the objects as the natural numbers and addition as the operation. When $r=s$, we define the morphisms $\text{Hom}(\oplus_r D_{d,1}^\infty, \oplus_s D_{d,1}^\infty) = \mathcal{B}\text{V}{d,r}$ which is the group of isotopy classes of asymptotically rigid homeomorphisms of $D{d,r}^\infty$; when $r \ne s$, let $\text{Hom}(\oplus_r D_{d,1}^\infty, \oplus_s D_{d,1}^\infty) = \emptyset$. Note that we did not universally define the morphisms to be the sets of isotopy classes of asymptotically rigid homeomorphisms as we want our category to satisfy cancellation, i.e., if $A \oplus C = A$ then $C = 0$, see [RWW17, Remark 1.11] for more information. The category $\mathcal{G}_d$ has a natural braiding as in the usual braid group case, see Figure 6(c).
Now applying [RWW17, Theorem 1.10], we have a homogeneous category $\mathcal{U}\mathcal{G}d$. The category $\mathcal{U}\mathcal{G}d$ has the same objects as $\mathcal{G}d$ and morphisms defined as following: For any $s \le r$, a morphism in $\text{Hom}(\oplus_s D{d,1}^\infty, \oplus_r D{d,1}^\infty)$ is an equivalence class of pairs $(\oplus{r-s} D_{d,1}^\infty, f)$ where $f: (\oplus_{r-s} D_{d,1}^\infty) \oplus (\oplus_s D_{d,1}^\infty) \to \oplus_r D_{d,1}^\infty$ is a morphism in $\mathcal{G}d$ and $(\oplus{r-s} D_{d,1}^\infty, f) \sim (\oplus_{r-s} D_{d,1}^\infty, f')$ if there exists an isomorphism $g: \oplus_{r-s} D_{d,1}^\infty \to \oplus_{r-s} D_{d,1}^\infty \in \mathcal{G}_d$ making the diagram commute up to isotopy.
We write $[\oplus_{s-r} D_{d,1}^{\infty}, f]$ for such an equivalence class. Now by Theorem 4.6, to prove the homological stability for the oriented ribbon Higman–Thompson groups, we only need to verify Condition LH3, i.e. the complex $W_r(D_{d,1}^{\infty}, D_{d,1}^{\infty})$ is highly connected. As a matter of fact, we will show that $W_r(D, D_{d,1}^{\infty})\cdot$ is $(r-3)$-connected in the next subsection. First, let us further characterize the morphisms in $\mathcal{U}\mathcal{G}d$. Call $0 = I^- \cap I^+$ the basepoint of $\oplus_r D{d,1}^\infty$.
Definition 4.7. Given $s < r$, an injective map $\varphi : (\oplus_s D_{d,1}^\infty, I^+) \to (\oplus_r D_{d,1}^\infty, I^+)$ is called an asymptotically rigid embedding if it satisfies the following properties:
(1) $\varphi(\partial D_{d,s}^\infty) \cap \partial D_{d,r}^\infty = I^+$.
(2) $\varphi$ maps $\oplus_s D_{d,1}^\infty$ homeomorphically to $\varphi(\oplus_s D_{d,1}^\infty)$ and there exists an admissible surface $A \subset \oplus_s D_{d,1}^\infty$ such that $\varphi: \oplus_s D_{d,1}^\infty \setminus A \to \varphi(\oplus_s D_{d,1}^\infty) \setminus \varphi(A)$ is rigid.