Definition 4.2 ([RWW17, Definition 1.5]). Let $(\mathcal{C}, \oplus, 0)$ be a monoidal category with 0 initial. We say that $\mathcal{C}$ is prebraided if its underlying groupoid is braided and for each pair of objects $A$ and $B$ in $\mathcal{C}$, the groupoid braiding $b_{A,B}: A \oplus B \to B \oplus A$ satisfies
Definition 4.3. [RWW17, Definition 2.1] Let $(\mathcal{C}, \oplus, 0)$ be a monoidal category with 0 initial and $(A, X)$ a pair of objects in $\mathcal{C}$. Define $W_n(A, X)_\bullet$ to be the semi-simplicial set with set of $p$-simplices
and with face map
defined by precomposing with $X^{\oplus i} \oplus \iota_X \oplus X^{\oplus p-i}$.
Also define the following property for a fixed pair $(A, X)$ and a slope $k \ge 2$.
LH3 For all $n \ge 1$, $W_n(A, X)_\bullet$ is $(\frac{n-2}{k})$-connected.
Quite often, we can reduce the semi-simplicial complex to a simplicial complex.
Definition 4.4 ([RWW17, Definition 2.8]). Let $A, X$ be objects of a homogeneous category $(\mathcal{C}, \oplus, 0)$. For $n \ge 1$, let $S_n(A, X)$ denote the simplicial complex whose vertices are the maps $f: X \to A \oplus X^{\oplus n}$ and whose $p$-simplices are $(p+1)$-sets ${f_0, \dots, f_p}$ such that there exists a morphism $f: X^{\oplus p+1} \to A \oplus X^{\oplus n}$ with $f \circ i_j = f_j$ for some order on the set, where
Definition 4.5. Let $\mathrm{Aut}(A \oplus X^{\oplus\infty})$ be the colimit of
Then any $\mathrm{Aut}(A \oplus X^{\oplus\infty})$-module $M$ may be considered as an $\mathrm{Aut}(A \oplus X^{\oplus n})$-module for any $n$, by restriction, which we continue to call $M$. We say that the module $M$ is abelian if the action of $\mathrm{Aut}(A \oplus X^{\oplus\infty})$ on $M$ factors through the abelianizations of $\mathrm{Aut}(A \oplus X^{\oplus\infty})$, or in other words if the derived subgroup of $\mathrm{Aut}(A \oplus X^{\oplus\infty})$ acts trivially on $M$.
We are now ready to quote the theorem that we will use.
Theorem 4.6 ([RWW17, Theorem 3.1]). Let $(\mathcal{C}, \oplus, 0)$ be a pre-braided homogeneous category satisfying LH3 for a pair $(A, X)$ with slope $k \ge 3$. Then for any abelian $\mathrm{Aut}(A \oplus X^{\oplus\infty})$-module $M$ the map
induced by the natural inclusion map is surjective if $i \le \frac{n-k+2}{k}$, and injective if $i \le \frac{n-k}{k}$.
4.2. Homogeneous category for the groups $RV_{d,r}^+$. The purpose of this section is to produce a homogeneous category for proving homological stability of the ribbon Higman–Thompson groups $RV_{d,r}^+$. Note that by Theorem 3.24, it is same as proving the asymptotic mapping class groups $BV_{d,r}$ have homological stability. This allows us to define our homogeneous category geometrically. The category is similar to the ones produced in [RWW17, Section 5.6]. Essentially, we replace the annulus or Möbius band by the infinite surface $D_{d,1}^\infty$.
Recall $D_{d,r}^\infty$ is an infinite surface equipped with a canonical asymptotic rigid structure. Let $I = [-1, 1] \subset \partial_b D_{d,r}^\infty$ be an embedded interval. Let $I^- = [-1, 0]$ and $I^+ = [0, 1]$ be subintervals of $I$. Let $D_{d,1}^\infty \oplus D_{d,1}^\infty$ be the boundary sum of two copies of $D_{d,1}^\infty$ obtained by identifying $I^+$ of the first copy with $I^-$ of the second copy. Inductively, we could define similarly $\oplus_r D_{d,1}^\infty$ for any $r \ge 0$. Here $\oplus_0 D_{d,1}^\infty$ is just the standard disk $D$. Abusing notation, when referring to $I^-$ and $I^+$ on $\oplus_r D_{d,1}^\infty$, we will mean the two copies of $I^-$ and $I^+$ which remain on the boundary. Thus we have an operation $\oplus$ on the set $\oplus_r D_{d,1}^\infty$ for any $r \ge 0$. See Figure 6(b) for a picture of $(\oplus_2 D_{d,1}^\infty) \oplus (\oplus_3 D_{d,1}^\infty)$. In fact, we have (($\oplus_r D_{d,1}^\infty$), $\oplus$) is the free monoid generated by $D_{d,1}^\infty$. Note