ScalingStacks

Now we turn to defining \(\mathrm{slide}_{B,\mathbf{1}_n}\) and \(\mathrm{slide}_{\mathbf{1}_m,B}\), where \(B\) is one of the generating Bott–Samelson bimodules. Here we place subscripts to distinguish the identity bimodules. We first consider the latter situation and reduce it to the case \(m=1\), where the cabled crossing is a Coxeter braid. Indeed, suppose that \(m>1\), then we use the first equality from Lemma 2.2.8 to define \(\mathrm{slide}_{\mathbf{1}_m,B}\) to be the composite: \[\begin{gathered} \nonumber \big((\mathrm{slide}_{\mathbf{1}_1,B}\boxtimes \mathrm{id}_{\mathbf{1}_{m-1}}) \circ_1\cdots \circ_1 \mathrm{id}_{\mathbf{1}_{m-1-i} \boxtimes X_{1,n}\boxtimes\mathbf{1}_{i}} \circ_1\cdots \circ_1 \mathrm{id}_{\mathbf{1}_{m-1}\boxtimes X_{1,n}}\big) \circ_2\cdots \\ \circ_2\big(\mathrm{id}_{X_{1,n}\boxtimes \mathbf{1}_{m-1}} \circ_1\cdots \circ_1 (\mathrm{id}_{\mathbf{1}_{m-1-i}} \boxtimes \mathrm{slide}_{\mathbf{1}_1,B}\boxtimes\mathrm{id}_{\mathbf{1}_{i}}) \circ_1\cdots \circ_1 \mathrm{id}_{\mathbf{1}_{m-1}\boxtimes X_{1,n}}\big) \circ_2\cdots\\ \nonumber \circ_2\big(\mathrm{id}_{X_{1,n}\boxtimes \mathbf{1}_{m-1}} \circ_1\cdots \circ_1 \mathrm{id}_{\mathbf{1}_{m-1-i} \boxtimes X_{1,n}\boxtimes\mathbf{1}_{i}} \circ_1\cdots \circ_1 (\mathrm{id}_{\mathbf{1}_{m-1}}\boxtimes \mathrm{slide}_{\mathbf{1}_1,B}) \big) \end{gathered}\] For the other case, we first choose chain maps \(\varphi\) and \(\varphi^{-1}\) realising the first homotopy equivalence in Lemma 2.2.8. Then we define \(\mathrm{slide}_{B,\mathbf{1}_n}\) as the composition: \[\begin{gathered} \nonumber \varphi^{-1} \circ_2\big((\mathbf{1}_{n-1}\boxtimes \mathrm{slide}_{B,\mathbf{1}_1}) \circ_1\cdots \circ_1 \mathrm{id}_{\mathbf{1}_{i} \boxtimes X_{m,1}\boxtimes\mathbf{1}_{n-1-i}} \circ_1\cdots \circ_1 \mathrm{id}_{X_{m,1}\boxtimes \mathbf{1}_{n-1}}\big) \circ_2\cdots \\ \circ_2\big(\mathrm{id}_{\mathbf{1}_{n-1}\boxtimes X_{m,1}} \circ_1\cdots \circ_1 (\mathrm{id}_{\mathbf{1}_{i}} \boxtimes \mathrm{slide}_{B,\mathbf{1}_1}\boxtimes\mathrm{id}_{\mathbf{1}_{n-1-i}}) \circ_1\cdots \circ_1 \mathrm{id}_{X_{m,1}\boxtimes \mathbf{1}_{n-1}}\big) \circ_2\cdots \\ \nonumber \circ_2\big(\mathrm{id}_{\mathbf{1}_{n-1}\boxtimes X_{m,1}} \circ_1\cdots \circ_1 \mathrm{id}_{\mathbf{1}_{i} \boxtimes X_{m,1}\boxtimes\mathbf{1}_{n-1-i}} \circ_1\cdots \circ_1 (\mathrm{slide}_{B,\mathbf{1}_1}\boxtimes \mathbf{1}_{n-1}) \big)\circ_2\varphi \end{gathered}\] It remains to construct \(\mathrm{slide}_{\mathbf{1}_1,B_i}\) and \(\mathrm{slide}_{B_j,\mathbf{1}_1}\) where \(B_i\) is a generating object of \(\mathrm{BSbim}_n\) and \(B_j\) is a generating object of \(\mathrm{BSbim}_m\). Now we reduce this problem to the cases when \(n=2\) and \(m=2\) respectively. We define \(\mathrm{slide}_{\mathbf{1}_1,B_i}\) as the composite: \[\begin{aligned} F(\sigma_{n}\cdots\sigma_{1}) \circ_1B_i =& F(\sigma_{n}\cdots\sigma_{i+1})\circ_1F(\sigma_{i}\sigma_{i-1})\circ_1F(\sigma_{i-2}\cdots\sigma_{1})\circ_1B_i \\ \rightarrow &F(\sigma_{n}\cdots\sigma_{i+1})\circ_1F(\sigma_{i}\sigma_{i-1}) \circ_1B_i \circ_1F(\sigma_{i-2}\cdots\sigma_{1}) \\ \xrightarrow{\mathrm{slide}} &F(\sigma_{n}\cdots\sigma_{i+1})\circ_1B_{i-1} \circ_1F(\sigma_{i}\sigma_{i-1}) \circ_1F(\sigma_{i-2}\cdots\sigma_{1}) \\ \rightarrow& B_{i-1} \circ_1F(\sigma_{n}\cdots\sigma_{i+1})\circ_1F(\sigma_{i}\sigma_{i-1}) \circ_1F(\sigma_{i-2}\cdots\sigma_{1})\\ =& B_{i-1} \circ_1F(\sigma_{n}\cdots\sigma_{1}) \end{aligned}\] where the unlabelled maps are far-commutativity isomorphisms and the labelled arrow is given by \(\mathrm{id}\circ_1\mathrm{slide}_{\mathbf{1}_1,B_1} \circ_1\mathrm{id}\), which is determined by the \(n=2\) case. The reduction of \(\mathrm{slide}_{B_j,\mathbf{1}_1}\) to the case \(m=2\) is completely analogous.