ScalingStacks

Theorem 2.5.4.

For any Bott–Samelson bimodule of the form \(Y=Y_1\boxtimes Y_2\) in \(\mathrm{BSbim}_m\boxtimes \mathrm{BSbim}_n\subset \mathrm{Sbim}_{m+n}\), there are homotopy equivalences of chain complexes in \(\mathrm{Ch}^b(\mathrm{Sbim}_{m+n})\): \[\begin{aligned} \mathrm{slide}_{Y_1,Y_2}\colon & X_{m,n} \circ_1Y \longrightarrow \mathrm{swap}_{m,n}(Y)\circ_1X_{m,n} \end{aligned}\]