ScalingStacks

Let \(h_1\mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) be the monoidal \(1\)-category whose objects are \(n \in \mathbb{N}_0\) and the morphisms between objects \(n, m\) are \[ \mathrm{Hom}_{h_1\mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})}(n, m) = \left\{ \begin{array}{ll} h_0\mathrm{K}^b(\mathrm{Sbim}_n) & \quad n = m \\ \{0\} & \quad n \neq m \end{array} \right.\] The monoidal product is induced by parabolic induction on chain complexes of Soergel bimodules and again we have \(\mathbb{Z}\)-actions on the morphism sets, inherited from grading shifts of bimodules. Since the inclusion \(\mathrm{BSbim}_n \rightarrow\mathrm{K}^b(\mathrm{Sbim}_n)\) of Bott–Samelson bimodules as chain complexes concentrated in homological degree zero is compatible with parabolic induction, this defines a monoidal functor \[ h_1 K_{\mathrm{loc}}\colon h_1\mathrm{BSbim}\rightarrow h_1\mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim}).\] In fact, this intertwines the \(\mathbb{Z}\)-actions on morphism sets.