We let \(h_1\mathrm{BSbim}\) denote the \(1\)-category, whose set of object is \(\mathbb{N}_0\) and whose morphism sets between objects \(n, m\) are \[\mathrm{Hom}_{h_1\mathrm{BSbim}}(n, m) = \left\{ \begin{array}{ll} h_0\mathrm{BSbim}_n & \quad n = m \\ \{0\} & \quad n \neq m \end{array} \right.\] where \(h_0 \mathrm{BSbim}_n\) denotes the set of isomorphism classes of objects in \(\mathrm{BSbim}_n\), and whose composition of morphisms \(n \rightarrow n\) is induced by the monoidal structure \(\otimes_{R_n}\) of \(\mathrm{BSbim}_n\). The category \(h_1\mathrm{BSbim}\) admits a monoidal structure with monoidal product \(\boxtimes\colon h_1\mathrm{BSbim}\times h_1\mathrm{BSbim}\rightarrow h_1\mathrm{BSbim}\) defined on objects by \(n \boxtimes m = n+m\) and on morphisms using parabolic induction: \[h_0 \boxtimes \colon h_0\mathrm{BSbim}_n \times h_0\mathrm{BSbim}_m \rightarrow h_0\mathrm{BSbim}_{n+m}.\]
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2