By construction, the isomorphism classes of objects of \(\mathrm{BSbim}\) are in bijection with the natural numbers \(n\in \mathbb{N}_0\). For every \(n \in \mathbb{N}_0\) we now fix a representing object in \(\mathrm{BSbim}\) in that isomorphism class and simply denote it by \(n\). For \(n,m \in \mathrm{BSbim}\), the hom-category in \(\mathrm{BSbim}\) is given by \[\underline{\mathrm{Hom}}_{\mathrm{BSbim}}(n,m)= \left\{ \begin{array}{lr} 0 & n \neq m\\ \mathrm{BSBim}_n & n=m \end{array} \right. ,\] where \(\mathrm{BSbim}_n\) is the category from definition 2.1.2. The monoidal functor \(\mathrm{BSbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) sends objects \(n\) to the polynomial algebra \(R_n\) and is given on hom-categories by the evident full inclusion of \(\mathrm{BSBim}_n\) into the category of graded \(R_n\)–\(R_n\)-bimodules that are graded-compact-projective as right \(R_n\)-modules.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2