Intuitively, \(\mathrm{Sbim}\) is the smallest locally \(k\)-linear additive and idempotent-complete ‘sub’-\((\infty,2)\)-category (mind warning 6.0.4) of \(\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) that is closed under the homwise \(\mathbb{Z}\)-action and contains \(\mathrm{BSbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\). Indeed, it follows from the definition that \(\mathrm{Sbim}\) together with its faithful monoidal \(\mathrm{add}_{k}^{B\mathbb{Z}}\)-enriched functor \(\mathrm{Sbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) is initial amongst factorizations of the monoidal \((\infty,2)\)-functor \(\mathrm{BSbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) through faithful monoidal \(\mathrm{add}_{k}^{B\mathbb{Z}}\)-enriched functors. More formally, it is the initial object of the pullback of the following span of \(\infty\)-categories: 
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2