We warn the reader that one should not think of \(\mathrm{BSbim}\) and \(\mathrm{Sbim}\) as sub-\((2,2)\)-categories of \(\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\). Indeed, \(\mathrm{BSbim}\) and \(\mathrm{Sbim}\) do not contain all \(1\)-equivalences between their objects that exists in \(\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\). Hence, the faithful functors \(\mathrm{BSbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) and \(\mathrm{Sbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) are not monomorphisms in \(\mathrm{Cat}_{(\infty, {2})}\); see also warning 5.3.9. In particular, for an \((\infty,2)\)-functor \(\mathcal X\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\), it is not merely a property to factor through \(\mathrm{BSbim}\) or \(\mathrm{Sbim}\) but additional data.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2