The hom-categories \({}_A\mathrm{grbmod}_B^{\mathrm{gr-cp}}\) of \(\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{Z})\) are ordinary \(1\)-categories, hence \(\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{Z})\) is a (2,2)-category. On the other hand, \(\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{Z})\) is a genuine \((\infty,2)\)-category with non-trivial higher morphisms.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2