ScalingStacks

By a graded \(k\)-linear category¹¹ we mean a category enriched in \(\mathbb{Z}\)-graded \(k\)-modules. As an example we can take as objects graded \(R_n\)-bimodules with all \(R_n\)-bimodule maps, denoted \(\mathrm{Hom}^{\mathrm{gr}}\). In this case, the grading shift functors are compatible with the grading on morphisms as follows: \[ \mathrm{Hom}^{\mathrm{gr}}(M\langle k\rangle,N\langle l\rangle) = \mathrm{Hom}^{\mathrm{gr}}(M,N)\langle l-k\rangle\]