Let \(k\) be an ordinary commutative ring and \(Z\) a discrete commutative monoid. We define \[\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{Z})\subseteq \mathrm{Morita}^{\mathrm{cp}}(\mathrm{Mod}_{Hk}^{\geq 0, Z})\] to be the full \(\mathrm{add}_{Hk}^{BZ}\)-enriched subcategory on the discrete flat \(Z\)-graded \(k\)-algebras.
Similarly, we define \[\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{Z})\subseteq \mathrm{Morita}^{\mathrm{c}}(\mathrm{Mod}_{Hk}^{Z})\] to be the full \(\mathrm{st}_{Hk}^{BZ}\)-enriched subcategory on the discrete flat \(Z\)-graded \(k\)-algebras.