Most of our constructions in this section are built on the symmetric monoidal \(k\)-linear \((2,2)\)-category with local shifts \[\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{\mathbb{Z}})\in \mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathrm{add}_{k}^{B\mathbb{Z}}])\] from definition 4.5.7. By corollary 4.5.8.([0093]), its objects are \(\mathbb{Z}\)-graded flat \(k\)-algebras, and its \(k\)-linear additive, idempotent-complete hom-categories between two such algebras \(A\) and \(B\) is given by the full subcategory of the \(1\)-category \({}_{A}\mathrm{grbmod}_{B}\) of (ordinary) \(\mathbb{Z}\)-graded \(A\)–\(B\)-bimodules on those bimodules which are graded-compact-projective (definition 3.6.8), as a right \(B\)-module. The homwise \(\mathbb{Z}\)-action is given by shifting the grading degree of these bimodules.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2