ScalingStacks

Remark 2.3.6.

In the \(\infty\)-categorical setting, we will replace \(h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) with a more natural target category with less restrictions on objects. Namely, the category \(h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) is a full symmetric monoidal subcategory of the category \(h_1\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\) whose objects are arbitrary flat graded algebras, and morphims are isomorphism classes of right-graded-perfect derived bimodules between them. By passing to module categories over these algebras, this can in turn be realized as a full subcategory of the category \(h_1 \mathrm{st}^{B\mathbb{Z}}_{k}\) of stable \(k\)-linear categories with a \(\mathbb{Z}\)-action and equivalence classes of \(k\)-linear exact \(\mathbb{Z}\)-equivariant functors between them. In the next sections, we will lift the composite functor \(h_1 \mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\rightarrow h_1 \mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\rightarrow h_1\mathrm{st}^{B\mathbb{Z}}_{k}\) to a functor of \((\infty,2)\)-categories.