Given a graded algebra \(A\), we call an object of the derived category \(\mathrm{D}(\mathrm{grmod}_A)\) of graded \(A\)-modules graded-perfect if it is quasi-isomorphic to a finite chain complex of finitely generated graded-projective \(A\)-modules (see reference [000S]).
Let \(h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) be the symmetric monoidal \(1\)-category whose objects are the graded algebras \(R_n= k[x_1, \ldots, x_n]\) for \(n \in \mathbb{N}_0\) and whose morphism sets between algebras \(R_n\) and \(R_m\) are given by the set \[h_0 \mathrm{D}\left( {}_{R_n} \mathrm{grbmod}_{R_m}\right)^{\mathrm{gr-perf}}\] of isomorphism classes of objects in the derived category of graded \(R_n\)–\(R_m\) bimodules which are graded-perfect as right (i.e. \(R_m\)-)modules; composition is the derived graded tensor product over the respective polynomial algebras. Similar to definition 2.1.5, the monoidal structure is given by the derived graded tensor product \(\otimes_k^L\) over the ground ring \(k\), under the identification \(R_n \otimes^{L}_k R_m \simeq R_n \otimes_k R_m \simeq R_{n+m}\).