1.2 Monoidality theorem[0003]
Our first result is an upgrade of \(\mathcal H\) to a monoidal \((\infty,2)\)-category whose hom-\(\infty\)-categories are \(k\)-linear, stable, idempotent-complete, and equipped with a \(\mathbb{Z}\)-action given by the grading shift action on \({\mathbf K}^b(\mathrm{Sbim}_n)\).
To formulate the result, we let \(\mathrm{st}_k\) denote the \(\infty\)-category of small stable idempotent-complete \(k\)-linear \(\infty\)-categories. Let \(\mathrm{st}^{B\mathbb{Z}}_{k}\) be the \(\infty\)-category \(\mathrm{Fun}(B\mathbb{Z}, \mathrm{st}_k)\) of such \(\infty\)-categories equipped with an additional compatible \(\mathbb{Z}\)-action. Then, Day convolution induces a symmetric monoidal structure on \(\mathrm{st}^{B\mathbb{Z}}_{k}\), and in turn on the \(\infty\)-category \(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\) of small \(\infty\)-categories enriched in \(\mathrm{st}^{B\mathbb{Z}}_{k}\) in the sense of Gepner-Haugseng [GH15].
[0004]
Theorem A. (proposition 6.4.2).
There is a monoidal \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) with objects labelled by natural numbers \(n \in \mathbb{N}_0\) and whose endomorphim \(\infty\)-categories are the \(k\)-linear, stable, idempotent-complete \(\infty\)-categories \({\mathbf K}^b(\mathrm{Sbim}_n)\) of chain complexes of Soergel bimodules, with a \(\mathbb{Z}\)-action by grading shift. More precisely, \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) defines an \(\mathbb E_1\)-algebra in the symmetric monoidal \(\infty\)-category \(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\).
The \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) from theorem A is constructed in section 6 via certain enriched variants of Morita \((\infty,2)\)-categories developed in section 4 following [Lur17], see also [Hau17, JS17].