There exists a unique braided monoidal (i.e., \(\mathbb E_2\)-algebra) structure on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\) that enhances its monoidal structure and satisfies the following conditions.
The fiber functor \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) is braided monoidal.
The braiding \(1 \otimes 1 \xrightarrow{\sim} 1 \otimes 1\) in \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) admits an equivalence with the Rouquier complex \(F(\sigma) \in \mathrm{End}_{{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})}(2) \coloneqq {\mathbf K}^b(\mathrm{Sbim}_2)\) corresponding to the braid group generator \(\sigma \in \operatorname{Br}_2\).