Corollary 2.5.3. (Relative prebraiding on \(h_1 K_{\mathrm{loc}}\)).
The Rouquier complexes of cabled crossings define a prebraiding on \(h_1 K_{\mathrm{loc}}\colon h_1 \mathrm{BSbim}\rightarrow h_1 \mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) over \(h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\).
Proof.
By Remark 2.2.6, the braiding complexes are quasi-isomorphic to the associated permutation bimodules, concentrated in homological degree \(0\). As the permutation bimodules implement the symmetric braiding, this means that the braiding complexes constructed as (shifted) Rouquier complexes of the cabled crossings become the canonical symmetric braiding in \(h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\). ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2