ScalingStacks

The space of braidings \[ \mathrm{Braid}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}}}\left({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\right)\] is equivalent to the set of prebraidings \[ \mathrm{PreBraid}_{/h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})}(h_1\mathrm{BSbim}\rightarrow h_1 \mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim}))\] over \(h_1H_{\mathrm{loc}}\colon h_1 \mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow h_1 \mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) as defined in definition 2.4.8.