ScalingStacks

Then, the map of spaces (constructed more formally in the proof below) \[ \mathrm{Braid}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathcal D}}({\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) \rightarrow\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal C\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)),\] which restricts a braiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\) to a prebraiding on the subcategory inclusion \(\mathcal C\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\) and then passes to the homotopy \(1\)-category \(h_1\), is an equivalence. (Here, we leave the evident maps to \(\mathcal D\) and \(h_1\mathcal D\) implicit.)