ScalingStacks

Given categories and functors as in part ([00H4]) of theorem 8.2.1, we will prove that the composite \[\begin{aligned} \mathrm{Braid}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathcal D}}({\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) &\rightarrow\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal C\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) \\ \nonumber &\rightarrow\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal B\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) \end{aligned}\] is an equivalence. Note that part ([00H2]) of theorem 8.2.1 then follows by taking \(\mathcal B\rightarrow\mathcal C\) to be the identity \(\mathcal C\rightarrow\mathcal C\) (which clearly satisfies the required conditions). Then, the second statement follows since the first map and the composite in ([00I7]) are equivalences, and hence so is the second map.