ScalingStacks

Recall that the functor \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) factors by definition through the small full subcategory \(\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\hookrightarrow \mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\hookrightarrow \mathrm{st}^{B\mathbb{Z}}_{k}\). Hence, the space ([00H6]) is equivalent to the space \[ \mathrm{Braid}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})}}\left({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\right).\] We now invoke theorem 8.2.1 for \(\mathcal C= \mathrm{Sbim}\), \(\mathcal D= \mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\), and \(\mathcal B=\mathrm{BSbim}\), the functor \(H_{\mathrm{loc}}\colon \mathrm{Sbim}\rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) from proposition 6.5.2 which is faitfhul by lemma 6.5.6, and the functor \(\iota\colon \mathrm{BSbim}\rightarrow\mathrm{Sbim}\) from ([00CU]) which satisfies the relevant conditions of theorem 8.2.1 by proposition 6.3.2. Thus, the space of braidings ([00H9]) is equivalent to the space of prebraidings \[\mathrm{PreBraid}_{(\mathrm{Cat}_{(1,1)})_{/h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})}}(h_1 \mathrm{BSbim}\rightarrow h_1 {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})).\] Both, \(h_1 \mathrm{BSbim}\rightarrow h_1 {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) and \(h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\), agree with the respective functors from section 2, namely with ([001D]) by corollary 6.4.3 and with  ([001H]) by  corollary 6.5.4 respectively. Hence, it follows from example 8.1.8 that this space ([00HA]) is equivalent to the set \(\mathrm{PreBraid}_{/h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})}\left(h_1 \mathrm{BSbim}\rightarrow h_1 \mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\right)\) from theorem 2.6.4.