[00H5]
Corollary 8.2.2.
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.
In particular, the space of pairs of an \(\mathbb E_2\)-algebra structure on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\) together with an \(\mathbb E_2\)-algebra structure on the functor \(H_{\mathrm{loc}}\colon{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\), which enhance their monoidal structures, and satisfy the condition that the positive braiding \[\sigma_{1,1} \colon 1 \otimes 1\rightarrow 1 \otimes 1 \in \underline{\mathrm{Hom}}_{{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})}(1 \otimes 1,1\otimes 1) = {\mathbf K}^b(\mathrm{Sbim}_2)\] agrees up to chain homotopy with the shifted Rouquier complex \(X_{1,1}= F(\sigma_{1,1})\langle -1\rangle\) from definition 2.2.7 and ([000Z]), is contractible.
[00H8]
Proof.
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.
The second half of corollary 8.2.2 follows directly from the first: By corollary 2.5.3, the condition on the positive braiding \(\sigma_{1,1}\) fixes an element of the set ([00H7]) and hence a point in the space ([00H6]). Thus, there is a contractible space of braidings on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) over \(\mathrm{st}^{B\mathbb{Z}}_{k}\) compatible with the given prebraiding on \(h_1\mathrm{BSbim}\rightarrow h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\). ◻