8.2 Statement of the main theorem[00H0]
By example 8.1.6, the space of compatible \(\mathbb E_2\)-structures on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) together with compatible \(\mathbb E_2\)-structures on its fiber functor \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) is precisely given by the space \[\mathrm{Braid}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}}}({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})),\] where \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) is seen as an \(\mathbb E_1\)-algebra in \(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}}\) via its \(\mathbb E_1\)-functor \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\).
Our main theorem will prove that this space is in fact a set, namely the set of prebraidings in the sense of §2.4 on a functor between certain ordinary monoidal \(1\)-categories. The key fact we use is that while the \((\infty,2)\)-‘fiber’-functor \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) is not faithful, the composite \(\mathrm{Sbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) is faithful by lemma 6.5.6, i.e. induces fully faithful functors on all hom-categories. Furthermore, \(\mathrm{Sbim}\) is generated by the \((2,2)\)-category \(\mathrm{BSbim}\) from definition 6.1.4, in the sense that the functor \(\mathrm{BSbim}\rightarrow\mathrm{Sbim}\) is surjective on objects and that any \(1\)-morphism in \(\mathrm{Sbim}\) is a retract of a finite coproduct of grading shifts of \(1\)-morphisms in the image of \(\mathrm{BSbim}\) as proven in proposition 6.3.2.
For future applications, we abstract this situation as follows:
[00H1]
Theorem 8.2.1.
Let \(\mathcal C\in \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\), \(\mathcal D\in \mathrm{Alg}_{\mathbb E_{\infty}}(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}])\) and let \(H \colon \mathcal C\rightarrow\mathcal D\) be a morphism in \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\) whose underlying \((\infty,2)\)-functor is faithful, i.e. induces fully faithful functors on hom-categories. Consider the monoidal functor \({\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \rightarrow\mathcal D\), induced by the adjunction ([00D4]), as an object of \(\mathrm{Alg}_{\mathbb E_1}\left(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathcal D}\right)\).
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.)
Further, assume there is a functor \(\iota \colon \mathcal B\rightarrow\mathcal C\) in \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{(\infty,2)})\) which is surjective on objects and such that for every two objects \(b,b' \in \mathcal B\), any object in \(\underline{\mathrm{Hom}}_{\mathcal C}(\iota b, \iota b') \in \mathrm{add}_{k}^{B\mathbb{Z}}\) is a retract of a finite coproduct of \(\mathbb{Z}\)-shifts of objects in the image of \(\underline{\mathrm{Hom}}_{\mathcal B}(b, b') \in \mathrm{Cat}_{(\infty,1)}\). Then, the pre-composition map \[\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal C\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) \rightarrow\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal B\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C))\] is an equivalence of spaces.
In other words, theorem 8.2.1 asserts that the space of pairs of an \(\mathbb E_2\)-structure on \({\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\) and an \(\mathbb E_2\)-structure on the functor \({\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \rightarrow\mathcal D\), compatible with their given \(\mathbb E_1\)-structures, is equivalent to the set of prebraidings on \(h_1\mathcal B\rightarrow h_1 {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\) over \(h_1\mathcal D\).
In the remainder of section 8, we prove theorem 8.2.1 by factoring ([00H3]) through various other spaces of prebraiding. Before discussing the proof, we immediately record how theorem 8.2.1 implies theorem B from the introduction:
[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 \(h_1\mathrm{BSbim}\rightarrow h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\). ◻