8.1 Spaces of braidings and prebraidings[00GP]
[00GQ]
Notation 8.1.1.
Let \(\mathcal V\) be a symmetric monoidal \(\infty\)-category, or more generally an \(\infty\)-operad.
For an \(\mathbb E_1\)-algebra \(A\) in \(\mathcal V\), we write \[\mathrm{Braid}_{\mathcal V}(A):= \mathrm{Hom}_{\mathrm{Op}}(\mathbb E_2, \mathcal V) \times_{\mathrm{Hom}_{\mathrm{Op}}(\mathbb E_1, \mathcal V)} \{A\}\] for the space of \(\mathbb E_2\)-algebra structures on \(A\) compatible with the given \(\mathbb E_1\)-structure and refer to this space as the space of braidings on \(A\).
For an \(\mathbb E_1\)-algebra \(A\) in \(\mathcal V\), we write \[\mathrm{PreBraid}_{\mathcal V}(A):= \mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal V)}(A\otimes A, A) \times_{\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal V)}(A, A)^{\times 2}}\{\mathrm{id}_A,\mathrm{id}_A\}\] and refer to this space as the space of prebraidings on \(A\).
For \(f \colon A \rightarrow B\) a morphism of \(\mathbb E_1\)-algebras in \(\mathcal V\), we write \[\mathrm{PreBraid}_{\mathcal V}(f) := \mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal V)}(A \otimes A, B) \times_{\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal V)}(A, B)^{\times 2}} \{ f, f\}\] and refer to this space as the space of prebraidings on \(f\).
For an \(\mathbb E_1\)-algebra \(A\), it follows by definition that \(\mathrm{PreBraid}_{\mathcal V}(A) = \mathrm{PreBraid}_{\mathcal V}(\mathrm{id}_A)\). Recall from corollary 7.2.6 that analogous to \(\mathrm{Braid}_{\mathcal V}\), the spaces of prebraidings are also corepresented by certain \(\infty\)-operads: \[\begin{aligned}
\mathrm{PreBraid}_{\mathcal V}(A) & = \mathrm{Hom}_{\mathrm{Op}}( \mathbb A_2 \otimes \mathbb E_1, \mathcal V) \times_{\mathrm{Hom}_{\mathrm{Op}}(\mathbb E_1, \mathcal V)} \{A\}\\
\mathrm{PreBraid}_{\mathcal V}(f:A \rightarrow B) &=\mathrm{Hom}_{\mathrm{Op}}(\mathbb T_2 \otimes \mathbb E_1, \mathcal V) \times_{\mathrm{Hom}_{\mathrm{Op}}([1] \otimes \mathbb E_1, \mathcal V)} \{f\}
\end{aligned}\] Moreover, for any \(\mathbb E_1\)-algebra \(A\), composing with the operad map \(\mathbb A_2 \otimes \mathbb E_1 \rightarrow\mathbb E_1 \otimes \mathbb E_1 \simeq \mathbb E_2\) from example 7.2.7 defines a ‘forgetful’ map of spaces \[
\mathrm{Braid}_{\mathcal V}(A) \rightarrow\mathrm{PreBraid}_{\mathcal V}(A).\]
[00GS]
Example 8.1.2.
For an ordinary monoidal \(1\)-category \(\mathcal A\in \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{({1}, {1})})\) it follows from corollary 7.7.8 that the map of spaces \[\mathrm{Braid}_{\mathrm{Cat}_{({1}, {1})}}(\mathcal A) \rightarrow\mathrm{PreBraid}_{\mathrm{Cat}_{({1}, {1})}}(\mathcal A).\] is an equivalence. By corollary 7.4.15([00FC]), these spaces are equivalent to the (discrete) set of classical braidings on \(\mathcal A\).
Similarly, it follows from corollary 7.4.15([00FB]) that for monoidal functors \(F \colon \mathcal A\rightarrow\mathcal B\) between ordinary monoidal \(1\)-categories, the spaces \(\mathrm{PreBraid}_{\mathrm{Cat}_{({1}, {1})}}(F)\) are equivalent to the (discrete) set of classical prebraidings on \(F\) in the sense of definition 2.4.8.
[00GT]
Warning 8.1.3.
example 8.1.2 is key to our paper, and is at the heart of an observation already encountered in remark 2.4.4: While braidings and prebraidings on ordinary monoidal \(1\)-categories coincide, the notions already diverge for monoidal \(2\)-categories; the map ([00GR]) is in general far from an equivalence.
As in §2.4, we would also like to consider spaces of prebraidings over a given fixed prebraiding.
The following generalizes definition 2.4.5 to the \(\infty\)-categorical setting.
[00GU]
Definition 8.1.4.
Let \(\mathcal V\) be a symmetric monoidal \(\infty\)-category, \(C\) an \(\mathbb E_2\)-algebra (or merely an \(\mathbb A_2 \otimes \mathbb E_1\)-algebra) and \(A\xrightarrow{f} B \xrightarrow{g}C\) be maps of \(\mathbb E_1\)-algebras. We define the space \(\mathrm{PreBraid}_{\mathcal V}(f)_{/C}\) of prebraidings on \(f\) over \(C\) to be the space \(\mathbb T^{\mathrm{Alg}_{\mathbb E_1}(\mathcal V)}_2(f)_{/C}\) from definition 7.3.1.
Unpacked, a prebraiding on \(f\) over \(C\) is therefore a prebraiding on \(f\) together with an identification of the induced prebraiding on \(g\circ f\) with the one induced by the \(\mathbb E_2\)-structure of \(C\).
[00GV]
Example 8.1.5.
It follows from example 8.1.2 that for an ordinary braided monoidal \(1\)-category \(\mathcal C\), and monoidal \(1\)-functors \(\mathcal A\xrightarrow{F} \mathcal B\xrightarrow{g} \mathcal C\) between ordinary monoidal \(1\)-category, the space \(\mathrm{PreBraid}_{\mathrm{Cat}_{({1}, {1})}}(F)_{/C}\) of prebraidings on \(F\) over \(C\) in the sense of definition 8.1.4 agrees with the set \(\mathrm{PreBraid}_{/C}(F)\) of prebraidings over \(C\) from definition 2.4.8.
Recall from §A.8.6 that for an \(\mathbb E_{\infty}\)-algebra \(C\) in a symmetric monoidal \(\infty\)-category \(\mathcal V\), the over-\(\infty\)-category \(\mathcal V_{/C}\) inherits a symmetric monoidal structure so that for any \(\infty\)-operad \(\mathcal O\), there is an equivalence of \(\infty\)-categories \(\underline{\mathrm{Alg}}_{\mathcal O}(\mathcal V_{/C}) \simeq \underline{\mathrm{Alg}}_{\mathcal O}(\mathcal V)_{/C}\).
[00GW]
Example 8.1.6.
It immediately follows from the defining property of \(\mathcal V_{/C}\), that for a given \(\mathbb E_1\)-algebra in \(\mathcal V_{/C}\), i.e. an \(\mathbb E_1\)-algebra \(A\) equipped with an \(\mathbb E_1\)-algebra map \(A\rightarrow C\), the space \(\mathrm{Braid}_{\mathcal V_{/C}}(A)\) encodes a compatible \(\mathbb E_2\)-structure on \(A\) together with \(\mathbb E_2\)-structure on the \(\mathbb E_1\)-morphism \(A\rightarrow C\).
Following example 8.1.6, to prove theorem B and to study \(\mathbb E_2\)-structures on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) together with \(\mathbb E_2\)-structures on its fiber functor \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\), we will therefore need to study the space \(\mathrm{Braid}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}}}\left({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \right)\) and relate it to certain spaces of prebraidings in certain over-categories. These spaces can be understood in terms of prebraidings over given prebraidings in the sense of definition 8.1.4:
[00GX]
Corollary 8.1.7.
Let \(C\) be an \(\mathbb E_{\infty}\)-algebra in a symmetric monoidal \(\infty\)-category \(\mathcal V\), and let \(F\) be a morphism of \(\mathbb E_1\)-algebras in \(\mathcal V_{/C}\), i.e. equivalently a commuting diagram 
of \(\mathbb E_1\)-algebras in \(\mathcal V\). Then, the spaces \[\mathrm{PreBraid}_{\mathcal V_{/C}}(F) \simeq \mathrm{PreBraid}_{\mathcal V}(F)_{/C}\] are equivalent.
[00GY]
Proof.
Apply proposition 7.3.2 to the symmetric monoidal \(\infty\)-category \(\mathrm{Alg}_{\mathbb E_1}(\mathcal V_{/C}) \simeq \mathrm{Alg}_{\mathbb E_1}(\mathcal V)_{/C}\). ◻
[00GZ]
Example 8.1.8.
Combining corollary 8.1.7 with example 8.1.5 we find that in the setup of example 8.1.5, the space \(\mathrm{PreBraid}_{(\mathrm{Cat}_{(1,1)})_{/C}}(F)\) agrees with the set \(\mathrm{PreBraid}_{/C}(F)\) of prebraidings over \(C\) from definition 2.4.8.