ScalingStacks

1.4 Context and proof outline for the braiding theorem[0009]

Here we further contextualize Theorem B while discussing key aspects of its setup and proof.

1.4.1 Braided monoidal 2-categories and their generalizations[000A]

The problem of finding the right definition of braided monoidal (even ordinary) \(2\)-categories has a long history, going back to [KV94, KV94, BN96], see [Sch09] for a survey. One motivation [BD95] was the construction of surface invariants in \(4\)-manifolds, in particular \(2\)-knots, just like braided monoidal \(1\)-categories are related to link invariants. A challenge hereby was the specification of the extra data required for a braiding on a \(2\)-category, for example with respect to the naturality of the braiding—which is not a property, but extra structure.

An even larger challenge is the construction of interesting concrete examples of braided monoidal \(2\)-categories, at least as rich as the theory of braided monoidal categories built from quantum groups. First hints of such a landscape came into view with the invention of Khovanov homology and its various associated invariants of tangles and tangle cobordisms [Kho06]. However, the idea of extracting a braided monoidal \(2\)-category from these invariants, see e.g. [BL11], turned out to be hard. Indeed, this idea drove the study of functoriality properties of categorified link and tangle invariants with respect to tangle cobordisms. At the state of the art, the most well-behaved link homology theories assign homotopy classes of chain maps to isotopy classes of tangle cobordisms. As discussed in [MWW22, § 6], this is sufficient to satisfy the (classical) axioms for a braided monoidal \(2\)-category in a toy model that is combinatorial and discrete up to the level of \(1\)-morphisms and truncated at the level of \(2\)-morphisms.

It is however desirable to go beyond such a toy model, so that the 2-hom-objects are no longer just sets or vector spaces, but are of a more homological nature: higher homotopies give invariants of higher isotopies between braids and links. Indeed, systematically incorporating such higher homotopies in link homology theories is also highly relevant beyond the goal of constructing an enhancement of a braided monoidal \(2\)-category, e.g. towards skein algebra categorification [QW21, Discussion after Conjecture 1.8], cabling operations [GHW22, § 1.3], as well as wrapping and flatting functors [GW23, § 6.4], [Eli18, MMV24].

To formulate answers to such questions, the classical axioms for braided monoidal \(2\)-categories are no longer sufficient: For example the Rouquier complexes corresponding to braid group generators satisfy the braid relation up to homotopy equivalence, [Rou06]. These equivalences must be provided as additional data, which then should be subject to further coherence conditions, requiring higher homotopies ad infinitum. Even worse, the compatibilities to be checked at each level quickly explode—in both complexity and in number,—and hence become unfeasible beyond the first two well-known stages of Reidemeister moves and Carter–Saito movie moves.

To be able to address these problems, we find it essential to work within the homotopy-theoretic context of \((\infty,2)\)-categories, as explained further in the following.

The modern formalism of higher algebra, as in [Lur17], offers a robust answer to the problem of modelling braided monoidal structures in such a homotopical context: Namely in terms of the notion of an \(\mathbb E_2\)-algebra [BV73]. This notion applies in any symmetric monoidal \(\infty\)-category \(\mathcal V\), and in the \((2,1)\)-category of ordinary categories recovers the notion of a braided monoidal category by a folklore result that we recover in remark 7.7.7. Namely, an \(\mathbb E_2\)-algebra structure on an object \(V \in \mathcal V\) consists of a suitably compatible system of maps \(\mathrm{Conf}_n(\mathbb{R}^2) \rightarrow\mathrm{Hom}_\mathcal V(V^{\otimes n},V)\) from configuration spaces of points in \(\mathbb{R}^2\). For example, a chosen basepoint of \(\mathrm{Conf}_2(\mathbb{R}^2)\) selects a multiplication map \(V \otimes V \xrightarrow{\mu} V\), and the generator of \(\pi_1(\mathrm{Conf}_2(\mathbb{R}^2)) \simeq \mathbb{Z}\) selects a braiding isomorphism \(\mu \xrightarrow{\sim} \mu \circ \tau\) (where \(\tau\) denotes the symmetry isomorphism in \(\mathcal V\)). More generally, the requisite compatibilites as \(n\) varies collectively encode the homotopy coherent associativity of \(\mu\) as well as its compatibility with the braiding.

To apply this formalism of \(\mathbb E_2\)-algebras to describe a braiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\), we use \(\mathcal V= \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\). Spelled out, \(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\) is the \(\infty\)-category of \(\mathrm{st}^{B\mathbb{Z}}_{k}\)-enriched \(\infty\)-categories, i.e. \((\infty,2)\)-categories whose hom-\((\infty,1)\)-categories are stable, idempotent complete, and equipped with a \(k\)-linear structure and an action by \(\mathbb{Z}\), all appropriately compatible with the composition operations. Indeed, a \(\mathrm{st}^{B\mathbb{Z}}_{k}\)-enriched \(\infty\)-category \(\mathcal C\in \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\) has composition morphisms \(\mathrm{Hom}_\mathcal C(c_0,c_1) \otimes \mathrm{Hom}_\mathcal C(c_1,c_2) \rightarrow\mathrm{Hom}_\mathcal C(c_0,c_2)\) in \(\mathrm{st}^{B\mathbb{Z}}_{k}\), where \(\otimes\) denotes the Day convolution symmetric monoidal structure on \(\mathrm{st}^{B\mathbb{Z}}_{k}\), rather than merely functors \(\mathrm{Hom}_\mathcal C(c_0,c_1) \times \mathrm{Hom}_\mathcal C(c_1,c_2) \rightarrow\mathrm{Hom}_\mathcal C(c_0,c_2)\): The tensor product in \(\mathrm{st}^{B\mathbb{Z}}_{k}\) implicitly enforces our desired compatibility.

theorem A and Theorem B construct \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) as an \(\mathbb E_2\)-algebra⁶ object in \(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\). In particular, its monoidal structure and braiding are automatically homotopy-coherently compatible with the \(k\)-linearity and \(\mathbb{Z}\)-actions on its hom-\((\infty,1)\)-categories.

To construct this \(\mathbb E_2\)-algebra structure, we take inspiration from the homotopy-theoretic machinery of obstruction theory, in which context one often finds it (somewhat paradoxically) easier to prove a stronger theorem. Specifically, we prove Theorem B by establishing not just the existence of an \(\mathbb E_2\)-algebra structure on the \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\), but also its homotopy-theoretic uniqueness along with its compatibility with \(k\)-linearity, \(\mathbb{Z}\)-action, and with the fiber functor \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\).

1.4.2 Higher-categorical notions of faithfulness and homotopy categories[000B]

To prove our main theorems, we develop machinery to reduce the construction of algebraic structures on \((\infty,2)\)-categories to corresponding structures on their underlying ordinary homotopy categories, as well as higher-dimensional variants. We outline some of these results, which might be of independent interest.

As a toy case, observe that given an \(\infty\)-category \(\mathcal C\), a full subcategory \(\mathcal C'\) is determined entirely by the subset \(h_0\mathcal C'\) of the set \(h_0\mathcal C\) of equivalences classes of objects in \(\mathcal C\) that are contained in \(\mathcal C'\). Furthermore, if \(\mathcal C\) is endowed with some multiplicative structure (e.g. monoidal, braided monoidal, or symmetric monoidal), then \(\mathcal C'\) inherits such a structure if and only if \(h_0\mathcal C'\) inherits the resulting structure from \(h_0\mathcal C\).

Given an \((\infty,2)\)-category \(\mathcal C\), we write \(h_1\mathcal C\) for the ordinary category obtained by first discarding its noninvertible 2-morphisms and then passing to the homotopy category, i.e. taking \(\pi_0\) of its hom-spaces. Moreover, we say that a functor between \((\infty,2)\)-categories is faithful if it is fully faithful on hom-\((\infty,1)\)-categories.⁷ Then, analogously to the above situation, we show that a faithful functor \(\mathcal C' \rightarrow\mathcal C\) is determined entirely by their corresponding faithful functor \(h_1\mathcal C' \rightarrow h_1\mathcal C\). Moreover, we show that if \(\mathcal C\) is braided monoidal equipped with the faithful functor \(\mathcal C' \rightarrow\mathcal C\), then endowing \(\mathcal C'\) together with a (compatible) braided monoidal structure is equivalent to endowing \(h_1\mathcal C'\) and \(h_1\mathcal C' \rightarrow h_1\mathcal C\) with such a structure, see corollary 5.5.5. The task of defining an \(\mathbb E_2\)-algebra structure on \(\mathcal C'\) therefore reduces in such a situation to the task of defining a braiding (in the classical sense!) on its homotopy 1-category \(h_1\mathcal C'\).

In fact, we prove the above results in much broader generality: In section 5, we study the notion of \(n\)-faithfulness for functors between \((\infty,k)\)-categories, and show in subsection 5.3 that they define the right classes in factorization systems on \(\mathrm{Cat}_{(\infty,k)}\), whose corresponding left classes are given by the \(n\)-surjective functors, which are surjective on objects, and on parallel morphisms up to level \((n+1)\), see definition 5.3.1. Using the iterative definition of higher categories, \(\mathrm{Cat}_{(\infty,k)} \coloneqq \mathrm{Cat}[\mathrm{Cat}_{(\infty,k-1)}]\), we deduce this from general results that we prove regarding factorization systems on enriched \(\infty\)-categories in subsection B.4, also see [Hau23] for related recent result. Generalizing the above, we establish in corollary 5.5.3 that for any \(n\) and any \(k\), \((n-1)\)-faithful functors to an \((\infty,k)\)-category \(\mathcal C\) are equivalently determined by \((n-1)\)-faithful functors to its homotopy \(n\)-category.

1.4.3 The fiber functor[000C]

The fiber functor \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) on which Theorem B is built, can be viewed as a categorified and graded analog of the forgetful functor from a category of quantum group representations to \(\mathrm{Vec}\). Since its target \(\mathrm{st}^{B\mathbb{Z}}_{k}\) is braided monoidal (in fact symmetric monoidal), it is tempting to apply the machinery of subsection 1.4.2 to obtain a braiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\).

However, this does not work because the analogy with the classical situation breaks down in an important way. In the classical setting, the fiber functor is faithful and monoidal, but in general not braided. By contrast, our categorified fiber functor will be braided monoidal but not faithful (in the sense of Subsection 1.4.2). This lack of faithfulness prevents us from directly using the reduction results from Subsection 1.4.2. In Subsection 1.4.4, we will discuss a restricted version of the fiber functor that is faithful, which allows us to leverage the results indicated in Subsection 1.4.2.

In the end, the non-faithfulness turns out to be an essential feature of our fiber functor. This feature is what allows \(H_{\mathrm{loc}}\) and its source category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) to be equipped with a non-symmetrically braided monoidal structure, even though the target category \(\mathrm{st}^{B\mathbb{Z}}_{k}\) is symmetric monoidal. A faithful braided fiber functor would force the source category to be symmetric!

In the classical situation, the existence of a fiber functor arises from the fact that quantum group representations are ultimately categories of modules for a quasitriangular Hopf algebra, which can be recovered from the fiber functor via Tannakian reconstruction. We would be very interested to see an application of Tannakian reconstruction to our categorified fiber functor.

We expect our fiber functor to be important for future applications and computations. For instance, the fact that it is braided may give a means of recursively computing all the algebraic data implicit in the braided monoidal structure on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) on a cell-by-cell basis.

1.4.4 The prebraiding[000D]

Another key ingredient of our proof of Theorem B is the following: \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) is generated, in an appropriate sense, by a monoidal sub-2-category, restricted to which the fiber functor to \(\mathrm{st}^{B\mathbb{Z}}_{k}\) is faithful.

We begin by considering the sub-2-category⁸ \(\mathrm{Sbim}\subset {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) described as follows: it contains all the same objects, but on hom-objects we pass to the subcategories \(\mathrm{Sbim}_n \subseteq {\mathbf K}^b(\mathrm{Sbim}_n)\) of Soergel bimodules (seen as complexes concentrated in degree 0). This is a monoidal sub-2-category: Soergel bimodules are closed under parabolic induction.

Consider now the restricted fiber functor, i.e. the composite \(\mathrm{Sbim}\hookrightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) of the inclusion followed by the fiber functor. Here, we arrive at an interesting tension. While the fiber functor \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) is not faithful but (will be) braided, this composite is faihtful but not braided: The braiding of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) does not restrict to one on \(\mathrm{Sbim}\). Indeed, Rouquier complexes are typically genuine complexes, not concentrated in degree 0.

However, we now make a key observation: the \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) is obtained by applying the left adjoint \({\mathbf K}^b\) to the hom-objects of the \((2,2)\)-category \(\mathrm{Sbim}\). In this sense, \(\mathrm{Sbim}\) generates \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\), which suggests that the braiding of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) might be uniquely determined by its values on \(\mathrm{Sbim}\). In fact, for each \(m\), there is a full additive monoidal subcategory \(\mathrm{BSbim}_m \subseteq \mathrm{Sbim}_m\) of Bott–Samelson bimodules ([000M]) which generates \(\mathrm{Sbim}_m\) under sums, retracts, and grading shifts. These Bott–Samelson bimodules assemble into a sub-2-category \(\mathrm{BSbim}\subset \mathrm{Sbim}\), which suggests that the braiding of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) might even be uniquely determined by its values on \(\mathrm{BSbim}\).

In order to explore this idea further, let us imagine constructing the braiding on the monoidal \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) of Theorem B by hand. This certainly requires, for every pair of objects \(m,n \in \mathbb{N}_0\), a braiding isomorphism \(m \otimes n \xrightarrow{\sim} n \otimes m\). Up to equivalence, this 1-morphism is already determined by condition ([0008]) of Theorem B: it must be given by the Rouquier complex for the positive \((m,n)\)-shuffle braid in \(\operatorname{Br}_{m+n}\) (see also figure [0016]). Of course, the braiding has to be natural: any complex \(C \in {\mathbf K}^b(\mathrm{Sbim}_m)\), seen as an endomorphism of \(m \in {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\), should “slide” along the \(m\) parallel strands through the braiding by means of a specified homotopy equivalence (see figure [0021]), and likewise for any \(C' \in {\mathbf K}^b(\mathrm{Sbim}_n)\). In fact, the discussion in the previous paragraph suggests that it suffices to specify these “slide” homotopy equivalences merely for objects \(C \in \mathrm{BSbim}_m \subseteq {\mathbf K}^b(\mathrm{Sbim}_m)\).

We formalize such a specification through the key notion of a prebraiding on the monoidal functor \(\mathrm{BSbim}\hookrightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\). We first describe such data in the simple setting of ordinary categories, i.e. in the \((2,1)\)-category \(\mathrm{Cat}\): a prebraiding on a monoidal functor \(F \colon \mathcal A\rightarrow\mathcal B\) between ordinary categories consists of equivalences \(F(x) \otimes F(y) \xrightarrow{\sim} F(y) \otimes F(x)\) that are natural in \(x,y \in \mathcal A\) and satisfy two appropriate analogs of the hexagon axioms for braidings; see definition 2.4.1. In particular, a prebraiding on the identity functor \(\mathrm{id}_\mathcal A\) is equivalent to a braiding on \(\mathcal A\). On the other hand, as we will see in subsection 1.4.5, in the \(\infty\)-categorical context, a prebraiding is a much sparser structure. Still, a prebraiding on a monoidal functor between \((\infty,2)\)-categories involves a substantial amount of coherence data.

Inspired by the results of Subsection 1.4.2, as a core input to our proof of Theorem B we construct a prebraiding on the ordinary monoidal functor \(h_1\mathrm{BSbim}\hookrightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) in subsection 2.5 whose prebraiding equivalence is given by the homotopy equivalence classes of the Rouquier complexes of positive \((m,n)\)-shuffle braids. This is based on explicit computations using the diagrammatic formulation of Soergel bimodules, [EW16, EK10]. Furthermore, this prebraiding is compatible with the fiber functor \(h_1H_{\mathrm{loc}}\colon h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\).

Finally, having constructed such a prebraiding at the level of homotopy categories, it remains to show that it lifts uniquely to a fully homotopy coherent braiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) and on its fiber functor \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\).

1.4.5 Prebraidings via \(\infty\)-operads[000E]

To lift our \(1\)-categorical prebraiding to braided monoidal structures on \((\infty,2)\)-categories, we first generalize prebraidings themselves to this \((\infty,2)\)-categorical setting.

We implement the notion of a prebraiding in the context of \(\infty\)-operads as indicated in the diagram \[\mathbb T_2 \otimes \mathbb E_1 \longrightarrow \mathbb A_2 \otimes \mathbb E_1 \longrightarrow \mathbb E_1 \otimes \mathbb E_1 \stackrel{\sim}{\longrightarrow} \mathbb E_2\] thereof, as we now explain. For more details see subsection A.8 and section 7.

• As indicated in Subsection 1.4.1, we formalize the notion of a braided monoidal structure via the notion of an \(\mathbb E_2\)-algebra (with \(k\)-ary operations parametrized by \(\mathrm{Conf}_k(\mathbb{R}^2)\)).

• Much simpler is the notion of an \(\mathbb E_1\)-algebra, which has \(k\)-ary operations parametrized by \(\mathrm{Conf}_k(\mathbb{R}^1)\). These spaces are discrete, and this \(\infty\)-operad is equivalent to the ordinary associative operad.

• The equivalence \(\mathbb E_1 \otimes \mathbb E_1 \xrightarrow{\sim} \mathbb E_2\) is an instance of Dunn additivity. Hence, an \(\mathbb E_2\)-algebra structure is equivalent to two compatible \(\mathbb E_1\)-algebra structures: \(\mathrm{Alg}_{\mathbb E_2}(\mathcal V) \simeq \mathrm{Alg}_{\mathbb E_1}(\mathrm{Alg}_{\mathbb E_1}(\mathcal V))\).

• Whereas \(\mathbb E_1\) parametrizes (homotopy-coherently) associative and unital binary operations, by forgetting associativity one arrives at the operad \(\mathbb A_2\), which parametrizes binary operations that are merely unital.

• The \(\infty\)-operad \(\mathbb T_2\) is a two-colored ordinary operad, and parametrizes pairs of pointed objects \(\mathcal A,\mathcal B\in \mathrm{Alg}_{\mathbb E_0}(\mathcal V)\)⁹ together with pointed morphisms \(\mathcal A\xrightarrow{F} \mathcal B\), and \(\mathcal A\otimes \mathcal A\xrightarrow{\mu} \mathcal B\), along with pointed homotopies \(\mu(\eta_\mathcal A\otimes (-)) \simeq F \simeq \mu((-) \otimes \eta_\mathcal A)\) in \(\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal V)}(\mathcal A,\mathcal B)\). In particular, we view a \(\mathbb T_2\)-algebra as having not an underlying object but having an underlying morphism, namely the morphism \(F\). The map of \(\infty\)-operads \(\mathbb T_2 \rightarrow\mathbb A_2\) corepresents the operation carrying an \(\mathbb A_2\)-algebra to the evident \(\mathbb T_2\)-algebra with \(\mathcal A= \mathcal B\) and \(F= \mathrm{id}_{\mathcal A}\).

We define a prebraiding on an \(\mathbb E_1\)-algebra morphism \(F\) in a symmetric monoidal \(\infty\)-category \(\mathcal V\) to be a \(\mathbb T_2\)-algebra structure on the correponding morphisms in \(\mathrm{Alg}_{\mathbb E_1}(\mathcal V)\), or equivalently a lift of the corresponding \([1] \otimes \mathbb E_1\)-algebra in \(\mathcal V\) to a \(\mathbb T_2 \otimes \mathbb E_1\)-algebra, see subsection 8.1. In the context of ordinary categories, we show in example 8.1.2 that this notion coincides with the one described in Subsection 1.4.4.

1.4.6 From prebraidings to braidings[000F]

In Subsection 1.4.4 we outlined the construction of a prebraiding on the monoidal functor \(h_1\mathrm{BSbim}\hookrightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) between ordinary monoidal \(1\)-categories, which is compatible with the fiber functor \(h_1 H_{\mathrm{loc}}: h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow h_1 \mathrm{st}^{B\mathbb{Z}}_{k}\). To complete the proof of Theorem B, it remains to show that this admits a unique lift to a braiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) such that the fiber functor is braided.

We prove this in steps, combining the machinery described above, as follows, see section 8.

Applying a version of the techniques described in Subsection 1.4.2, it follows that our prebraiding on \(h_1\mathrm{BSbim}\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) over \(h_1\mathrm{st}^{B\mathbb{Z}}_{k}\) lifts uniquely to a prebraiding on the \((\infty,2)\)-functor \(\mathrm{BSbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) over \(\mathrm{st}^{B\mathbb{Z}}_{k}\).

In order to proceed, we note a crucial property of prebraidings, see corollary 8.3.7: given an adjunction \(F \colon \mathcal C\rightleftarrows \mathcal D\colon G\) in which the left adjoint is symmetric monoidal, the data of a prebraiding on an \(\mathbb E_1\)-algebra morphism \(c \rightarrow G(d)\) is equivalent to the data of a prebraiding on its adjunct \(F(c) \rightarrow d\). We use this to extend our prebraiding over \(\mathrm{st}^{B\mathbb{Z}}_{k}\) from one on \(\mathrm{BSbim}\hookrightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) to one on \(\mathrm{Sbim}\hookrightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\), and then we use it again to extend the latter to one on the defining equivalence \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \xrightarrow{\sim} {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\).

So, we have obtained a prebraiding, i.e. a \(\mathbb T_2\)-structure on the identity morphism of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}})\). Because the identity morphism is invertible, this is equivalent to an \(\mathbb A_2 \otimes \mathbb E_1\)-structure on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}}\).

A final task is to lift this \(\mathbb A_2 \otimes \mathbb E_1\)-structure to an \(\mathbb E_2 \simeq \mathbb E_1 \otimes \mathbb E_1\) structure. Recall that prebraidings on identity functors between ordinary monoidal \(1\)-categories, i.e. \(\mathbb A_2\otimes \mathbb E_1\)-structures in \(\mathrm{Cat}_1\), are precisely the same as braidings, i.e. \(\mathbb E_2\)-structures. More generally, we show in subsection 7.7 that \(\mathbb A_2 \otimes \mathbb E_1\)- and \(\mathbb E_2\)-algebras agree in general \((2,1)\)-operads.

This observation applies to our situation due to a crucial truncatedness result regarding the endomorphism \(\infty\)-operad of the object \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}}\). Namely, while we expect it to be quite complicated in general, we prove in corollary 8.4.4 that the maximal sub-\(\infty\)-operad in the image of its \(\mathbb E_1\)-structure is in fact just a \((2,1)\)-operad. This suffices for our purposes since the map of \(\infty\)-operads \(\mathbb E_1 \rightarrow\mathbb E_2\) is surjective on path components of mapping spaces.

Thus, our prebraiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) over \(\mathrm{st}^{B\mathbb{Z}}_{k}\) extends uniquely to an \(\mathbb E_2\)-algebra structure establishing our main goal.

After now having introduced the key ideas and concepts, we finish this introduction by giving an outline of the organization of the paper.