ScalingStacks

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 \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\).