ScalingStacks

1.1 Overview[0002]

Braided monoidal categories of representations, particularly quantum group representations, serve as a powerful framework for understanding invariants of knots, links, and tangles. Via skein theory [Wal06, BK01, MW12] or (relatedly) factorization homology [AF15, AFR18], such braided monoidal categories are also at the heart of 3- and 4-dimensional topological quantum field theories, such as those by Witten–Reshetikhin–Turaev [Wit89, RT91] and Crane–Yetter–Kauffman [CKY97].

Following the paradigm of [CF94], our work is motivated by the desire to construct higher-dimensional TQFTs by categorifying these theories. To this end, we will provide a construction of a “categorified braided monoidal category”, which we believe encompasses all the necessary data to form the basis for future constructions of 4- and 5-dimensional analogs, respectively, of Witten–Reshetikhin–Turaev and Crane–Yetter–Kauffman theories of Lie type A, see also [MWW22, Str23] and references therein.

We begin our discussion in the decategorified context. The Reshetikhin–Turaev link invariants [RT90] give an interpretation of the Jones polynomial [Jon85] in terms of morphisms in the braided monoidal category of representations of quantum \(\mathfrak{sl}_2\). In fact, all of the Reshetikhin–Turaev invariants of type A are controlled via Schur–Weyl duality by a single braided monoidal category \(H\), just as the \(\mathfrak{sl}_N\) link polynomials are specializations of the HOMFLYPT link invariant. Recall for \(n \in \mathbb N_0 \coloneqq \{ 0 , 1, 2, \ldots \}\) the Hecke algebra \(H_n\), which is a quotient of the group algebra over \(\mathbb{Z}[q^{\pm 1}]\) of the Artin braid group \(\operatorname{Br}_n\) for the symmetric group \(S_n\). The braided monoidal category \(H\) consists of the following data:

1. The objects are given by natural numbers \(n \in \mathbb N_0\).

2. The endomorphism algebra of each object \(n \in H\) is \(H_n\). All other hom-sets are trivial.

3. The monoidal structure is given on objects by addition, i.e. \(m \otimes n \coloneqq m + n\), and on morphism as the map \(H_m \times H_n \rightarrow H_{m+n}\) corresponding to the parabolic subgroup \(S_m\times S_n\hookrightarrow S_{m+n}\).

4. The braiding—a natural isomorphism \(m \otimes n \xrightarrow{\sim} n \otimes m\) for each \(m,n \in H\)—is given by the image in \(\mathrm{End}_H(m+n) \coloneqq H_{m+n}\) of the positive \((m,n)\)-shuffle braid in \(\operatorname{Br}_{m+n}\).

In the pioneering work [Kho00], Khovanov defined a homology theory for knots and links that categorifies the Jones polynomial, leading to a plethora of categorifications of many other polynomials invariants; see e.g. [Kho06] for a survey. Despite the diverse flavors and contexts of these constructions (e.g. involving perverse or coherent sheaves, matrix factorizations, symplectic geometry, Lie theory, and diagrammatic calculus), these categorifications are all—whether explicitly, implicitly, or a posteriori—shadows of a universal categorification involving Soergel bimodules [Soe92]; see [Str23]. Specifically, there is a monoidal additive category \(\mathrm{Sbim}_n\) of Soergel bimodules, which categorifies the Hecke algebra \(H_n\), i.e. whose split Grothendieck ring is \(H_n\). Concretely, \(\mathrm{Sbim}_n\) is a certain full additive subcategory of the category of graded bimodules of the polynomial algebra \(R_n \coloneqq k[x_1, \ldots, x_n]\) over a \(\mathbb{Q}\)-algebra \(k\) with each \(x_i\) in degree \(2\). The \(\mathbb{Z}\)-action by grading shift categorifies the \(\mathbb{Z}[q^{\pm 1}]\)-action on \(H_n\). The image of braids in \(H_n\) can only be categorified to objects in the (monoidal) bounded chain homotopy category \(\mathrm{K}^b(\mathrm{Sbim}_n)\) of \(\mathrm{Sbim}_n\), the so called Rouquier complexes [Rou06], see subsection 2.2.

The natural next step is to assemble these categorifications into a “categorified braided monoidal category”, in which categorified braid invariants can be seen as morphisms. Our main Theorems A and B yield this as a consequence. Namely, they provide a braided monoidal 2-category \(\mathcal H\)—more precisely, an \(\mathbb E_2\)-monoidal \((2,2)\)-category—which decategorifies to \(H\) upon taking Grothendieck groups.

1. The objects are given by natural numbers \(n \in \mathbb N_0\).

2. The endomorphism monoidal category of each object \(n \in \mathcal H\) is \(\mathrm{K}^b(\mathrm{Sbim}_n)\). All other hom-categories are trivial.

3. The monoidal structure is given on objects by addition, i.e. \(m \otimes n \coloneqq m + n\), and on morphism by a parabolic induction functor \(\mathrm{K}^b(\mathrm{Sbim}_m)\times \mathrm{K}^b(\mathrm{Sbim}_n) \rightarrow\mathrm{K}^b(\mathrm{Sbim}_{m+n})\).

4. The braiding isomorphism \(m \otimes n \xrightarrow{\sim} n \otimes m\) is given by the Rouquier complex in \(\mathrm{K}^b(\mathrm{Sbim}_{m+n})\) corresponding to the \((m,n)\)-shuffle braid in \(\operatorname{Br}_{m+n}\).

In fact, we go much further: we work in a homotopy-theoretic context based on higher algebra in the sense of Lurie [Lur09, Lur17] and construct a braided monoidal \((\infty,2)\)-category¹ \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\); a truncation of which recovers the braided monoidal 2-category \(\mathcal H\) described above. Explicitly, the hom triangulated categories \(\mathrm{K}^b(\mathrm{Sbim}_n)\) are replaced by stable \(\infty\)-categories \({\mathbf K}^b(\mathrm{Sbim}_n)\) of chain complexes, chain maps and chain homotopies with the full hierarchy of higher homotopies. In contrast to \(\mathrm{K}^b(\mathrm{Sbim}_n)\), using the stable \(\infty\)-categories \({\mathbf K}^b(\mathrm{Sbim}_n)\) has an essentially advantage, which we crucially use in the construction of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) and its braided monoidal structure. Namely, \({\mathbf K}^b\) has a universal property: It constructs the free idempotent-complete stable \(\infty\)-category on an idempotent-complete additive category, see subsection 3.4.

The braided monoidal \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) yields, in the spirit of Rouquier [Rou17], a suitable derived setting for defining invariants not just of braids but also of braid cobordisms with their isotopies and higher isotopies. Hints of such (coherent) braided monoidal 2-categories from link homology have appeared in the literature, see e.g. [MR20]².

A further motivation for such a derived setting, and for working \(\infty\)-categorically comes from TQFTs: passing from the triangulated categories \(\mathrm{K}^b(\mathrm{Sbim}_n)\) to their underlying stable \(\infty\)-categories \({\mathbf K}^b(\mathrm{Sbim}_n)\) yields better finiteness properties. These finiteness properties can be essential for extending TQFTs to manifolds of higher dimensions, and for ensuring that the resulting invariants are small enough to have well-defined decategorifications. See [MWW23, Thm. 1.5 and § 4.7] for an example of this phenomenon.