A braided monoidal $(\infty,2)$-category of Soergel bimodules

Abstract

The Hecke algebras for all symmetric groups taken together form a braided monoidal category that controls all quantum link invariants of type A and, by extension, the standard canon of topological quantum field theories in dimension 3 and 4. Here we provide the first categorification of this Hecke braided monoidal category, which takes the form of an $\mathbb{E}_2$-monoidal $(\infty,2)$-category whose hom-$(\infty,1)$-categories are $k$-linear, stable, idempotent-complete, and equipped with $\mathbb{Z}$-actions. This categorification is designed to control homotopy-coherent link homology theories and to-be-constructed topological quantum field theories in dimension 4 and 5. Our construction is based on chain complexes of Soergel bimodules, with monoidal structure given by parabolic induction and braiding implemented by Rouquier complexes, all modelled homotopy-coherently. This is part of a framework which allows to transfer the toolkit of the categorification literature into the realm of $\infty$-categories and higher algebra. Along the way, we develop families of factorization systems for $(\infty,n)$-categories, enriched $\infty$-categories, and $\infty$-operads, which may be of independent interest. As a service aimed at readers less familiar with homotopy-coherent mathematics, we include a brief introduction to the necessary $\infty$-categorical technology in the form of an appendix.