  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.
