2024/01/05 by Yu Leon Liu, Aaron Mazel-Gee, Liu, Yu Leon +7
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2401.02956
openalex publication_date 2024/01/05 · openalex created_date 2024/01/13 · openalex updated_date 2026/08/01
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 𝔼2-monoidal (∞,2)-category whose hom-(∞,1)-categories are k-linear, stable, idempotent-complete, and equipped with 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 ∞-categories and higher algebra. Along the way, we develop families of factorization systems for (∞,n)-categories, enriched ∞-categories, and ∞-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 ∞-categorical technology in the form of an appendix.