Following the theory of principal ∞-bundles of Niklaus-Schreiber-Steveson, we develop a homotopy categorification of Hopf algebras, which model quantum groups. We study their higher-representation theory in the setting of 2Vect^hBC, which is a homotopy refinement of the notion of 2-vector spaces due to Baez-Crans that allows for higher coherence data. We construct in particular the 2-quantum double as a homotopy double crossed product, and prove its duality and factorization properties. We also define and characterize "2-R-matrices", which can be seen as an extension of the usual notion of R-matrix in an ordinary Hopf algebra. We found that the 2-Yang-Baxter equations describe the braiding of extended defects in 4d, distinct from but not unlike the the Zamolodchikov tetrahedron equations. The main results we prove in this paper is that the 2-representation 2-category of a weak 2-bialgebra is braided monoidal if it is equipped with a universal 2-R-matrix, and that our homotopy quantization admits the theory of Lie 2-bialgebras as a semiclassical limit.
Keywords: quantum groups, representation theory, categorification, quantum field theory
2020 MSC: 16T10 (primary), 18M15 (secondary), 18N70
Theory and Applications of Categories, Vol. 45, 2026, No. 36, pp 1426-1514.
Published 2026-06-05.
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