Hopf 2-algebras and Braided Monoidal 2-Categories

Following the theory of principal $\infty$-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 $\mathsf{2Vect}^{hBC}$, 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 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.