Categorified Quantum Groups and Braided Monoidal 2-Categories

We develop a systematic categorificationof the theory of quantum groups/bialgebras, and study their higher-representation theory. By following closely the generalized quantum double construction of Majid, we construct in particular the 2-quantum double $D(\mathcal{G})$ associated to a 2-bialgebra $\mathcal{G}$, and prove its duality and factorization properties, when $\mathcal{G}$ is strict or weak and skeletal. We also characterize a notion of (quasitriangular) 2-R-matrix $\mathcal{R}$ which can be seen as a categorification of the usual notion of R-matrix. We identify the associated 2-Yang-Baxter equations. The main result we prove in this paper is that the 2-representation 2-category $\operatorname{2Rep}^{\mathcal{T}}(\mathcal{G})$ of a quasitriangular 2-bialgebra $(\mathcal{G},\mathcal{T},\mathcal{R})$ -- when monoidally weakened by a Hochschild 3-cocycle $\mathcal{T}$ -- forms a braided monoidal 2-category.