Categorified Drinfel'd double and BF theory: 2-groups in 4D

The gauge symmetry and shift/translational symmetry of a 3D BF action, which are associated to a pair of dual Lie algebras, can be combined to form the Drinfel'd double. This combined symmetry is the gauge symmetry of the Chern-Simons action, which is equivalent to the BF action, up to some boundary term. We show that something similar happens in four dimensions when considering a 2-BF action (also known as BFCG action), whose symmetries are specified in terms of a pair of dual strict Lie 2-algebras (i.e., crossed modules). Combining these symmetries gives rise to a 2-Drinfel'd double, which becomes the gauge symmetry structure of a four-dimensional BF theory, up to a boundary term. Concretely, we show how, using 2-gauge transformations based on dual crossed modules, the notion of 2-Drinfel'd double defined by Bai et al. [Commun. Math. Phys. 320, 149 (2013).] appears. We also discuss how, similarly to the Lie algebra case, the symmetric contribution of the r-matrix of the 2-Drinfel'd double can be interpreted as a quadratic 2-Casimir, which allows us to recover the notion of duality.