Measurable Categories and 2-Groups

Using the theory of measurable categories developped by Yetter in work in preparation, we provide a notion of representations of 2-groups more well-suited to physically and geometrically interesting examples than that proposed in unpublished work of Barrett and Mackaay using 2-VECT of Kapranov and Voevodsky. Using this theory we sketch a 2-categorical approach to the state-sum model for Lorentzian quantum gravity proposed in by the authors in previous work and suggest state-integral constructions for 4-manifold invariants.