While I’m in the flow of posting old talk notes, here are my notes for a talk (with the above title) that I gave a year ago, at the conference Gauge theory and higher geometry, which was a birthday conference for my PhD supervisor Michael Murray. Here’s the abstract:
Bundle gerbes have been very fruitful structures due to the number of different constructions available. When one passes to more general higher geometric structures, even those just beyond abelian gerbes, a stunning paucity of analogous constructions presents itself. I will review several down-to-earth ideas that give rise to prototype constructions in order to fill this gap, focussing on crossed modules of Lie groups as key objects
And, just for fun, a picture of a crossed module from an old talk
theHigherGeometer