Bundle 2-gerbes

We make the category BGrb M of bundle gerbes on a manifold M into a 2-category by providing 2-cells in the form of transformations of bundle gerbe morphisms. This description of BGrb M as a 2-category is used to define the notion of a bundle 2-gerbe. To every bundle 2-gerbe on M is associated a class in H 4 (M ; Z). We define the notion of a bundle 2-gerbe connection and show how this leads to a closed, integral, differential 4-form on M which represents the image in real cohomology of the class in H 4 (M ; Z). Some examples of bundle 2-gerbes are discussed, including the bundle 2-gerbe associated to a principal G bundle P → M. It is shown that the class in H 4 (M ; Z) associated to this bundle 2-gerbe coincides with the first Pontryagin class of P — this example was previously considered from the point of view of 2-gerbes by Brylinski and McLaughlin.