  Usually bundle gerbes are considered as objects of a 2-groupoid, whose
1-morphisms, called stable isomorphisms, are all invertible. I introduce new
1-morphisms which include stable isomorphisms, trivializations and bundle gerbe
modules. They fit into the structure of a 2-category of bundle gerbes, and lead
to natural definitions of surface holonomy for closed surfaces, surfaces with
boundary, and unoriented closed surfaces.
