Abstract: A principal bundle is called `natural' if diffeomorphisms of the base lift to automorphisms of the bundle in a way that respects composition. We propose to call a principal bundle `infinitesimally natural' if the infinitesimal version of this requirement holds, that is if vector fields on the base lift to equivariant vector fields on the total space in a way that respects the Lie bracket. This talk will be about how and why to classify infinitesimally natural principal bundles. Explaining the `how' will take most of the talk, but the `why' part is easy: in classical field theory, infinitesimal naturality is precisely what makes the energy-momentum tensor well defined.
