The fundamental notion of symmetry can be expressed as the action of a group on a set, or more generally in terms of a transformation groupoid. In this talk I will show how a group action on a set may be generalized to a 2-group action on a category C, and how such an action gives rise to a transformation double groupoid (in the case when C is a groupoid), or a transformation double category, in general. I will describe some of the different perspectives from which this construction can be viewed, and illustrate it with a simple example: the adjoint action of a 2-group on itself.
I will then talk about the application which motivated this work: the study of moduli spaces of flat connections in higher gauge theory. No prior knowledge of gauge theory or higher gauge theory will be assumed. Both ordinary gauge theory and higher gauge theory become much simpler to describe if one passes from locally defined connection 1- and 2-forms to holonomies or transports along specified 1- and 2-cells in the underlying manifold M. In higher gauge theory, the moduli space is the transformation double groupoid coming from the action of a 2-group of gauge transformations on a groupoid of connections. I will give some examples of this construction for simple manifolds M, and if time permits, I will mention related work on (higher) gauge theory for surfaces with boundary, when the group or 2-group is finite.
This talk is based on collaborations with Jeffrey Morton, João Faria Martins and Diogo Bragança.