The idea to consider a higher or categorified version of gauge theories goes in part back to John Baez, but can be also motivated as arising from String Theory in form of an effective theory. Standard gauge theories, as they appear in the description of the elementary particles in our current understanding of fundamental interactions, are of the Yang-Mills type: The interaction particles are described by (the quantization of) connections in a principal bundle P over the spacetime manifold M and matter by sections of associated vector bundles. Locally, these connections correspond to (Lie algebra valued) 1-forms on spacetime M. One way of seeing higher gauge theories is that there are not only 1-form gauge fields, but also higher form degree ones, all interacting with one another in an appropriately defined consistent manner.
I pursue a concrete task — the construction of the Batalin-Vilkoviskiy (BV) extension of the Lie algebroid Yang-Mills (LAYM) gauge theories. These can be understood as strongly perturbed topological field theories of the BF type. It is an interesting question of how to construct such extensions, also since on the Hamiltonian level, the theories contain second class constraints and for those not much has been developed in this context yet.
Another objective is to use the BV technique for studying higher gauge theories with supersymmetry as they are naturally arise from String Theory. Looking at some examples of such theories in the physics literature, I want to understand the underlying total graded geometry of the supersymmetric extension of the higher gauge theories. An interesting further investigation could be to construct the BV formulation of such theories, which, in addition to the gauge symmetry also encodes the supersymmetry.