In this talk I will give a brief explanation of higher categorical structures in gauge theory, and I will discuss the simplest case, linear gauge theory, which can be modelled by a chain complex. We study the derived critical locus of the linear Yang-Mills action and find that the usual shifted Poisson structure is exact on globally hyperbolic spacetimes. It therefore allows for trivializations, which are chain complex analogues of (advanced and retarded) Green's operators. These trivializations give rise to an unshifted Poisson structure which is central to quantizing the theory. We find that pleasingly the usual quantization already is compatible with quasi-isomorphisms.
This is joint work with M. Benini and A. Schenkel.