  We propose a general reduction procedure for classical field theories
provided with abelian gauge symmetries in a Lagrangian setting. These ideas
come from an axiomatic presentation of the general boundary formulation (GBF)
of field theories, mostly inspired by topological quantum field theories
(TQFT). We construct abelian Yang-Mills theories using this framework. We treat
the case for space-time manifolds with smooth boundary components as well as
the case of manifolds with corners. This treatment is the GBF analogue of
extended TQFTs. The aim for developing this classical formalism is to
accomplish, in a future work, geometric quantization at least for the abelian
case.
