Cylinder topological quantum field theory: A categorical presentation of classical field theory and its symmetries

We use geometric ideas coming from certain classic algebraic constructions to associate, to every classical field theory, a symmetric monoidal double functor from the double category of cobordisms with corners to a certain symmetric monoidal double category. We call symmetric monoidal double functors so constructed cylinder topological quantum field theories. Our initial formulation of cylinder topological quantum field theory is presented in a purely topological context. We present appropriate refinements in order to accommodate information relevant to classical field theory.