  Field Theories in Physics can be formulated giving a local Lagrangian
density. Locality is imposed using the infinite jet bundle. That bundle is
viewed as a pro-finite dimensional smooth manifold and that point of view has
been compared to different topological and Frech\'et structures on it. A
category of local (insular) manifolds has been constructed. Noether's second
theorem is reviewed and the notion of Lie pseudogroups is explored using these
concepts. The L-infinity algebra of local observables is defined depending only
on the cohomology of the Lagrangian (using a result in multisymplectic manifold
which has been extended to the local category). That local pre-multisymplectic
form, called the Poincar\'e-Cartan can be thought of as a coordinate free,
cohomological version of other similar structures in the field.
