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Abstract

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 Frechét 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∞-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 Poincaré-Cartan can be thought of as a coordinate free, cohomological version of other similar structures in the field.