I briefly review variational local Lagrangian field theory from a comonadic point of view and then use this to construct higher codimension prequantizations. After resolving the critical locus this shows for instance how the BV "shifted symplectic form" is the homotopy that witnesses the conservation of the canonical pre-symplectic current on covariant phase space. The talk is based on joint work with Igor Khavkine.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/3946