  The classical Poisson reduction of a given Lagrangian system with (local)
gauge symmetries has to be done before its quantization. We propose here a
coordinate free and self-contained mathematical presentation of the covariant
Batalin-Vilkovisky Poisson reduction of a general gauge theory. It was
explained in physical terms (DeWitt indices) in Henneaux and Teitelboim's book
on quantization of gauge theories. It was studied in coordinates using jet
spaces by Barnich-Brandt-Henneaux and Stasheff, among others. The main idea of
our approach is to use the functor of point approach to spaces of fields to
gain coordinate free geometrical insights on the spaces in play, and to focus
on the notion of Noether identities, that is a simple replacement of the notion
of gauge symmetry, harder to handle algebraically. Our main results are a
precise formulation and understanding of the optimal finiteness hypothesis
necessary for the existence of a solution of the classical master equation, and
an interpretation of the Batalin-Vilkovisky construction in the setting of
homotopical geometry of non-linear partial differential equations.
