  We elaborate on the recently proposed notion of a weak presymplectic gauge
PDE. It is a $\mathbb{Z}$-graded bundle over the space-time manifold, equipped
with a degree $1$ vector field and a compatible graded presymplectic structure.
This geometrical data naturally defines a Lagrangian gauge field theory.
Moreover, it encodes not only the Lagrangian of the theory but also its
full-scale Batalin-Vilkovisky (BV) formulation. In particular, the respective
field-antifield space arises as a symplectic quotient of the super-jet bundle
of the initial fiber bundle. A remarkable property of this approach is that
among the variety of presymplectic gauge PDEs encoding a given gauge theory we
can pick a minimal one that usually turns out to be finite-dimensional, and
unique in a certain sense. The approach can be considered as an extension of
the familiar AKSZ construction to not necessarily topological and
diffeomorphism-invariant theories. We present a variety of examples including
$p$-forms, chiral Yang-Mills theory, Holst gravity, and conformal gravity. We
also explain the explicit relation to the non-BV-BRST version of the formalism,
which happens to be closely related to the covariant phase space and the
multisymplectic approaches.
