  We provide a presymplectic characterization of Liouville sectors introduced
by Ganatra-Pardon-Shende in terms of the characteristic foliation of the
boundary, which we call Liouville $\sigma$-sectors. We extend this definition
to the case with corners using the presymplectic geometry of null foliations of
the coisotropic intersections of transversal coisotropic collection of
hypersurfaces which appear in the definition of Liouville sectors with corners.
We identify its automorphism group which enables one to give a natural
definition of bundles of Liouville sectors. As a byproduct, we affirmatively
answer to a question raised by Ganatra-Pardon-Shende, which asks about the
optimality of their definition of Liouville sectors.
