  Tropical polytopes are images of polytopes in an affine space over the
Puiseux series field under the degree map. This viewpoint gives rise to a
family of cellular resolutions of monomial ideals which generalize the hull
complex of Bayer and Sturmfels, instances of which improve upon the hull
resolution in the sense of being smaller. We also suggest a new definition of a
face of a tropical polytope, which has nicer properties than previous
definitions; we give examples and provide many conjectures and directions for
further research in this area.
