  We introduce a sheaf-theoretic approach to tropical homology, especially for
tropical homology with potentially non-compact supports. Our setup is suited to
study the functorial properties of tropical homology, and we show that it
behaves analogously to classical Borel-Moore homology in the sense that there
are proper push-forwards, cross products, and cup products with tropical
cohomology classes, and that it satisfies identities like the projection
formula and the K\"unneth theorem. Our framework allows for a natural
definition of the tropical cycle class map, which we show to be a natural
transformation. Finally, we prove Poincar\'e-Verdier duality over the integers
on tropical manifolds.
