Tropical geometry is often regarded recording the cycle of a variety. For example, the cohomology class of a subvariety of a toric
variety can be recovered from its tropicalization. In the recent preprint arXiv.1308.0042 Jeff and Noah Giansiracusa introduced a
notion of scheme structure for tropical varieties. In the affine case this is a congruence on the tropical semiring. They show that the
tropical variety as a set is determined by these tropical scheme structure. I will outline how to also recover the tropical cycle from
this information. This is joint work with Felipe Rincon.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/4811