  We show how to equip the cone complexes of toroidal embeddings with
additional structure that allows to define a balancing condition for weighted
subcomplexes. We then proceed to develop the foundations of an intersection
theory on cone complexes including push-forwards, intersections with tropical
divisors, and rational equivalence. These constructions are shown to have an
algebraic interpretation: Ulirsch's tropicalizations of subvarieties of
toroidal embeddings carry natural multiplicities making them tropical cycles,
and the induced tropicalization map for cycles respects push-forwards,
intersections with boundary divisors, and rational equivalence. As an
application we prove a correspondence between the genus 0 tropical descendant
Gromov-Witten invariants introduced by Markwig and Rau and the genus 0
logarithmic descendant Gromov-Witten invariants of toric varieties.
