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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.

Intersection theory on tropicalizations of toroidal embeddings