Definition
Let \((X,\alpha_X:\mathcal M_X\to \mathcal O_X)\) be a log scheme. The tropicalization of \(X\), denoted \(\Sigma(X)\), is a generalized cone complex defined \[\Sigma(X) = \left(\coprod_{x\in \underline X}\Hom(\overline{\mathcal M}_{X,x}, \mathbb R_{\geq 0})\right)\Bigg/ \sim\] where the disjoint union is taken over all scheme theoretic points of \(X\) and the equivalence relation is generated by the identification of faces given by dualizing generization maps \(\overline{\mathcal M}_{X,x} \to \overline {\mathcal M}_{X,\eta}\) when \(x\in \overline{\{\eta\}}\).
You can also write this as a direct limit of the
A good reference for this is Appendix C in the Punctured Paper.