(3.2) Let $n$ be a positive integer, and let $T$ be a split algebraic $K$-torus of dimension $n$ with character module $M$ and cocharacter module $N=M^{\vee}$. We define the tropicalization map of $T$ by

$$\rho_{T}\colon T^{\mathrm{an}}\to N_{\mathbb{R}}\colon x\mapsto(M\to\mathbb{R}\colon m\mapsto-\ln|m(x)|).$$

This map is continuous, and its fibers are (not necessarily strictly) $K$-affinoid tori. The tropicalization map $\rho_{T}$ has a canonical continuous section $s\colon N_{\mathbb{R}}\to T^{\mathrm{an}}$ that maps each $n\in N_{\mathbb{R}}$ to the Gauss point of the affinoid torus $\rho^{-1}_{T}(n)$. The image of $s$ is called the canonical skeleton of $T$, and denoted by $\Delta(T)$. The map $s$ induces a homeomorphism $N_{\mathbb{R}}\to\Delta(T)$, which we will use to tacitly identify $\Delta(T)$ with $N_{\mathbb{R}}$.