(3.1.1) Let $\mathscr{Y}$ be a connected regular flat separated $R$-scheme of finite type such that the special fiber $\mathscr{Y}_{k}$ is a divisor with strict normal crossings. Then, as explained in [MN13, §3.1], one can associate to $\mathscr{Y}$ its skeleton $\mathrm{Sk}(\mathscr{Y})$, which is a topological subspace of the generic fiber $\widehat{\mathscr{Y}}_{\eta}$ of the formal $t$-adic completion of $\mathscr{Y}$. It is the set of points of $\widehat{\mathscr{Y}}_{\eta}$ that correspond to a real valuation on the function field of $\mathscr{Y}_{K}$ that is monomial with respect to the strict normal crossings divisor $\mathscr{Y}_{k}$. The skeleton $\mathrm{Sk}(\mathscr{Y})$ is canonically homeomorphic to the dual intersection complex $\mathcal{D}((\mathscr{Y}_{k})_{\mathrm{red}})$ of $\mathscr{Y}_{k}$, and there exists a canonical continuous retraction

$$\rho_{\mathscr{Y}}:\widehat{\mathscr{Y}}_{\eta}\to\mathrm{Sk}(\mathscr{Y}).$$

Moreover, $\mathrm{Sk}(\mathscr{Y})$ carries a canonical piecewise $\mathbb{Z}$-affine structure [MN13, §3.2].