Let $\mathcal{X}$ be a proper $S$-variety. Its generic fiber $\mathcal{X}_{K}$ is in particular a proper $K$-scheme. As a topological space, its $K$-analytification $\mathcal{X}_{K}^{\mathrm{an}}$ in the sense of Berkovich is compact and can be described as follows (cf. [Ber90, Theorem 3.4.1]). Choose a finite cover of $\mathcal{X}_{K}$ by Zariski open subsets of the form $U=\spec A$ where $A$ is a $K$-algebra of finite type. The Berkovich space $U^{\mathrm{an}}$ is defined as the set of all multiplicative seminorms $|\cdot|_{x}:A\to\mathbf{R}_{+}$ extending the given absolute value of $K$, endowed with the topology of pointwise convergence. It is common usage to write $|f(x)|:=|f|_{x}$. The space $\mathcal{X}_{K}^{\mathrm{an}}$ is then obtained by gluing together the open sets $U^{\mathrm{an}}$. There is a canonical continuous map $s_{\mathcal{X}}:\mathcal{X}_{K}^{\mathrm{an}}\to\mathcal{X}_{K}$, locally defined on $U^{\mathrm{an}}$ by setting

$$s_{\mathcal{X}}(x)=\left\{f\in A,\,|f(x)|=0\right\}.$$

The seminorm $|\cdot|_{x}$ defines a norm on the residue field $\kappa(s_{\mathcal{X}}(x))$, extending the given absolute value on $K$. The completion of $\kappa(s_{\mathcal{X}}(x))$ with respect to this norm is denoted $\mathcal{H}(x)$. It is the residue field at $x$ of the natural structure sheaf of $\mathcal{X}_{K}^{\mathrm{an}}$, that we will however not explicitely use.

Given $x\in\mathcal{X}_{K}^{\mathrm{an}}$ denote by $R_{x}$ the corresponding valuation ring in $\kappa(s_{\mathcal{X}}(x))$. By the valuative criterion of properness, the map $T_{x}:=\spec R_{x}\to S$ admits a unique lift $T_{x}\to\mathcal{X}$ mapping the generic point to $s_{\mathcal{X}}(x)$. In line with valuative terminology [Va00], we call the image of the closed point of $T_{x}$ in $\mathcal{X}$ the center of $x$ on $\mathcal{X},$ and denote it by $c_{\mathcal{X}}(x)$. It is a specialization of $s_{\mathcal{X}}(x)$ in $\mathcal{X}$. It also belongs to $\mathcal{X}_{0}$ since it maps to the closed point of $S$ by construction. The map $c_{\mathcal{X}}:\mathcal{X}_{K}^{\mathrm{an}}\to\mathcal{X}_{0}$ so defined is anti-continuous. It is referred to as the reduction map in rigid geometry.