For $n\in\mathbb{N}_{>0}$ and $a\in K^{\circ\circ}$ we define

For tuples $\boldsymbol{n}=(n_{0},...,n_{p})\in\mathbb{N}_{>0}^{p+1}$ and $\boldsymbol{a}=(a_{0},...,a_{p})\in(K^{\circ\circ})^{p+1}$ we define $\mathfrak{X}(\boldsymbol{n},\boldsymbol{a}):=\mathfrak{X}(n_{0},a_{0})\times_{K^{\circ}}...\times_{K^{\circ}}\mathfrak{X}(n_{p},a_{p})$ and for $m\in\mathbb{N}$ we set $\mathfrak{X}(m):=\mathfrak{X}(m,1)$. A strictly polystable formal scheme over $K^{\circ}$ is an admissible formal scheme $\mathfrak{X}$ over $K^{\circ}$ which can be covered by formal open sets $\mathfrak{U}$ with étale morphisms

where $\boldsymbol{n}$, $\boldsymbol{a}$ and $m$ may depend on $\mathfrak{U}$. We say that $\mathfrak{X}$ is strongly nondegenerate strictly polystable if all $a_{i}$ can be chosen nonzero.

To a strongly nondegenerate strictly polystable formal scheme $\mathfrak{X}$ over $K^{\circ}$ Berkovich introduced in [Ber99] a canonical polytopal subset $S(\mathfrak{X})$ of $\mathfrak{X}^{\textup{an}}$ called the skeleton. It is a closed subset of $\mathfrak{X}^{\textup{an}}$ which is locally given by canonical polysimplices and can be described as follows. Let $\psi:\mathfrak{U}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)$ be an étale morphism as above. The generic fibre of the right hand side is given as $\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)^{\textup{an}}=\mathscr{M}(A)$ where $A=(K\langle T_{0}^{\pm},...,T_{m}^{\pm}\rangle)\langle T_{00},...,T_{p,n_{p}}\rangle/(T_{00}...T_{0,n_{0}}-a_{0},...,T_{p0}...T_{p,n_{p}}-a_{p})$. The elements of $A$ can be expressed as $\sum_{\mu}a_{\mu}T^{\mu}$ with $a_{\mu}\in K\langle T_{0}^{\pm},...,T_{m}^{\pm}\rangle$ and $a_{\mu}=0$ if there is an $i\in\{0,...,p\}$ such that $\mu_{i,k}\geq 1$ for all $k\in\{0,...,n_{i}\}$. Now to an element $\boldsymbol{t}$ in the polysimplex $\left\{\boldsymbol{t}\in\mathbb{R}_{\geq 0}^{\boldsymbol{n}+\boldsymbol{1}}\;\Big|\;t_{i0}+...+t_{in_{i}}=-\log(|a_{i}|),0\leq i\leq p\right\}$ we associate a seminorm on $A$ by sending a power series as above to $\max_{\mu}\{|a_{\mu}|\exp(-\boldsymbol{t}\cdot\mu)\}$. This gives an embedding of the polysimplex into $\mathscr{M}(A)$ whose image is denoted by $\Delta$. The skeleton $S(\mathfrak{U})$ of $\mathfrak{U}$ is defined to be $(\psi^{\textup{an}})^{-1}(\Delta)$. One can show that $\psi^{\textup{an}}$ induces a homeomorphism from $(\psi^{\textup{an}})^{-1}(\Delta)$ to $\Delta$ if $\mathfrak{U}$ has a unique minimal stratum which maps to the minimal stratum of $\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)$. The skeleton $S(\mathfrak{X})$ of $\mathfrak{X}$ is the union of all $S(\mathfrak{U})$ and is independent of all choices.

To a stratum $S$ of $\mathfrak{X}$ one can associate a canonical polysimplex $\Delta_{S}$ in the skeleton such that the interiors of the $\Delta_{T}$ form a disjoint cover of $S(\mathfrak{X})$ where $T$ ranges over all strata of $\tilde{\mathfrak{X}}$. In order to do so, we choose a refinement of the cover of $\mathfrak{X}$ as described in the Proposition below and choose $\mathfrak{U}$ such that $S$ is its distinguished stratum. We then define $\Delta_{S}:=S(\mathfrak{U})$.

An admissible formal scheme $\mathfrak{X}$ is called strongly nondegenerate polystable if there exists a strongly nondegenerate strictly polystable formal scheme $\mathfrak{X}^{\prime}$ and a surjective étale morphism $\mathfrak{X}^{\prime}\rightarrow\mathfrak{X}$. The skeleton of $\mathfrak{X}$ is defined to be the image of the skeleton of $\mathfrak{X}^{\prime}$ under the map $\mathfrak{X}^{\prime\textup{an}}\rightarrow\mathfrak{X}^{\textup{an}}$.

One can endow the skeleton with a piecewise linear structure, see [Ber04, §6]. We will define piecewise affine linear functions on the skeleton of a strongly nondegenerate strictly polystable formal scheme in Definition 2.10. There is a canonical continuous retraction map $p_{\mathfrak{X}}:\mathfrak{X}^{\textup{an}}\rightarrow S(\mathfrak{X})$ which restricts to the identity on $S(\mathfrak{X})$. For details see [Ber99, §4], [Ber04, §4] or [Gub10, 5.3].