The purpose of this appendix is to extend some of the constructions and results in this paper to the following situation: let $X$ be a smooth family of $n$-dimensional projective varieties over a punctured disk $\mathbb{D}^{*}\subset\mathbb{C}$ centered at $0$, and let $\mathscr{X}$ be an snc proper regular algebraic space over $\mathbb{D}$ extending the family $X$ to $0$. In particular, the special fiber $\mathscr{X}_{0}$ is strict normal crossing, the morphism $\mathscr{X}\rightarrow\mathbb{D}$ is a proper holomorphic submersion, and $\mathscr{X}$ is not assumed to be projective; we still call such space $\mathscr{X}$ an snc model of $X$. Our goal is to define a skeleton $\Sk(\mathscr{X})\subset X^{\an}$ and a retraction $\rho_{\mathscr{X}}:X^{\an}\rightarrow\Sk(\mathscr{X})$ associated with $\mathscr{X}$.

We start by recalling a construction from [BJ17, 4.2]. Let $\mathscr{X}^{\prime}$ be another snc model of $X$ that dominates $\mathscr{X}$; we write $h:\mathscr{X}^{\prime}\longrightarrow\mathscr{X}$ and the special fibers $\mathscr{X}_{0}=\sum_{i\in I}a_{i}D_{i}$ and $\mathscr{X}^{\prime}_{0}=\sum_{i\in I^{\prime}}a^{\prime}_{i}D^{\prime}_{i}$. Then there exists an integral affine retraction

$$r_{\mathscr{X}^{\prime}\mathscr{X}}:\mathcal{D}(\mathscr{X}^{\prime}_{0})\longrightarrow\mathcal{D}(\mathscr{X}_{0}),$$

as follows. Let $\tau^{\prime}$ be the simplex in $\mathcal{D}(\mathscr{X}^{\prime}_{0})$ corresponding to a stratum $Y^{\prime}\subseteq D^{\prime}_{0}\cap\ldots\cap D^{\prime}_{q}$. Let $Y$ be the minimal stratum of $\mathscr{X}_{0}$ such that $h(Y^{\prime})\subset Y$; we denote by $\tau$ the simplex in $\mathcal{D}(\mathscr{X}_{0})$ corresponding to $Y$ and we write $D_{0},\ldots,D_{p}$ the irreducible components of $\mathscr{X}_{0}$ containing $Y$. Then

$$h^{*}D_{i}=\sum_{j=0}^{q}a_{ij}D^{\prime}_{j}+\sum_{h\in I^{\prime}\setminus\{0,\ldots,q\}}a_{ih}D^{\prime}_{h}$$

and we define the map $r_{\mathscr{X}^{\prime}\mathscr{X}}$ on $\tau^{\prime}$ by the formula:

$$\tau^{\prime}\ni w=(w_{0},\ldots,w_{q})\mapsto r_{\mathscr{X}^{\prime}\mathscr{X}}(w^{\prime})=\Big(\sum_{j=0}^{q}a_{ij}w^{\prime}_{j}\Big)_{0\leqslant i\leqslant p}\in\tau.$$

This yields a continuous integral affine map. Furthermore, we have the following transitivity property: if $\mathscr{X}^{\prime\prime}\longrightarrow\mathscr{X}^{\prime}\longrightarrow\mathscr{X}$ are three snc models of $X$, then $r_{\mathscr{X}^{\prime\prime}\mathscr{X}}=r_{\mathscr{X}^{\prime}\mathscr{X}}\circ r_{\mathscr{X}^{\prime\prime}\mathscr{X}^{\prime}}$.

We write $A_{\mathscr{X}^{\prime}\mathscr{X}}$ for the union of active faces in $\mathcal{D}(\mathscr{X}^{\prime}_{0})$. It follows from [BJ17, Proposition 4.3] that $r_{\mathscr{X}^{\prime}\mathscr{X}}$ induces a homeomorphism from $A_{\mathscr{X}^{\prime}\mathscr{X}}$ onto $\mathcal{D}(\mathscr{X}_{0})$.

It follows directly from the transitivity property that this does not depend on the choice of a projective model $\mathscr{X}^{\prime}$.