The most explicit version of the tropicalization operation is given by embedded tropicalization: over a non-archimedean base field \((K, \operatorname{val}_K)\), choose an embedding \(\iota : X \hookrightarrow \operatorname{Tor}(\Sigma)\) of your variety \(X\) into a toric variety and then, in each torus orbit, apply the valuation \(\operatorname{val}_K\) coordinatewise. The images of these valuation maps are then the strata of the tropicalization \(\operatorname{Trop}(X, \iota)\) of \(X\) with respect to \(\iota\). This tropical object highly depends on \(\iota\) and it was observed by [S. Payne, Math. Res. Lett. 16, No. 2–3, 543–556 (2009; Zbl 1193.14077)] that removing this dependence via a projective limit construction yields the Berkovich analytification of \(X\), i.e. \[ X^\mathrm{an} \cong \lim\limits_{\overset{\longleftarrow}{\iota}} \operatorname{Trop}(X, \iota) \] which suggests to view \(X^\mathrm{an}\) as a universal tropicalization.
In the paper under review, the authors restrict attention to the subproblem of describing a universal tropicalization of \(\mathbb{P}^r\) with respect to linear embeddings. The first theorem states that this role is played by the (bordified) Goldman-Iwahori space \(\overline{\mathcal{X}}_r\) (which in case of spherically complete base field is nothing but a compactification of the affine Bruhat-Tits building of \(\operatorname{PGL}_{r+1}\)), i.e. \[ \overline{\mathcal{X}}_r \cong \lim\limits_{\overset{\longleftarrow}{\iota \text{ linear}}} \operatorname{Trop}(\mathbb{P}^r, \iota). \] Moreover, there is a faithful tropicalization result, which says that the tropicalization of a linear space \(\iota : \mathbb{P}^r \hookrightarrow \mathbb{P}^n\) can be embedded into the building of \(\operatorname{PGL}_{r+1}\) such \(\operatorname{Trop}(\mathbb{P}^r, \iota)\) is given as the restriction of some universal tropicalization from the building of \(\operatorname{PGL}_{n+1}\) to \(n\)-dimensional tropical projective space.
Finally, the authors tie their universal tropicalization of linear spaces to the familiar description of tropical linear spaces via valuated matroids: in their third theorem the authors consider the universal valuated matroid, which is a valuated matroid on the uncountable set of \((r+1)\)-element subsets of \(K^{r+1} \setminus \{0\}\). The authors show that the construction of a tropical linear space from a valuated matroid can be carried out on this infinite object as well and as a result recovers the Goldman-Iwahori space, i.e. the universal tropical linear space.