Measures on Cameron's treelike classes and applications to tensor categories

Measures on Fra"iss'e classes are a key input in the Harman–Snowden (2022) construction of tensor categories. Treelike Fra"iss'e classes provide a particularly tractable source of examples. In this paper, we complete the classification of measures on Cameron’s elementary treelike classes. In particular, for the class $\mathrm{\partial }{\mathfrak{T}}_3(n)$ of node-colored rooted binary tree structures with $n$ colors, we classify measures by an explicit bijection with directed rooted trees edge-labeled by $\{1,\dots ,n\}$ with a distinguished vertex, yielding $(2n+2{)}^n$ distinct $\mathbb{Z}\left[\frac{1}{2}\right]$-valued measures. For each $n\ge 1$, we use a family of measures ${\mu }_n^I$ and their supports $\mathrm{\partial }{\mathfrak{T}}_3(n{)}_I^{\mathrm{o}\mathrm{r}\mathrm{d}}$ (where $I\subseteq \{1,\dots ,n\}$) to construct the Karoubi envelopes $\mathbf{R}\mathbf{e}\mathbf{p}(\mathrm{\partial }{\mathfrak{T}}_3(n{)}_I^{\mathrm{o}\mathrm{r}\mathrm{d}};{\mu }_n^I)$, producing infinite families of semisimple tensor categories with superexponential growth that cannot be obtained via Deligne’s interpolation of representation categories. We also prove the nonexistence of measures on the $n$-colored tree class $C_n\mathfrak{T}$ for $n\ge 2$ and the labeled tree class $L\mathfrak{T}$, extending Snowden’s results for uncolored trees.