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

Abstract

Measures on Fraïssé classes are a key input in the Harman--Snowden (2022) construction of tensor categories. Treelike Fraïssé 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 $\partial {\mathfrak{T}}_3(n)$ 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 $\hat{I}{¼}_n^I$ and their supports $\partial {\mathfrak{T}}_3(n{)}_I^{\mathrm{ord}}$ (where $I\subseteq \{1,\dots ,n\}$) to construct the Karoubi envelopes $\mathbf{Rep}(\partial {\mathfrak{T}}_3(n{)}_I^{\mathrm{ord}};\hat{I}{¼}_n^I)$, 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.

48 pages; comments are welcome