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

TL;DR

This paper classifies measures on Cameron's treelike classes, constructs new tensor categories from these measures, and extends nonexistence results for measures on certain colored and labeled tree classes.

Contribution

It completes the classification of measures on Cameron's elementary treelike classes and constructs novel tensor categories with superexponential growth.

Findings

Classified measures on node-colored rooted binary trees with explicit bijection.

Constructed infinite families of semisimple tensor categories from these measures.

Proved nonexistence of measures on certain colored and labeled tree classes.

Abstract

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 $\partial \mathfrak{T}_3(n)$ 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 $n \geq 1$, we use a family of measures $\mu_n^I$ and their supports $\partial \mathfrak{T}_3(n)^{\mathrm{ord}}_I$ (where $I \subseteq \{1, \dots, n\}$) to construct the Karoubi envelopes $\mathbf{Rep}(\partial \mathfrak{T}_3(n)^{\mathrm{ord}}_I;\mu^I_n)$, 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 $n \geq 2$ and the labeled tree class $L \mathfrak{T}$, extending Snowden's results for uncolored trees.