Arboreal tensor categories

TL;DR

This paper introduces new symmetric tensor categories derived from tree combinatorics, including a discrete family and a complex parameterized family, expanding the landscape of pre-Tannakian categories with superexponential growth.

Contribution

It constructs novel tensor categories based on trees, proves key properties like measure classification and semi-simplicity, and demonstrates a unique family not obtainable by existing interpolation methods.

Findings

Defined the discrete family (n)nd continuous family (t)ategories.

Established measure classification for trees and proved semi-simplicity.

Showed (t)ategories are the first of their kind with superexponential growth not from interpolation.

Abstract

We introduce some new symmetric tensor categories based on the combinatorics of trees: a discrete family $\mathcal{D}(n)$, for $n \ge 3$ an integer, and a continuous family $\mathcal{C}(t)$, for $t \ne 1$ a complex number. The construction is based on the general oligomorphic theory of Harman--Snowden, but relies on two non-trivial results we establish. The first determines the measures for the class of trees, and the second is a semi-simplicity theorem. These categories have some notable properties: for instance, $\mathcal{C}(t)$ is the first example of a 1-parameter family of pre-Tannakian categories of superexponential growth that cannot be obtained by interpolating categories of moderate growth.