Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Arboreal tensor categories

Nate Harman, Ilia Nekrasov, Andrew Snowden

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.

Arboreal tensor categories

Abstract

We introduce some new symmetric tensor categories based on the combinatorics of trees: a discrete family , for an integer, and a continuous family , for 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, 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.
Paper Structure (45 sections, 45 theorems, 93 equations, 1 figure)

This paper contains 45 sections, 45 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Measures on trees
  3. Arboreal algebras and categories
  4. Motivation
  5. Open problems
  6. Outline
  7. Notation
  8. Measures on Fraïssé classes
  9. Overview
  10. Fraïssé classes
  11. Measures
  12. A presentation for $\Theta$
  13. Separated elements
  14. A refined presentation
  15. The induced regular measure
  16. ...and 30 more sections

Key Result

Theorem 1.1

We have an isomorphism $\Theta(\mathfrak{T}) \cong \mathbf{Z}[u,v]/(uv)$.

Figures (1)

  • Figure 1: Trees used in § \ref{['s:example']}. We label these $T_1, \ldots, T_{15}$, starting at the top left and reading left to right.

Theorems & Definitions (98)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • ...and 88 more