Measures on Cameron's treelike classes and applications to tensor categories
Thanh Can, Thomas R\"ud

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 of node-colored rooted binary tree structures with colors, we classify measures by an explicit bijection with directed rooted trees edge-labeled by with a distinguished vertex, yielding distinct -valued measures. For each , we use a family of measures and their supports (where ) to construct the Karoubi envelopes $\mathbf{Rep}(\partial…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Advanced Operator Algebra Research
