Oligomorphic groups and tensor categories

doi:10.1007/s00222-026-01452-2

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Oligomorphic groups and tensor categories

2026-09-29

Nate Harman

Andrew Snowden *

DOI：10.1007/s00222-026-01452-2

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Abstract

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Given an oligomorphic group $G$ and a measure $\mu $ for $G$ (in a sense that we introduce), we define a rigid tensor category $\operatorname{\underline{Perm}}(G; \mu )$ of “permutation modules,” and, in certain cases, an abelian envelope $\operatorname{\underline{\textup{Rep}}}(G; \mu )$ of this category. When $G$ is the infinite symmetric group, this recovers Deligne’s interpolation category. Other choices for $G$ lead to fundamentally new tensor categories. For example, we construct the first known semi-simple pre-Tannakian categories in positive characteristic with super-exponential growth. One interesting aspect of our construction is that, unlike previous work in this direction, our categories are concrete: the objects are modules over a ring, and the tensor product receives a universal bi-linear map. Central to our constructions is a novel theory of integration on oligomorphic groups, which could be of more general interest. Classifying the measures on an oligomorphic group appears to be a difficult problem, which we solve in only a few cases.

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