Universal properties of Delannoy categories
Kevin Coulembier, Nate Harman, Andrew Snowden
TL;DR
The paper develops a universal framework for tensor categories arising from oligomorphic groups, with a focus on Delannoy categories $\\mathfrak{C}_i$ built from the Delannoy group $\\mathbb{G}=\\mathrm{Aut}(\\mathbf{R},<)$ and four measures $\\mu_i$. It proves a general mapping property: tensor functors out of $\\underline{\mathrm{Perm}}(G,\\mu)$ correspond to additive, left-exact functors from the category of finitary $G$-sets to the opposite category of étale algebras, compatible with the measure; this is refined for $G=\\mathbb{G}$ via ordered étale algebras and Delannic algebras. The Delannoy universality result shows that tensor functors from $\\mathfrak{C}_i$ correspond to Delannic algebras of type $i$, providing a concrete and highly structured description of the Delannoy categories and their interrelationships. These universal properties enable the construction of many tensor functors between the Delannoy categories and yield information on abelian envelopes, showing, for instance, that $\\mathfrak{C}_2$ and $\\mathfrak{C}_3$ admit at least two local envelopes while $\\mathfrak{C}_4$ has at least four. The work thereby advances the understanding of infinite-rank tensor categories arising from oligomorphic symmetry, with potential implications for new pre-Tannakian and envelope phenomena.
Abstract
Recently, the second and third authors introduced a new symmetric tensor category $\underline{\mathrm{Perm}}(G, μ)$ associated to an oligomorphic group $G$ with a measure $μ$. When $G$ is the group of order preserving self-bijections of the real line there are four such measures, and the resulting tensor categories are called the Delannoy categories. The first Delannoy category is semi-simple, and was studied in detail by Harman, Snowden, and Snyder. We give universal properties for all four Delannoy categories in terms of ordered étale algebras. As a consequence, we show that the second and third Delannoy categories admit at least two local abelian envelopes, and the fourth admits at least four. We also prove a coarser universal property for $\underline{\mathrm{Perm}}(G, μ)$ for a general oligomorphic group $G$.