Classification of simple commutative algebras in the Delannoy category

TL;DR

This paper classifies all simple commutative algebras in the Delannoy category, a pre-Tannakian category linked to automorphisms of the ordered real set, using novel methods beyond interpolation categories.

Contribution

It provides the first complete classification of simple commutative algebras in the Delannoy category, which cannot be analyzed using previous interpolation-based techniques.

Findings

All simple commutative algebras correspond to certain transitive G-sets.

Previous classification methods do not apply to the Delannoy category.

New methods are developed to handle categories not obtained by interpolation.

Abstract

The Delannoy category is an interesting pre-Tannakian category associated to the oligomorphic group $\mathbb{G}$ of automorphisms of the totally ordered set $(\mathbf{R}, <)$. By construction, it admits some obvious simple commutative algebras, corresponding to certain transitive $\mathbb{G}$-sets. We show that these account for all of the simple commutative algebras in the Delannoy category. Previous results of this kind have been limited to interpolation categories; since the Delannoy category cannot be obtained by interpolation, new methods are required.