The Delannoy category is an interesting pre-Tannakian category associated to the oligomorphic group \(\mathbb {G}\) of automorphisms of the totally ordered set \((\textbf{R}, <)\) . 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.
