Oligomorphic groups, categories of partial bijections, and ultrahomogeneous cubic spaces over finite fields

We show that for certain class of oligomorphic groups there is a version of multiplication of double cosets in the Ismagilov–Olshanski sense. Categories of (reduced) double cosets are realized as certain categories of partial bijections. As an example, we consider the ultrahomogeneous linear space $\mathcal{Z}$ of countable dimension over afinite field equipped with a cubic form. The group of automorphisms of $\mathcal{Z}$ is an oligomorphic group, we describe its open subgroups. According Tsankov, this gives a classification of its unitary representations. The category of reduced double cosets in this case is the category of partial linear bijections of finite-dimensional cubic spaces preserving cubic forms.