Regular categories, oligomorphic monoids, and tensor categories

Knop constructed a tensor category associated to a finitely-powered regular category equipped with a degree function. In recent work with Harman, we constructed a tensor category associated to an oligomorphic group equipped with a measure. In this paper, we explain how Knop's approach fits into our theory. The first, and most important, step describes finitely-powered regular categories in terms of oligomorphic monoids; this may be of independent interest. We go on to examine some aspects of this construction when the regular category one starts with is the category of $G$-sets for an oligomorphic group $G$, which yields some interesting examples.