Hereditarily finite sets

Hello, I would like to know if anybody is doing research on applying category theory to hereditarily-finite sets, e.g. where an object is a hereditarily set with some kind of structure to it and morphism that preserves that structure. Obviously, the subcategory of SET where objects are just hered. finite sets and morphisms are functions between hered. finite sets is not "interesting". Also, the case where an object is a hered. set together with an endomorphism doesn't sound "interesting". I would like URL's of papers if possible. Thank you. Regards, Bill