Minimal support and families for the semantics of calculi with structured resources

Calculi that feature resource-allocating constructs (e.g. the pi-calculus or the fusion calculus) require special kinds of models. The best-known ones are presheaves and nominal sets. But named sets have the advantage of being finite in a wide range of cases where the other two are in finite. The three models are equivalent. Finiteness of named sets is strictly related to the notion of finite support in nominal sets and the corresponding presheaves. We generalise previous equivalence results by introducing a notion of minimal support in presheaf categories indexed over small categories of monos. We show that nominal sets are generalisd by families, that is, free coproduct completions, indexed by symmetries.