About permutation algebras and sheaves (and named sets, too!)

In recent years, many general presentations (metamodels) for calculi dealing with names, e.g. process calculi with name-passing, have been proposed. (Pre)sheaf categories have been proved to satisfy classical properties on the existence of initial algebras/final coalgebras. Named sets are a theory of sets with permutations, introduced as the basis for the operational model of HD-automata. Permutation algebras are more in the line of algebraic specifications, where the direct axiomatization of equivalence under name permutation allows for the development of a theory of structured coalgebraic models. In this paper, we investigate the connections among these proposals, with the aim of establishing a bridge between different approaches to the