SYMMETRIC MONOIDAL COMPLETIONS AND THEEXPONENTIAL PRINCIPLE AMONG LABELED COMBINATORIAL STRUCTURES.

ABSTRACT. We generalize Dress and M"uller's main result in [5]. We observe thattheir result can be seen as a characterization of free algebras for certain monad on the category of species. This perspective allows to formulate a general exponentialprinciple in a symmetric monoidal category. We show that for any groupoid G, the category c!G of presheaves on the symmetric monoidal completion!G of G satisfies theexponential principle. The main result in [5] reduces to the case G = 1. We discuss twonotions of functor between categories satisfying the exponential principle and expres