Symmetric Monoidal Completions and the Exponential Principle among Labeled Combinatorial Structures

We generalize Dress and Müller’s main result in [5]. We observe that their 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 exponential principle in a symmetric monoidal category. We show that for any groupoid G, the category !̂G of presheaves on the symmetric monoidal completion !G of G satisfies the exponential principle. The main result in [5] reduces to the case G = 1. We discuss two notions of functor between categories satisfying the exponential principle and express some well known combinatorial identities as instances of the preservation properties of these functors. Finally, we give a characterization of G as a subcategory of !̂G.