We generalize Dress and Müller's main result in Decomposable functors and the exponential principle. 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 of presheaves on the symmetric monoidal completion !G of G satisfies the exponential principle. The main result in Dress and Müller 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 presheaves on !G.
Keywords: symmetric monoidal categories, combinatorics
2000 MSC: 05A99, 18D10, 18D35
Theory and Applications of Categories,
Vol. 11, 2003,
No. 18, pp 397-419.