Symmetric monoidal completions and the exponential principle among labeled combinatorial structures

Matias Menni

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.

TAC Home