Symmetric monoidal completions and the exponential principle among labeled combinatorial structures

Matias Menni

Theory and applications of categories, Tome 11 (2003), pp. 397-419

Résumé

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.

Détail

Classification : 05A99, 18D10, 18D35
Keywords: symmetric monoidal categories, combinatorics