SYMMETRIC MONOIDAL COMPLETIONS AND THE EXPONENTIAL PRINCIPLE AMONG LABELED COMBINATORIAL STRUCTURES

We generalize Dress and Muller’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.

[1]  Miguel A. Méndez.  Species on Digraphs , 1996 .

[2]  S. Lane.  Categories for the Working Mathematician , 1971 .

[3]  Edward A. Bender,et al.  THE ENUMERATIVE USES OF GENERATING FUNCTIONS. , 1969 .

[4]  Peter T. Johnstone.  Cartesian monads on toposes , 1997 .

[5]  Gilbert Labelle,et al.  An extension of the exponential formula in enumerative combinatorics , 1995, Electron. J. Comb..

[6]  Tomoyuki Yoshida.  Categorical aspects of generating functions (I) Exponential formulas and Krull-Schmidt categories , 1998 .

[7]  François Bergeron,et al.  Une combinatoire du pléthysme , 1987, J. Comb. Theory, Ser. A.

[8]  Stephen Lack.  On the monadicity of finitary monads , 1999 .

[9]  Gian-Carlo Rota,et al.  Plethysm, categories, and combinatorics , 1985 .

[10]  G. Rota,et al.  On the foundation of combinatorial theory. X : A categorical setting for symmetric functions , 1992 .

[11]  Gérard Viennot,et al.  Combinatorial resolution of systems of differential equations, I. Ordinary differential equations , 1986 .

[12]  Andreas W. M. Dress,et al.  Decomposable Functors and the Exponential Principle , 1997 .

[13]  G. M. Kelly,et al.  A universal property of the convolution monoidal structure , 1986 .

[14]  Dayanand S. Rajan.  The Adjoints to the Derivative Functor on Species , 1993, J. Comb. Theory, Ser. A.

[15]  Miguel A. Méndez,et al.  Colored Species, c-Monoids, and Plethysm, I , 1993, J. Comb. Theory, Ser. A.

[16]  P. T. Johnstone,et al.  Adjoint Lifting Theorems for Categories of Algebras , 1975 .

[17]  A. Joyal.  Foncteurs analytiques et espèces de structures , 1986 .

[18]  Gilbert Labelle,et al.  Combinatorial species and tree-like structures , 1997, Encyclopedia of mathematics and its applications.

[19]  A. Joyal.  Une théorie combinatoire des séries formelles , 1981 .

[20]  Pierre Leroux,et al.  Combinatorial resolution of systems of differential equations. IV. separation of variables , 1988, Discret. Math..

[21]  François Bergeron.  A combinatoric of plethysm , 1987 .