#
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.

http://www.tac.mta.ca/tac/volumes/11/18/11-18.dvi

http://www.tac.mta.ca/tac/volumes/11/18/11-18.ps

http://www.tac.mta.ca/tac/volumes/11/18/11-18.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/18/11-18.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/18/11-18.ps

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