Towards a description of weakly symmetric monoidal n-categories
by
Mark Weber
Department of Mathematics and Statistics, University of Ottawa

In [1], by using trees to parametrise certain pasting diagrams, Batanin was able to give an explicit description of the monad on the category of globular sets whose algebras are strict omega-categories, which formed the basis of a definition of weak omega-category. The goal of this work is to extend this approach to provide precise yet conceptual combinatorial descriptions of higher dimensional monoidal categorical structures with some sort of weak symmetry.

Low dimensional observations suggest that such structures arise as one k-cell weak n-categories (where k < n). For instance, braided monoidal categories may be regarded as one arrow tricategories. Unfortunately, it is not clear from this how to describe free higher-dimensional monoidal structures, which are important topologically. We provide a formal categorical construction, called monad suspension, that codifies this process of taking one k-cell algebras. We analyse some of its properties, and motivate the description of a new operad notion, namely that of symmetric globular operad, which provides a common generalisation of symmetric operads for the category of sets, and the operads that arise in Batanin's work. It is expected that the higher dimensional monoidal structures of interest are all algebras of symmetric globular operads.

The description of these new operads requires some new abstract notions, namely of generic morphisms and generic factorisations, which were motivated by Joyal's work on combinatorial species [2]. These notions enable symmetric operads for sets to be described in a way that facilitates their generalisation to the world of globular sets and trees.

By extending Kelly's abstract club notion [3] (a club in this context being a ``categorically well-behaved'' monad) from monads to monad distributive laws, we are able to demonstrate that all the one-dimensional examples, such as braided monoidal categories, are algebras of symmetric globular operads.

References

[1] M. Batanin, Monoidal globular categories as a natural environment for the theory of weak n-categories, Advances in Mathematics, 136:39-103, 1998.

[2] A. Joyal, Foncteurs analytiques et espèces de structures, SLNM 1234:126-159, 1991.

[3] G. M. Kelly, On clubs and data-type constructors, in Applications for categories to computer science, pages 163-190. Cambridge University Press, 1992.

Date received: May 29, 2002