Subject: categories: preprints: higher-dimensional categories

From: Eugenia Cheng <[email-masked]>
Date: Wed, 10 Jan 2001 15:13:24 +0000 (GMT)

The following papers are available at my website:

[1] The relationship between the opetopic and multitopic approaches to
weak n-categories

[2] Equivalence between approaches to the theory of opetopes

These papers cover the work I presented at PSSL 73 in Braunschweig, and
CT2000 in Como.  The following paper covers the work I presented at PSSL
74 in Cambridge:

[3] Equivalence between the opetopic and classical approaches to
bicategories

This currently has hand-drawn diagrams and so is available only on paper;
I will be happy to send copies to anyone interested.

The website is 

http://www.dpmms.cam.ac.uk/~elgc2 

Summaries follow below.

Thank you,
Eugenia Cheng

The problem of defining a weak n-category has been approached in various
ways, but so far the relationship between these approaches has not been
fully understood.  The subject of the above papers is the approaches given
by Baez/Dolan, Hermida/Makkai/Power and Leinster ([BD, HMP, Lei]); we
exhibit a relationship between them.

In each case the definition has two components. First, the language for
describing k-cells is set up.  Then, a concept of universality is
introduced, to deal with composition and coherence.  Any comparison of
these approaches must therefore begin at the construction of k-cells.  
This, in the language of Baez/Dolan, is the theory of opetopes.  Hermida,
Makkai and Power use an analogous construction of 'multitopes'.  In [1] we
exhibit a relationship between the constructions of opetopes and
multitopes.  In [2] we exhibit a relationship between Baez/Dolan opetopes
and Leinster opetopes.

It must be pointed out that we do not use the opetopic definitions
precisely as given in [BD], but rather, we develop a generalisation along
lines which Baez and Dolan began but chose to abandon, for reasons unknown
to the present author.  Baez and Dolan work with operads having an
arbitrary *set* of types (objects), but at the beginning of the paper they
use operads having an arbitrary *category* of objects, before restricting
to the case where the category of objects is small and discrete.

In fact, the use of a *category* of objects is a crucial aspect of our
work.  The morphisms in this category keep account of the successive
layers of symmetry arising from the Baez/Dolan use of symmetric operads.
Abandoning this information destroys the relationship between the
approaches; by retaining it, a clear relationship can be seen.

In [3] we begin to examine the complete opetopic definition of weak
n-category, with the modifications necessitated by our previous work.  We
show how this modified definition is equivalent to the classical
definitions for n <= 2.

REFERENCES

[BD] John Baez and James Dolan.  Higher-dimensional algebra III:
n-categories and the algebra of opetopes.  Adv. Math., 135(2):145--206,
1998.  Also available via http://math.ucr.edu/home/baez.

[HMP] Claudio Hermida, Michael Makkai, and John Power.  On weak higher
dimensional categories, 1997.  Available via
http://triples.math.mcgill.ca.

[Lei] Tom Leinster.  Structures in higher-dimensional category theory,
1998.  Available via http://www.dpmms.cam.ac.uk/~leinster