Michael Makkai:
	A duality involving Joyal's disks and omega-categories
		(joint work with Marek Zawadowski)

Abstract. 

	Andre Joyal has, in an extended abstract entitled "Disks,
Duality amd Theta-categories", introduced the combinatorial concept of
"disk". He denotes the category of finite disks by Disks. "Cellular sets"
or "theta-sets" are (covariant) set-valued functors on Disks;
"theta-categories" are cellular sets satisfying a "horn-filling"
condition.
	We prove that Disks is contravariantly equivalent to another
category named SimpleOmegaCat, the category of simple omega categories,
defined as follows. An omega-graph G (also called a globular set, by Ross
Street and Micheal Batanin)  is said to be *composable* if G "has a unique
composite", where an element a of <G> , the omega-category freely
generated by G , is called "a composite of G" if for all proper
sub-omega-graphs H of G , a does not belong to <H> (construed naturally as
a sub-omega-category of <G> ). An omega-category is simple if it is of the
form <G> for a composable omega-graph G ; SimpleOmegaCat is a full
subcategory of the category of omega-categories with ordinary, strict,
structure-preserving omega-functors as morphisms.

	The stated equivalence arises as a duality generated by an
explicit schizophrenicject, a disk-object in SimpleOmegaCat, which is a
simple-omega-category-object in Disks at the same time.

	The work involves, as a somewhat minor ingredient, a simple
characterization of composable omega-graphs; this characterization seems
implicit in Michael Batanin's work.

	The "duality" in Joyal's paper is the "bottom level" case of our
result; this is a well-known fact, and it is mentioned in Joyal's paper
for expository purposes. 

	The speaker has announced, for instance at the St John, New
Brunswick, CMS meeting (July 1998), the concept of "protocategory". This
is intended as a *finitary* equivalent to the notion of multitopic
category (the latter is a version of the Baez/Dolan concept of opetopic
category; the main part of the definition of "multitopic category" is
described in a paper by Claudio Hermida, the speaker and John Power,
entitled "On higher dimensional categories I"). Joyal's "theta category"
is a well-identifble, albeit proper, part of the notion of
"protocategory". For the identification involved, and also for the further
work on protocategories (intended to show that they are equivalent (in a
specific see defined and described e.g. by the speaker at the St. John
meeting) to multitopic categories), the du theorem stated above is an
important tool.