ABSTRACT:

Tuesday, 27 May 2003 
2:30 - 4:00   Thorsten Palm 

Dendrotopic Sets for Weak Infinity-Categories

The concept of what is called a _multitopic set_ arises from that of a
globular set by allowing cells to have arbitrarily many source facets.
A definition is given in the three-part paper [1]. In the not
officially published paper [2], Makkai equips multitopic sets with
inherent compositions. He then defines weak infinity-categories (under
the name `multitopic omega-categories') as those multitopic sets in
which composition can always be carried out.

I start my talk by presenting a definition of multitopic sets that is
shorter and more elementary than that of [1]. I prefer to call these
structures _dendrotopic sets_, alluding to a property (tree structure
of source facets) rather than an unnecessary means of definition
(_multi_categories). I then introduce the concept of a _universality
system_ in a dendrotopic set: a set of cells that behaves as if each
element were a universal arrow. Finally, I sketch the proof of my

Theorem. A dendrotopic set containing a universality system is a
Makkai infinity-category.

References:

[1] C. Hermida, M. Makkai, A. J. Power: `On weak higher dimensional
categories'; Journal of Pure and Applied Algebra. Part 1: 154 (2000),
pp. 221--246; part~2: 157 (2001), pp. 247--277; part~3: 166 (2002),
pp. 83--104

[2] M. Makkai: `The multitopic omega-category of all multitopic
omega-categories'; mystic.biomed.mcgill.ca/Makkai (1999)