The multitopic universe

My talk will focus on the contents of the manuscript
entitled "The multitopic omega-category of all multitopic omega-categories"
posted on my website in 1999; the version posted in
2004 is a minor corrected variant. The proposal is for a notion
of weak omega-dimensional (or infinity) category, called multitopic
category, together with the definition of the structure that (small)
multitopic categories form, which structure is itself a (large) multitopic
category, here called the multitopic universe. The structure
contains the definition of the concept of n-dimensional multitopic
category for finite n, and, most importantly, the specication of
the (large) n+1-dimensional multitopic category of the (small) n-
dimensional multitopic categories. The proposal is inspired by John
Baez's and James Dolan's previous introduction of the notion of
opetopic category. In a three-part paper that appeared in the early
2000's, but was already done in 1997, with Claudio Hermida and
John Power we worked out in detail a concept called multitopic set,
which was meant to be a variant of opetopic set of Baez and Dolan.
A multitopic category is a multitopic set with additional properties,
much like opetopic categories are opetopic sets with properties, or
as elementary toposes are categories with properties. The definition
of multitopic category also uses what I called FOLDS equivalences
in my work on first-order logic with dependent sorts. The main new
items in the structure of the multitopic universe are what Sjoerd
Crans called the transfors, the lowest-dimensional transfors being
the morphisms between multitopic categories, which are infinity-
dimensional anafunctors. Anafunctors of ordinary (1-) categories
are introduced and shown to be a viable weak replacement of ordinary
functors in a paper of mine in the middle '90's. The talk
will point to further work on multitopic categories, the most important
one being Thorsten Palm's work, and also to comparisons
with other concepts of weak higher dimensional category.