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 infinity) category, called multitopic

category, together with the definition 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 definition of the concept of n-dimensional multitopic

category for finite n, and, most importantly, the specication 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 definition

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 infinity-

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.

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