Rafael Borowiecki wrote:
I am not an expert on oo-categories but i am sure there is a structure to the "class" of all omega-categories. I hope all this will not depend on the definition of an oo-category.
Given any notion of higher category, usually considerable interesting information is already encoded in the collection of all of these - with all suitable morphisms between them - and with all suitable invertible transformations and invertible higher transformations between these. Notably this is sufficient to talk about equivalence of the higher categories in question. In other words, given any notion of higher category, their collection should at least form an (oo,1)-category. http://ncatlab.org/nlab/show/%28infinity%2C1%29-category This should be the truncation of a more general structure, but should already contain a considerable amount of the relevant information and structure. For various flavors of higher categories the corresponding (oo,1)-categories "of all of them" are well known. These are "presented" by what is known as "folk model structures": http://ncatlab.org/nlab/show/folk+model+structure . More generally and more recently, Jacob Lurie has used unpublished work by Clark Barwick to define and study (oo,1)-categories of collections of (infty,n)-categories for n in N http://ncatlab.org/nlab/show/(infinity,n)-category
Are there different strict/weak n-categories with n any infinite ordinal number omega? omega does remind of an ordinal number.
One should beware that in practice the difference between the usage "oo-category" and "omega-category" is usually more due to tradition than being of intrinsic meaning. Ross Street originally introduced "omega-category" to explicitly denote a notion where cells of non-finite degree exist, but later authors didn't stick to that use of the work. Compare the remark by Sjoerd Crans that is reproduced here: http://ncatlab.org/nlab/show/strict+omega-category [For admin and other information see: http://www.mta.ca/~cat-dist/ ]