Hi all Note my new e-mail. I had to get a new e-mail to exclude html. 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. 6> Is the "class" of oo-categories of a certain recursive depth always an oo-category of depth one higher than the previous depth? I think yes for both strict and weak oo-categories. What should the "class" of all n-categories in Makkais foundation be called to describe it technically accurately? An oo-cosmos? in the categorical sense of a cosmos. I am not sure but this "class" maby also contain all oo-categories. Are there different strict/weak n-categories with n any infinite ordinal number omega? omega does remind of an ordinal number. The category need not to be accessible by forming categories of categories, just satisfy some axioms of an strict/weak oo-category for oo=omega. There might be a better definition of an omega-category if it is necessary at all, i don't know. Best regards Rafael Borowiecki [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
A partial answer to Rafael Borowiecki's question about the class of n-categories is given in a 30 years old paper I had published with Charles Ehresmann: "Multiple categories, II: The monoidal closed category of multiple categories", Cahiers de Top. et GD XIX-3 (1978), 295-333. Il is freely accessible on the NUMDAM site:http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1978__19_3/CTGDC_1978__19_3_29... Andrée [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
participants (3)
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Andree Ehresmann -
Rafael Borowiecki -
Urs Schreiber