Re: Existence of very high categories

Re: Existence of very high categories

[Andree Ehresmann]

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_295_0/CTGDC_1978__19_3_295_0.pdf

Andrée



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Re: Existence of very high categories

[Urs Schreiber]

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More generally and more recently, Jacob Lurie has used unpublished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


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Existence of very high categories

[Rafael Borowiecki]

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


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Existence of very high categories

[Hasse Riemann]

 

Hi all categorists
 
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



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