Makkai's suggestion
In "Towards a categorical foundation of mathematics", Makkai wrote that: "This suggests that, possibly, the right approach to the definition of weak n-category is to aim at formulating all coherence conditions at once, regardless the fact that this might give a very "theoretical" definition. It would then be a separate, and still very important, project to find a (hopefully) finite and concise set of coherence conditions that would be enough to imply all coherence conditions." It was 15 years ago! Did this approach give something? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
David Leduc wrote: In "Towards a categorical foundation of mathematics", Makkai wrote that:
"This suggests that, possibly, the right approach to the definition of weak n-category is to aim at formulating all coherence conditions at once, regardless the fact that this might give a very "theoretical" definition. It would then be a separate, and still very important, project to find a (hopefully) finite and concise set of coherence conditions that would be enough to imply all coherence conditions."
It was 15 years ago! Did this approach give something?
Yes: most or all of the successful approaches to infinity-categories follow a philosophy of this general sort. You can find a lot of references here: http://ncatlab.org/johnbaez/show/Towards+Higher+Categories Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
John Baez <baez@math.ucr.edu> wrote:
Yes: most or all of the successful approaches to infinity-categories follow a philosophy of this general sort. You can find a lot of references here:
Thank you for the reference. But I don't know where to start. Each author seems to be working with his own favorite special case of infinity-categories: (inf,1)-categories, opetopic and multitopic categories, simple omega-categories, theta-categories, protocategories... and so on. Is there a definitive definition of omega-categories somewhere in the literature or is it still unknown? Can it be stated in elementary terms (i mean in terms of object, arrows, ... without references to simplicial sets or topology) ? In the definition of a bicategory, one could replace the coherence axioms by the statement that all diagrams built from the canonical ismorphisms commute. Can it be generalized to n=3, ... , omega. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Is there a definitive definition of omega-categories somewhere in the literature or is it still unknown? Can it be stated in elementary terms
There is a very short (few lines), very elementary (slight variation on Kan filler conditions) definition of what is supposed to be a model for fully general omega-categories by Verity http://ncatlab.org/nlab/show/weak+complicial+set .
i mean in terms of object, arrows, ... without references to simplicial sets
A simplicial set is nothing but a way to collect a bunch of objects, arrows, 2-arrows, etc. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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David Leduc -
John Baez -
Urs Schreiber