Re: Makkai's suggestion

[John Baez <baez@math.ucr.edu>]

David wrote:

> http://ncatlab.org/johnbaez/show/Towards+Higher+Categories
>
> Thank you for the reference. But I don't know where to start.


Start by reading the above book together with Cheng and Lauda's "Higher
categories: an illustrated guidebook":

http://www.cheng.staff.shef.ac.uk/guidebook/

and Leinster's "A Survey of Definitions of n-Category":

http://arxiv.org/abs/math/0107188

Then try Lurie's "Higher Topos Theory":

http://arxiv.org/abs/math/0608040

They're all free online!

Expect to spend a decade on this stuff.  Or, wait two decades for people to
polish it up, and then spend half a decade learning the basics and half a
decade learning what people have done in the next two decades.  That may be
more efficient.

Is there a definitive definition of omega-categories somewhere in the
> literature or is it still unknown?


There are *several* definitions that are almost surely "right" and likely to
be studied for many years hence.  There is no particular reason to expect
that one definition will be best for all applications - but there's a lot of
reason to expect that all the "right" definitions will be shown to be
equivalent (in a rather subtle sense).


> Can it be stated in elementary terms (I mean in terms of object, arrows,
> ... without references to simplicial sets or topology) ?
>

You should learn to love simplicial sets - they're way too important to
avoid!

If for some reason you're allergic to simplicial sets, you might like
Batanin's definition of omega-categories.   But then you need to like
operads.  You could state it without operads, but then it becomes quite
long.

The book by Cheng and Lauda takes various definitions and makes them less
scary by illustrating how they work with lots of pictures.


> 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.
>

You could say that's the basic idea behind Batanin's definition.

Best,
jb


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