Dear John, i have always taken this situation 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). to be a pretty clear critique of category theory's basic formulation. This may be too idealistic, but i've always felt that an ideally robust formulation would admit a meta-theory that makes n-categories "just fall out". The fact that they are so hard to formulate suggests that the basic design of the original presentation misses something crucial. i confess that taken together with the fact that it is exceptionally hard to get categorical composition to line up with parallel composition (in the sense of concurrent computation) in a manner that respects Curry-Howard, has really made me evaluate category theory as still very much a work in progress. Personally, i have wondered if there is a presentation that takes monad as the fundamental building block. i think this might not be too much of a stretch goal, actually, as monad as polymorphic comprehension is now well-established. Best wishes, --greg On Mon, Aug 30, 2010 at 9:34 PM, John Baez <baez@math.ucr.edu> wrote:
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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