Re: Makkai's suggestion

[Greg Meredith <lgreg.meredith@biosimilarity.com>]

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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