Subject: categories: n-categorical miscellany

From: john baez <[email-masked]>
Date: Tue, 24 Nov 1998 15:28:09 -0800 (PST)

1) Carlos Simpson writes:

>A. Joyal writes:

>>In particular, the new theory should help understanding
>>homotopy groups of spheres.

>I think it is safe to say that the ``n-category crowd'' (myself included)
>would love to make some progress on this.  The main thing I am wondering 
>about is: what exactly does one want to know about the homotopy groups of 
>spheres? for example, is there a concrete question which needs answering?

There are a lot of complicated and interesting patterns in the stable 
homotopy groups of spheres.   A lot of these patterns are currently 
studied using a generalized cohomology theory called complex cobordism 
theory, together with an intricate mess of spectral sequences:

Complex cobordism and stable homotopy groups of spheres / Douglas C.
Ravenel.  Orlando: Academic Press, 1986.

Nilpotence and periodicity in stable homotopy theory / by Douglas C.
Ravenel.  Princeton, N.J. : Princeton University Press, 1992.

It's all far too technical for me to understand, so I don't have much
hope of suddenly answering some question that homotopy theorists are 
stuck on!  However, I think that the n-category crowd can give more
conceptual explanations of some facts that appear as miracles in the 
current approach.  For example, there's an important relation between 
complex cobordism theory and formal group laws: the complex cobordism 
of a point is "the universal formal group law".  There should be an 
elegant conceptual proof of this, but as far as I know, there are just 
elegant *heuristics* for why it should be true, followed by a grungy 
computation.  

2) Carlos Simpson writes:

>On an introduction to n-categories: it would be great if J. Baez could
>bundle together his ``This week's finds in math. phys.'' which concern
>n-categories. This would make a really nice introduction specially for
>friends and family.

Especially during the holiday season!  I should try to bundle them together
more nicely at some point, but right now you can start at:

http://math.ucr.edu/home/baez/week73.html ,

skip the bit about left-right asymmetry in amino acids, read the stuff
on n-categories, and then keep clicking on the thing that says:

"To continue reading the `Tale of n-Categories', click here."

It fizzles out around week100, though I plan to resume it with an
issue discussing Carlos Simpson's paper on the stabilization hypothesis 
and Mark Hovey's book on model categories.   

3) Some people have noted that

http://xxx.lanl.gov/abs/math.QA/9811139

doesn't give them the paper "HDA4 - 2-Tangles".  Sorry! - the delay before 
the preprint server makes papers publicly available is longer than it
used to be.  It should be working by tomorrow.   If you still have trouble 
after that (it's a big file and may bust your printer), feel free to 
email me and ask me to send you a copy.