categories: re: Tangle, Braid... related category?

[Jules Bean <jules@jellybean.co.uk>]

On Fri, Sep 21, 2001 at 10:47:24AM -0700, baez@math.ucr.edu wrote:
> First of all, Jules Bean leaves it quite open-ended exactly which category 
> he is talking about.  He is actually talking about a large number of 
> interesting categories each with their own description.  Secondly, the 
> usual definition of bimonoid involves structures and laws that are not 
> so natural from the topological viewpoint - i.e., certain morphisms are 
> decreed to be equal even when their corresponding embedded graphs are not 
> isotopic.  Whether this is good or bad depends on what you're trying to
> do.  

Thank you to everyone for the wealth of helpful answers!

I appreciate that my description was not 'tight': in actual fact,
there is probably more than one category I'm interested in in the family.

I've followed up the references to the category 2COB (as encountered
in TQFT, in Abrams' paper as well as Baez + Dolan), and that is quite
similar to the category I'm describing.  However, it's not quite the
one I have.  In 2COB, the following are equivalent (Abrams labels this 
relation 'F')

*  *      *  *
 \/       |\ |
  |   =   | \|
 /\       |  |
*  *      *  *

I suppose you might call that equation X = N .

The way I've implemented my category is not as a 2-mfd, but as a
1-complex, embedded 'sensibly'. There is a distinction between some
points of the boundary being the 'top', and the other points of the
boundary being the 'bottom'. (Which my diagrams have been assuming).

And, obviously, X and N are different as one-complexes, even though they 
are the deformation retracts of homeomorphic 2-mfds. (Actually, the
above diagram isn't even an X, it's an X-like shape with an extended
vertical section; that's a different one-complex too).

I have an intuitive justification for wanting these to be different,
if people aren't offended by slightly silly analogies. Think of the
networks (which is what I call them) as river networks. They have to
flow downhill (down the page). They can join as tributaries do, or
split into distributaries. Then in the 'X' all the water has possibly
mixed; we can't assume it will divide the same way.  In the 'N' on the 
other hand, all of the water which came in on the right, has
definitely gone out on the right.

The other helpful lead I was given was a category (sometimes) called
Vine, see Lavers [Comm. Algebra 25(4) pp1257-84], or Solomon 'A
Category of Concrete Monoids' at :
http://www.maths.usyd.edu.au:8000/res/Algebra/Sol/1996-07.html 

This is closely related to what I'm trying to do, but Vine is
different in two ways.  Firstly, the threads only join in Vine, never
split; secondly, Vine only has morphisms from n --> n, whereas my
category has morphisms from n --> m for all n and m. For example, in
Vine, the morphism diagram which looks like a capital 'V' is in fact a 
morphism from 2 --> 2, with one node at the bottom unconnected
(something like 'V.'), whereas in my category it's naturally a morphism 
from 2 --> 1.

The principle point of uncertainty is whether or not I allow the
threads to cross: this corresponds to whether some underlying monoid
is commutative or not.  Both possibilities are interesting.

Thanks again to everyone for their help. If anyone has any further
pointers to a category like the one I'm describing, I'm very
interested.

Yours,

Jules Bean