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

[Tom Leinster <T.Leinster@dpmms.cam.ac.uk>]

John Baez wrote, concerning categories whose morphisms look like this:

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

> 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.  

Agreed on all counts.

It's also interesting (to me) to consider the dual diagrams, which look
something like computads (depending on exactly which version of the category
above you're using); this gives different geometric intuitions.  There's
more about this, and higher-dimensional generalizations, in

Ross Street, Categorical structures, in Handbook of Algebra I,
ed. M. Hazewinkel, North-Holland, 1996, pp. 529-577.

Tom