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

[baez@math.ucr.edu]

Tom Leinster wrote:

> Jules Bean wrote:

> > Related to these two these is a category whose objects are again the
> > natural numbers, and whose morphisms are pieces of string which are
> > allowed to split into multiple strands, and join together into single
> > strands, such as the following morphism 3 --> 2:
> > 
> >  *   *   *
> >   \ /   /
> >    |   /\
> >    \  /  |
> >     \/   |
> >      *   *
> > 
> > (excuse the crude drawing which will only look OK if you have a
> > monospaced font).
> > 
> > There are various ways this category could be formulated (are the
> > strings allowed to cross each other? are they allowed to double back?
> > etc), but my question is: has anything been written about it?  Does it 
> > have a name? Does it remind anyone of another category which has been
> > studied?
 
> I don't know if it has a name, but it's the free strict monoidal category
> containing a bimonoid.  By a bimonoid I mean an object which has both the
> structure of a monoid and a comonoid, with the two structures compatible with
> each other.

This answer is a bit more definite-sounding than the one I would give.
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.  

But anyway: there are lots of interesting categories along these
general lines!   Tom has described one, and like his example they
all tend to have nice universal properties - i.e. they tend to be 
the "free ..... category on a .....".
 
As described here:

Higher-dimensional algebra and topological quantum field theory, with
James Dolan, Jour. Math. Phys. 36 (1995), 6073-6105.

Higher-dimensional algebra II: 2-Hilbert spaces,
Adv. Math. 127 (1997), 125-189.

the category of framed tangles in 2/3/4 dimensions is the "free 
monoidal/braided/symmetric category with duals on one object". 
We can enhance these categories to obtain various categories of 
embedded framed graphs by throwing in extra morphisms involving our 
object, which give vertices in our graph.  We can also get rid of 
the framing or "doubling back" by eliminating various clauses buried 
within the phrase "with duals".  

I don't know of anyone who attempted to write about *all* these 
variations - there are just too many to handle individually, and 
people haven't yet tackled the general theory of such categories 
(though such a theory does exist).  However, you can find a lot
of examples treated in Yetter's book "Functorial Knot Theory", 
Turaev's book on "Quantum Invariants of Knots and 3-Manifolds", 
and the references in my papers above.