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

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

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.  So multiplication looks like 

*   *
 \ /
  |
  *

and comultiplication is the other way up.  The unit looks like

 |
 *

(a string coming out of nowhere); if you find this unpleasant then don't have
units or counits, in other words, take the free strict monoidal category
containing a "bisemigroup" (now there's a daft name).  Crossings could be
allowed by introducing (co)commutativity, and doubling back by introducing
duality (or nondegenerate bilinear forms, in the world of vector spaces).  

Similarly, Brd is the free braided strict monoidal category on one object,
and Tng (tangles) has a similar description (doesn't it?).

Tom