categories: Tangle, Braid... related category?

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

There's a category Braid, or Brd, whose objects are the natural
numbers and morphisms "n parallel pieces of string twisted around each 
other".  And a related one Tng where the objects distinguish
between 'string going in' and 'string going out', and strings are
allowed to double back on themselves.

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?

Yours,

Jules Bean