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 21-Sep-2001 17:54:59 -0300,3285;000000000001-0000001d