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. 23-Sep-2001 13:41:47 -0300,3994;000000000000-00000023