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

[Joachim Kock <kock@math.unice.fr>]

Jules Bean:

[...]
>  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:
>
>   *   *   *
>    \ /   /
>     |   /\
>     \  /  |
>      \/   |
>       *   *
>
>  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?

Tom Leinster:

>  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).


Once you have units and counits you automatically get duality (doubling
back): just compose the multiplication with the counit:

   *   *       *  *
    \ /         \/
     |     =
     *
     |


To complement Tom's good description with some more names: With crossings
(commutativity), we've got the skeleton of the category 2COB (objects:
compact oriented 1-manifolds, arrows: (diffeomorphism classes of)
2-cobordisms).  In the drawings, the 'particles' are then replaced by
'closed strings'; we get those 'pair-of-pants' for the (co)multiplication,
and 'caps' for (co)unit.  The representations of 2COB are called 2D
topological quantum field theories, and the category of those is equivalent
to the category of (commutative) Frobenius algebras.

A detailed reference for this is

@article{Abrams:tqft,
    author =  {Lowell Abrams},
    title =   {Two-dimensional topological quantum field theories
              and Frobenius algebras},
    journal = {J.~Knot Theory and its Ramifications},
    volume =  5,
    year =    1996,
    pages =   {569--587},
}

(available on his home page, I think.)

Cheers,
Joachim.

----------------------------------------------------------------------
Joachim KOCK
Laboratoire de Mathématiques J.A.Dieudonné    Tél.  +33 04.92.07.62.40
Université de Nice Sophia-Antipolis           Fax   +33 04.93.51.79.74
Parc Valrose - 06108 Nice cédex 2 - FRANCE    Mél.  kock@math.unice.fr
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