Tangle, Braid... related category?
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 20-Sep-2001 20:06:37 -0300,6821;000000000001-0000001a
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
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
On Fri, Sep 21, 2001 at 10:47:24AM -0700, baez@math.ucr.edu wrote:
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.
Thank you to everyone for the wealth of helpful answers! I appreciate that my description was not 'tight': in actual fact, there is probably more than one category I'm interested in in the family. I've followed up the references to the category 2COB (as encountered in TQFT, in Abrams' paper as well as Baez + Dolan), and that is quite similar to the category I'm describing. However, it's not quite the one I have. In 2COB, the following are equivalent (Abrams labels this relation 'F') * * * * \/ |\ | | = | \| /\ | | * * * * I suppose you might call that equation X = N . The way I've implemented my category is not as a 2-mfd, but as a 1-complex, embedded 'sensibly'. There is a distinction between some points of the boundary being the 'top', and the other points of the boundary being the 'bottom'. (Which my diagrams have been assuming). And, obviously, X and N are different as one-complexes, even though they are the deformation retracts of homeomorphic 2-mfds. (Actually, the above diagram isn't even an X, it's an X-like shape with an extended vertical section; that's a different one-complex too). I have an intuitive justification for wanting these to be different, if people aren't offended by slightly silly analogies. Think of the networks (which is what I call them) as river networks. They have to flow downhill (down the page). They can join as tributaries do, or split into distributaries. Then in the 'X' all the water has possibly mixed; we can't assume it will divide the same way. In the 'N' on the other hand, all of the water which came in on the right, has definitely gone out on the right. The other helpful lead I was given was a category (sometimes) called Vine, see Lavers [Comm. Algebra 25(4) pp1257-84], or Solomon 'A Category of Concrete Monoids' at : http://www.maths.usyd.edu.au:8000/res/Algebra/Sol/1996-07.html This is closely related to what I'm trying to do, but Vine is different in two ways. Firstly, the threads only join in Vine, never split; secondly, Vine only has morphisms from n --> n, whereas my category has morphisms from n --> m for all n and m. For example, in Vine, the morphism diagram which looks like a capital 'V' is in fact a morphism from 2 --> 2, with one node at the bottom unconnected (something like 'V.'), whereas in my category it's naturally a morphism from 2 --> 1. The principle point of uncertainty is whether or not I allow the threads to cross: this corresponds to whether some underlying monoid is commutative or not. Both possibilities are interesting. Thanks again to everyone for their help. If anyone has any further pointers to a category like the one I'm describing, I'm very interested. Yours, Jules Bean 1-Oct-2001 18:40:03 -0300,2592;000000000001-0000002a
From cat-dist@mta.ca Mon Oct 01 18:40:03 2001
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 ---------------------------------------------------------------------- 24-Sep-2001 11:36:51 -0300,1905;000000000001-00000024
John Baez wrote, concerning categories whose morphisms look like this: * * * \ / / | /\ \ / | \/ | * *
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.
Agreed on all counts. It's also interesting (to me) to consider the dual diagrams, which look something like computads (depending on exactly which version of the category above you're using); this gives different geometric intuitions. There's more about this, and higher-dimensional generalizations, in Ross Street, Categorical structures, in Handbook of Algebra I, ed. M. Hazewinkel, North-Holland, 1996, pp. 529-577. Tom 26-Sep-2001 13:10:01 -0300,1644;000000000001-00000025
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 vaguely recall these being referred to as "vines", and I think a text by Joan Birman dealt with them a little. I'm not sure as to how far these things were "categorified", though. Sorry for being so vague, Duraid 23-Sep-2001 13:22:35 -0300,2528;000000000000-00000020
participants (5)
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baez＠math.ucr.edu -
Duraid Madina -
Joachim Kock -
Jules Bean -
Tom Leinster