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September 24, 2007

The Catsters Strike Again: “String Diagrams”

Posted by John Baez

You may have thought the Catsters was a funny stage name for Eugenia Cheng… but now the other Catster, Simon Willerton, steps in front of the camera and demonstrates his star power: This guy is full of energy. Watch how he literally bursts onstage at the beginning of the second act!
Category theory is great stuff, but it can easily seem ‘too abstract’ before you see how much of it can be done by wiggling around pieces of string! I tried to explain this back in the early days of the Quantum Gravity Seminar — in track 1 of the Fall 2000 and Winter 2001 notes. But, for explaining how to do math with pictures, videos are better.
Posted at September 24, 2007 9:29 PM UTC

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Re: The Catsters Strike Again: “String Diagrams”

Category theory is great stuff, but it can easily seem ‘too abstract’ before you see how much of it can be done by wiggling around pieces of string!

You know, it sometimes bothers me when people complain that category theory is too abstract – to me the modes of reasoning seem no more or less abstract than a lot of what goes on in algebra. But perhaps it’s the perceived lack of geometric “hooks” that explains people’s discomfort – the sort of thing that makes string diagrams come as such a relief.

The yanking moves associated with triangular identities of an adjunction are just the tip of an iceberg: they are the lowest-dimensional examples of cancellation of critical points in Morse theory. Interchange equalities give rearrangements of such critical points. There is a general yoga of cancellation and rearrangement of critical points in Morse theory (used in proofs of the h-cobordism theorem, among other things), and it seems to me it would be desirable to give a full n-categorical account of this yoga. Surely this sort of thing has been considered by many people, but has there been serious work on this? (There’s been some work in low dimensions, and surely this sort of thing is implicit in the cobordism conjecture, but I don’t know of much beyond that.)

Posted by: Todd Trimble on September 25, 2007 2:07 AM | Permalink | Reply to this

Re: The Catsters Strike Again: “String Diagrams”

I think it’s also seen as “too abstract” because for the longest time categories weren’t seen as algebraic structures on the level of groups and rings and fields. They were used to study those algebraic structures, and so were thought to be “even higher” sorts of structures.

Posted by: John Armstrong on September 25, 2007 3:10 AM | Permalink | Reply to this

Re: The Catsters Strike Again: “String Diagrams”

Todd wrote:

You know, it sometimes bothers me when people complain that category theory is too abstract – to me the modes of reasoning seem no more or less abstract than a lot of what goes on in algebra.

It bothers me too. I think John Armstrong gave a good explanation: a category like the category of all groups seems like a much bigger, scarier thing to comprehend as a totality than a single group. One remedy is to place more emphasis on examples of categories that are small enough to draw on the blackboard — categories with just a few objects and a few morphisms.

Sometimes when giving a talk to non-category-theorists I draw such a category and say “Look: it’s just a few dots and arrows, and a rule for composing arrows. If you think this is ‘too abstract’, you really shouldn’t be doing mathematics.” That usually gets a laugh.

Of course, dots and arrows also provide a ‘geometric hook’ for the visualizers in the crowd… as do string diagrams.

Posted by: John Baez on September 25, 2007 3:58 AM | Permalink | Reply to this

Re: The Catsters Strike Again: “String Diagrams”

This is exactly why I’m running my seminar on “Applications of Category Theory” the way I am. These first few weeks I throw out just enough theory of categories qua categories to get our hands dirty, and then move directly to applications to semantics, logic, foundations, physics, universal algebra, and (of course) knot theory.

If we get as many of these nice “concrete” (as opposed to concrete) categories in students’ hands as soon as possible, maybe we can convince the next generation that these things are really useful, and not just abstract nonsense.

Posted by: John Armstrong on September 25, 2007 4:31 AM | Permalink | Reply to this

Re: The Catsters Strike Again: “String Diagrams”

Todd, is this an appropriate place to ask about your paper with Mcintyre? I don’t seem to be able to get hold of it anywhere. Do you do geometric higher categories there?

Posted by: Simon Willerton on September 25, 2007 12:08 PM | Permalink | Reply to this

Re: The Catsters Strike Again: “String Diagrams”

I think it’s a perfectly appropriate time and place to ask, Simon, especially since I myself brought up this business of rearrangement and cancellation of critical points! In fact, I owe you an apology: you did ask me before about this. Some other people have asked, too.

The last working title for the paper was “The Geometry of Gray Categories”, and had as its main aim the development of surface diagrams, with emphasis on progressive surface diagrams in the 3-cube, as a means to constructing the free Gray category generated from a Gray computad. The generating 3-cells of the computad are represented by “coning together” pairs of string diagrams (assumed to be in generic position in order to properly account for Gray interchange), and the 3-cells of the free Gray-category are isotopy rel boundary classes of surface diagrams appropriately labeled in the Gray computad. As you can see, this was supposed to be analogous to the progressive string diagrams of Joyal and Street, but raised by one categorical dimension.

The paper (such as it is) is in a somewhat scattered and incomplete state: at the first round of writing it (1996-1997!), there were technical difficulties that had not been properly addressed. It’s clear that surface diagrams in dimension 3 and beyond should be certain stratified spaces, but the critical questions about which geometric category to work in (e.g., topological, PL, or smooth, etc.), and how to circumvent noisome pathologies in a satisfactory way, sort of bogged down the project for a long time.

I guess around 1999 or 2000 I began to take seriously “tame geometry”, as developed by model theorists who work on so-called “o-minimal structures”, as the proper way to take care of issues of pathology. To deal properly with isotopy rel boundary, one of the things I needed was a version of Thom’s first isotopy lemma for stratified spaces that could be proven in the o-minimal framework. It was only in the last few years (after reading around and consulting some experts) that I was convinced that all this could indeed be carried out. But by then, the project had languished for a long time, and so far I haven’t gotten back into it to bring it to completion. Too many distractions…

Thanks for asking. If enough people hold my feet to the fire, I might finish the damned thing once and for all!

Posted by: Todd Trimble on September 25, 2007 6:23 PM | Permalink | Reply to this

Re: The Catsters Strike Again: “String Diagrams”

I would like to add some fuel to the fire. I noticed a reference to this paper `to appear in Advances in Math.’ on Ross Street’s web page a while back. I have been wanting to have a look at it ever since.

Posted by: Aaron Lauda on September 25, 2007 9:17 PM | Permalink | Reply to this

Re: The Catsters Strike Again: “String Diagrams”

it might be me, but it seems Natural transformations 1 and Open-Closed Cobordism 1, both by the brilliant miss Cheng seem to be missing here.

Posted by: ericv on September 25, 2007 9:18 PM | Permalink | Reply to this

Re: The Catsters Strike Again: “String Diagrams”

He literally bursts? I can’t wait to see it!

Posted by: Tom Leinster on September 26, 2007 2:08 PM | Permalink | Reply to this

Re: The Catsters Strike Again: “String Diagrams”

He literally bursts? I can’t wait to see it!

That makes two of us!

Posted by: Bruce Bartlett on September 26, 2007 10:25 PM | Permalink | Reply to this

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