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There is an idea I've recently gotten interested in that doesn't seem to have a good agreed-upon name ("diagrammatic algebra?"). It centers around the use of two-dimensional diagrams of dots, possibly boxes, and arrows, and is related to (in no particular order) knot theory, braided monoidal categories, quantum groups and Hopf algebras, subfactors, planar algebras, and (topological) quantum field theory. However, it also has a more accessible aspect: it can be used as an elegant notation for working with $\text{Vect}$ (a particularly ubiquitous braided monoidal category; see question #6139), and at least one textbook has used a variant of it to develop the basics of Lie theory. There is also John Baez's Physics, Topology, Logic, and Computation: a Rosetta Stone, and another accessible introduction to some of these ideas is Kock's Frobenius Algebras and 2D Topological Quantum Field Theories. These ideas have also been used to understand quantum mechanics.

This is all pretty fascinating to me. These are elegant and beautiful ideas, and it seems to me that they are badly in need of unification and accessible exposition (something like Selinger's A survey of graphical languages for monoidal categories, but maybe with a more historical and/or expository bent). Beyond Baez's paper, does anyone know of any resources like that? Where can I learn more about what you can do with these diagrams that doesn't necessarily require a lot of background?

Related: how should I TeX these diagrams?

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8 Answers 8

I find the area your question covers very interesting and am looking forward to seeing what answers people come up with. I thought I would supplement my rather minimal answer from a few days ago. I'm not really providing any work that links all these ideas together, rather I'm giving some useful references for your quest.

  • Peter Freyd and David Yetter (1989). "Braided Compact Closed Categories with Applications to Low-Dimensional Topology". Advances in Mathematics 77: 156–182.

  • A. Joyal and R. Street, The geometry of tensor calculus, Advances in Math. 88 (1991) 55-112

  • André Joyal and Ross Street. "The Geometry of Tensor calculus II". Synthese Lib 259: 29–68.

  • André Joyal, Ross Street, Dominic Verity (1996). "Traced monoidal categories". Mathematical Proceedings of the Cambridge Philosophical Society 3: 447–468.

  • In fact, many things by Ross Street

  • Functorial Knot Theory: Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants (Series on knots & everything) by David N. Yetter (Author)

  • Many works by Bob Coecke and Samson Abramsky (eg. Temperley-Lieb algebra: From knot theory to logic and computation via quantum mechanics) are certainly accessible.

  • The Catsters youtube channel, especially the presentations on string diagrams

  • Functorial boxes in string diagrams by Paul-Andre Mellies Invited talk at the Computer Science Logic 2006 conference in Szeged, Hungary. Lecture Notes in Computer Science 4207, Springer Verlag.

  • Masahito Hasegawa, Martin Hofmann and Gordon Plotkin Finite Dimensional Vector Spaces are Complete for Traced Symmetric Monoidal Categories

Mostly these reference cover the Tensor/Monoidal Category line. I've certainly seen other work, but am less familiar with it.

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I recall now receiving a preprint "Traced monoidal categories" from Ross Street which I foolishly put in the recycling when I moved countries. Now I see it cited all over the place. –  supercooldave Jun 3 '10 at 11:16
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For LaTeXing the diagrams, Tikz is extremely versatile and usable. I'd start there.

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Discussion here: tex.stackexchange.com/questions/97507/… –  Ben Crowell Feb 10 '13 at 19:54
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I'll echo supercooldave's vote for TikZ.

For applications to knot theory, Kassel's longer book is great, as is his short book with Rosso and Turaev on quantum groups and knots. Kauffman's book Knots and Physics also has some nice parts.

You should be sure to read Penrose's original paper where he introduced the notation: R. Penrose. Applications of Negative Dimensional Tensors. Combinatorial mathematics and its applications. D.J.A. Welsh, ed. Mathematical Institute, Oxford, Academic Press, London, 1971. pp. 221-244.

Of course, for (linear) algebra, there's Cvitanovic's book. I call it a "linear algebra" book in the same way that the right way to understand much of Lie algebra theory is as graduate level linear algebra.

I think that Joyal and Street are the correct names for proving that planar diagrams exactly capture monoidal category theory. So diagrammatic languages are not just for Vect, but for almost any category.

To make the last comment more precise, I recently went to a talk by Bruno Vallette on "properads". One of his points was that such things (a blend between props and operads) on the one hand can basically capture any algebraic structure that you like, and on the other hand are naturally described with these graphs and diagrams.

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I've been working on a GUI for typesetting tensor/monoidal diagrams in TikZ.

http://tikzit.sourceforge.net/

It's especially geared at applications to quantum mechanics, namely "dot"-style diagrams of Frobenius algebras for complementary observables (Coecke, Duncan, arXiv:0906.4725) and entangled states (Coecke, me, arXiv:1002.2540).

Given Dave's already quite extensive list of what's out there on the monoidal side of things, I can only really refine what he's said.

Bob Coecke's short book (or long paper :-P) "Categories for the Practising Physicist" gives a pretty gentle buildup from physical principals, through Dirac notation for QM, to graphical notation, explaining some of the intuitions along the way.

http://web.comlab.ox.ac.uk/people/Bob.Coecke/ctfwp1_final.pdf

I found Ross Street's slides on Frobenius algebras to be a quick and easy (though sketchy) intro to the topic:

http://www.maths.mq.edu.au/~street/FAMC.pdf

The contemporary paper (Street. Frobenius monads and pseudomonoids. J. Math. Phys. (2004) vol. 45 (10) p. 3930) is very good, but considerably more technical, as it works in the language of higher categories.

** edit

I just realised that John's paper, "A Prehistory of n-Categorical Physics" hasn't been mentioned. This one puts the whole monoidal/graphical physics thing in a historical context starting from Maxwell and going through Feynman, Penrose, Mac Lane, Joyal, and all the other usual suspects. This is a long one, but it seems quite comprehensive.

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Just downloaded tikzit. Very cool. –  supercooldave May 20 '10 at 16:08
    
Cool! If you find any bugs (and I'm sure there are plenty), I'd be much obliged if you send 'em my way. –  Aleks Kissinger May 20 '10 at 16:18
    
Thanks for the links! These all look great. –  Qiaochu Yuan May 21 '10 at 5:40
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Part III of Kassel's "Quantum Groups" is an almost completely self-contained introduction to monoidal categories and knot theory. Diagrams of the sort you are describing play an increasing role throughout. I taught a course based of Part III this past term, and really enjoyed it.

Part III assumes some sophistication on the part of the reader, but the specific requirements are pretty slight. In particular, you definitely don't need to have read the first two parts of the book, and you don't need to know what a quantum group is. (On the other hand, if you want to know what a quantum group is, maybe you would want to read the whole book. I've put much less effort into understanding parts I and II, and haven't looked at part IV, so I'm not going to comment on how successful those parts are.)

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I'm not sure whether this write-up adds anything on top of the existing resources, but Jim Blinn, a graphics researcher at Microsoft Research has written up some course notes on tensor diagrams:

http://research.microsoft.com/apps/pubs/default.aspx?id=79791

Update: I suspected that these diagrams might also be lurking in Bayesian networks and other kinds of graphical models used in machine learning. Although I'm not 100% sure about that, it looks like Dan Piponi (aka. sigfpe from "A Neighborhood of Infinity") mentions Bayesian networks as a possible connection during a talk at the 2009 International Conference on Functional Programming.

http://vimeo.com/6590617

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Here is a recent talk by Micah McCurdy on how to extend string diagrams to express (monoidal) functors, and monads.

Since monads and functors are ubiquitous in computer science (e.g. denotational semantics, functional programming etc.) McCurdy's work has immediate practical applications.

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Other sources of possible interest

Redfield, J.H. (1927), "The Theory of Group-Reduced Distributions", Amer. J. Math. 49, 433–455.

Roberts, D.D. (1973), The Existential Graphs of Charles S. Peirce, Mouton, The Hague.

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