Categories: History of string diagrams (thread, 2017may02-...)Quick index:
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1. Pawel Sobocinski
From: Pawel Sobocinski Date: Tue, 2 May 2017 15:50:06 +0100 Subject: categories: History of string diagrams Dear Categorists, I would like to ask for comments about the history of string diagrams as graphical notation for the arrows of higher and monoidal categories. For the sake of precision, I mean the (various kinds of) graphical notation where there is a "dimension flip", i.e. given a (weak) n-category, the n-cells are drawn as points (0-dimension), the n-1 cells as lines (1-dimension) etc. This includes, as a special case, string diagrams as notation for the arrows of symmetric monoidal categories (Joyal and Street), which have found a number of applications (quantum mechanics, computer science, engineering, linguistics, ...) in recent years. We now also have impressive online tools, such as Jamie Vicary's Globular, that allow both type-setting and computing with string diagrams. It seems to me that there aren't very many historical notes available: Peter Selinger's "A survey of graphical languages for monoidal categories" is a nice survey but it's quite terse on the historical aspects. In the historical notes that I've come across, string diagrams are often mentioned in the same breath with Penrose tensor diagrams, Feynman diagrams, and proof nets, but while there are of course similarities, there are also clear differences owing to the categorical nature of string diagrams; for example, string diagrams are usually quite strictly "typed" with domain and codomain determined by dangling wires in the case of monoidal categories (or, in higher dimensions, surfaces). I'm interested in the history of the use of the notation, as well as the surrounding "sociological" aspects. Through overheard gossip, I believe that the notation was a quasi-secret "house style" in some groups, used for calculations, but carefully exided from formal publications. But maybe this is a bit overblown, and the printing technology simply wasn't there? Or were there particularly conservative editors who were not comfortable with publishing diagrammatic calculations? In any case, it seems strange that we have had to wait until the 1990s for this notation to actually start making it into papers. Many calculations in earlier works were quite clearly worked out with string diagrams, then painstakingly copied into equations. Sometimes, clearly graphical structures were described in some detail without actually being drawn: e.g. the construction of free compact closed categories in Kelly and Laplazas 1980 "Coherence for compact closed categories". From personal experience, some papers become much more readable after being redrawn into almost comic books: Carboni and Walters' 1987 "Cartesian bicategories I" comes to mind. I'm reminded of quote by E.J. Aiton from his biography of Leibniz (which I came across in Peter Gabriel's Matrices, géométrie, algèbre linéaire): "Owing to the reluctance of printers to accept books on mathematics, because of the difficulties of type-setting and the small number of potential readers, the statement of results in letters, especially when these were registered in the Royal Society or the Paris Academy, provided a means of establishing a claim to invention, rending possible publication at a later date. The most precious possessions of a mathematician were, of course, the original methods by which new results could be obtained. While communicating results, in order to establish his possession of a general method, to which he might refer in impenetrably opaque terms, he took pains to eliminate any dues that would enable his correspondent to guess the method..." I'd appreciate any comments -- both personal and more summative. I'll be happy to compile any information sent to me personally, or to the list, and make it available online. I'm especially interested in: * Who came up with the notation? When was it first used? Was it rediscovered independently by several groups? * Was there an effort to keep it a "house secret"? * Was there any institutional resistance to the use/publishing of string diagrams? Finally, I'd like to take the opportunity to advertise the 1st Workshop on String Diagrams in Computation, Logic, and Physics, which I'm organising with Aleks Kissinger, and which will take place at the Jericho Tavern in Oxford, September 8-9, 2017. More information is available at http://string2017.cs.ru.nl, and we will soon send out a formal call for papers. Best wishes, Pawel.
2. Aleks Kissinger
From: Aleks Kissinger Date: Wed, 3 May 2017 17:19:34 +0200 A short note: This idea that string diagrams are, due to technical issues, only useful for private calculation, is said explicitly by Penrose. Penrose and Rindler's book "Spinsors and Spacetime" (CUP 1984) has an 11-page appendix full of all sorts of beautiful, carefully hand-drawn graphical notation for tensors and various operations on them (e.g. anti-symmetrization and covariant derivative). On the second page, he says the following: "The notation has been found very useful in practice as it grealy simplifies the appearance of complicated tensor or spinor equations, the various interrelations expressed being discernable at a glance. Unfortunately the notation seems to be of value mainly for private calculations because it cannot be printed in the normal way." Best, Aleks
3. Bob Coecke
From: Bob Coecke Date: Thu, 4 May 2017 13:20:08 +0100 Hi Pawel, we definitly have had plenty of experiences at the quantum computing end making it clear that diagrams reduce the respect when preducing actual results, and that translating to the Hilbert space model, which causes loosing generality, actually increases likeliness of acceptance. I also felt that in Theoretical Computer Science symbolic categories also improve the acceptance likeliness as compared to diagrams, although this may have been changing recently in part thanks to your efforts. That said, a lot of people do appreciate the idea of an entirely diagrammatic formalisms, in particular researchers with a multi-disciplinary tendency, and in quantum foundations a diagrammatic default for crafting theories is in the process of becoming the standard, so there they have been very succesful, artly thanks to leading researchers (without a category theory background) such as Lucien Hardy picking them up.
4. John Baez
From: John Baez Date: Thu, 4 May 2017 10:20:41 +0800 HI - Pavel wrote: > I would like to ask for comments about the history of string diagrams as > graphical notation for the arrows of higher and monoidal categories. There's a lot of history in this paper: John C. Baez and Aaron D. Lauda, A prehistory of n-categorical physics, in Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, ed. Hans Halvorson, Cambridge U. Press, Cambridge, 2011, pp. 13-128. Also at https://arxiv.org/abs/0908.2469 The most important people in the early history of string diagrams are Feynman and Penrose, and I give references and a discussion of their key papers. Some of Penrose's papers are a bit hard to find, but he gave me permission to put them here: http://math.ucr.edu/home/baez/penrose/ His approach to constructing space from spin networks (a certain kind of string diagrams) later became part of loop quantum gravity, and I give the story of how that happened, along with the more mathematical side of the story involving the Jones polynomial, the work of Joyal and Street, etc. Best, jb
5. Ross Street
From: Ross Street Date: Fri, 5 May 2017 22:48:41 +0000 On 4 May 2017, at 1:19 AM, Aleks Kissinger <...@gmail.com> wrote: A short note: This idea that string diagrams are, due to technical issues, only useful for private calculation, is said explicitly by Penrose. Penrose and Rindler's book "Spinsors and Spacetime" (CUP 1984) has an 11-page appendix full of all sorts of beautiful, carefully hand-drawn graphical notation for tensors and various operations on them (e.g. anti-symmetrization and covariant derivative). Some random comments: The person who told me of the Penrose-Rindler reference and the earlier R. PENROSE, Applications of negative dimensional tensors, in ``Combinatorial Mathematics and its Applications,'' (D.J.A. Welsh, Ed., Academic Press, 1971) 221--244 was Iain Aitchison who found a coloured string-diagram Pascal-triangle-like algorithm for producing the n-cocycle condition arising from the orientals and their cubical analogues. While Iain's more recent The geometry of oriented cubes, arXiv:1008.1714v1 [math.CT] has incredible diagrams in comparison with 1984 technology, the string versions are not there. Speaking of Roger Penrose, Max Kelly used to tell the following story about their time (mid 1950s) in Cambridge. Max thought Roger must be very visually impaired. Two reasons: 1. When Max first met him he was wearing very thick glasses. It turned out Roger was conducting an experiment to test whether one would adapt to wearing lenses that inverted the world. After a few days apparently the brain adjusts and it believes everything is the right way up. 2. Looking over Roger's shoulder on lectures using tensors, Max noticed that Penrose was not using the usual notation at all. He was using the string notation instead. When Max asked why, Roger said that all the i_1, j_2, 1_1, . . . sub- and super-scripts were impossible to read, whereas the connecting strings made it clear. Who knows what lies in one's subconscience! However, I think the string notation Max used when talking about his work with Eilenberg on extraordinary natural transformations (not the more general Set-based dinatural transformations Dubuc and I wrote about) arose quite independently of Max's Penrose experience. Sometimes when Graeme Segal was in Sydney, I was around while he and Max discussed comparisons of the Eilenberg-Kelly string diagrams (which do not appear in their paper: A generalization of the functorial calculus, Jour. Algebra 3 (1966) 366--375) and string diagrams in physics. Best wishes, Ross
6. André Joyal
From: Joyal, André Date: Sat, 6 May 2017 16:45:02 +0000 Some more comments. I always imagined that Penrose was inspired by Feynman's diagrams, but Max's story is casting doubts on this idea; Penrose may have been chiefly concerned with the syntax of tensor calculus. Let me point out that Eduardo Dubuc invented an "elevator calculus" in the early 70's which is a form of the string diagram notation. It was never published. Best regards, André
7. Eduardo Dubuc
From: Eduardo Julio Dubuc Date: Sun, 7 May 2017 16:03:24 -0300 Thank you Andre for pointing out to my elevator calculus, that I actually invented in 1968 - 1969 while working in my Thesis (published as SLN 145). The same thing happened with elevators that with string diagrams, when I suggested to Mac Lane to write the thesis with the elevators he said to me that elevators were fine for private calculations, but not for publishing. He gave me two reasons: One was that notations were important things, and to introduce a new notation you should be an established mathematician. The second, more important, that he considered unquestionable, was that printers will not accept manuscripts with elevators. I continuously used elevator calculus instead of diagrams in order to find proofs of equations in tensor categories. Or instead of pasting diagrams when calculating in 2-categories (1-arrows in the role of objects, composition in the role of tensor, and 2-cells in the role of arrows). But translate to diagrams or pasting diagrams for publishing. With augmented experience with LateX, I and my students started to publish with elevators. For those that may be curious or interested, here are three links where elevators are explained and used: https://arxiv.org/abs/1406.5762v1 https://arxiv.org/abs/1110.6411v2 https://arxiv.org/abs/1110.5293 Best regards, Eduardo.
8. Ross Duncan
From: Ross Duncan Date: Mon, 15 May 2017 16:04:12 +0100 Sorry to arrive so late to the party. The following may be wrong, because I am working from memory alone. Regarding this: On 2 May 2017, at 15:50, Pawel Sobocinski <...@gmail.com> wrote: > Sometimes, clearly graphical structures were described in some > detail without actually being drawn: e.g. the construction of free > compact closed categories in Kelly and Laplazas 1980 "Coherence for > compact closed categories". Indeed the diagrams which are the core of that paper have been bafflingly excised. However if you take a look at: G. Kelly. Many-variable functorial calculus I. In G. Kelly, M. Laplaza, G. Lewis, and M. L. S., editors, Coherence in Categories, volume 281 of Lecture Notes in Mathematics, pages 66-105. Springer, 1972. DOI : 10.1007/BFb0059556 You'll find the hand-drawn diagrams still intact. (The volume that this paper appeared in a treasure trove of good stuff, so I was very pleased to discover it is now available electronically - http://link.springer.com/book/10.1007/BFb0059553 ) -r