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I expect this is because it’s only very recently that functionality like tikz was available on the nlab for drawing string diagrams. Before that they would have had to be included images or svg, and probably no one wanted to put in the effort.
Wow, why is someone surprised that a wiki constructed by a very few guys in their spare time doesn’t have every desirable feature?
The only “accidental” thing is that there are so few people who can be bothered to contribute.
No, there is no reason that diagrams have been left out. It’s just a question of the changing interests and energy levels of the very few people here, and the functionality available.
I can’t imagine what you took a lack of string diagrams to be a symptom of. Mike Shulman has publications using them, such as Traces in symmetric monoidal categories. So do I for that matter, Ch. 10 of my book.
I see you added one at double category. Keep it up. Plenty of opportunities obviously on this page, string diagram.
“Style of the house.” Sigh.
There is no house style. Each contributor has his/her own style of writing. (There are a few conventions, but they do not dictate style.)
I don’t think we should be surprised that people think there is a “house style”, or that decisions about what or what not to include were made in some centralized or intentional way, or that by adding something to scratch their own itch they might be violating some norms or expectations. The idea of a truly decentralized wiki is still alien to the way people are used to thinking about the way things are written and created. Most people’s experience with wikis is limited to wikipedia, which is nowadays much more centralized and controlled than we are, and probably looks even more so than it is to the casual reader.
(I’m not necessarily claiming that any of this applies to any particular person such as a_delpeuch, just responding to the general feeling of frustration with this misconception that David and Todd are expressing — which I do share, but I’m more or less resigned to it by now.)
Some of my annoyance is a carry-over from reading some ill-informed comments on Twitter. Some people seem to think that unless an entry is swimming in $(\infty, 1)$-tags that we don’t want to know. No doubt it would be better for me not to read this, but it has made me aware that there are people under the banner of ’Applied Category Theory’ who have the impression that it will require a radical shift to have their ideas represented here.
While some scepticism was expressed about the ACT-movement in the corresponding discussion, the one explicitly anti-ACT view aired on the nForum that I know about was given anonymously by ’Guest’ at game theory.
Ah, now I see that a_delpeuch participated in the discussion Applied Category Theory on the nLab. The idea that we’re ultra-abstract and unapplied is in the air.
Can’t people just come along with a modicum of humility, find out a little bit about how things work around here, and then get on with providing some content?
I’m tempted to start writing “Myths About the nLab”, but I’ll hold off on that.
IIRC, Urs said something in the other thread (linked to by David C in the last comment) about not letting all these meta-thoughts about the nLab get in the way. I agree. There’s a lot to be done, so if you have something useful you’d like to add, please just go ahead. Personally, I’m really glad that a_delpeuch is putting in string diagrams. We need them, and needed them. The lack had nothing to do with any supposed philosophical or stylistic opposition to them.
All very odd when you consider that of the authors of the last three comments, two have done important mathematical/logical work with string diagrams and the other has discussed their philosophical importance.
Re #12: very interesting! I’ve only just taken a quick glance, because German is a significant obstacle for me in reading mathematics, but why is this history not better known?
Perhaps that question contains its own answer? (-:O
added pointer to
in the process, I have merged the References-subsection that used to be separate as “Introductions” and “Surveys”.
There is an old query box in the entry, essentially challenging the truth of the assertion that Kelly-Laplaza’s articles speaks about string diagrams at all:
Where in Kelly-Laplaza do string diagrams appear? I can’t find any picture of a string diagram in the paper. Perhaps they are described somewhere in the text, but I can’t see it.
This should be easy to settle…
Although it seems the revision where the claim first appears was made by Urs, I might have had something to do with the spread of such a rumor. Anyway, if someone looked and didn’t see it, I’m sure the claim is false. I don’t have the article to hand.
Thanks for checking. I don’t remember having made that edit, but apparently then I have. Certainly I never looked at that book.
So now I did a search for “history of string diagrams” and found this helpful message by Pawel Sobocinski, where it seems to clarify what’s going on:
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”.
Will edit the entry accordingly…
so I added a bunch of references, and missing publication data for previously existing references, to the section References – Original articles.
Now is starts out as follows (there was and is overlap with the subsequent section References – Details, which I have not tried to deal with):
The development and use of string diagram calculus pre-dates its graphical appearance in print, due to the difficulty of printing non-text elements at the time.
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”.
(Pawel Sobocinski, 2 May 2017)
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 “Spinors 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.”
The first formal definition of string diagrams in the literature appears to be in
Application of string diagrams to tensor-calculus in mathematical physics (hence for the case that the ambient monoidal category is that of finite dimensional vector spaces equipped with the tensor product of vector spaces) was propagated by Roger Penrose, whence physicists know string diagrams as Penrose notation for tensor calculus:
Roger Penrose, Applications of negative dimensional tensors, Combinatorial Mathematics and its Applications, Academic Press (1971) (pdf)
Roger Penrose, Angular momentum: An approach to combinatorial spacetime, in Ted Bastin (ed.) Quantum Theory and Beyond, Cambridge University Press (1971), pp.151-180 (PenroseAngularMomentum71.pdf:file)
Roger Penrose, On the nature of quantum geometry, in: J. Klauder (ed.) _Magic Without Magic, Freeman, San Francisco, 1972, pp. 333–354 (spire:74082, PenroseQuantumGeometry.pdf:file)
Roger Penrose, Wolfgang Rindler, appendix (p. 424-434) of: Spinors and space-time – Volume 1: Two-spinor calculus and relativistic fields, Cambridge University Press 1984 (doi:10.1017/CBO9780511564048)
See also
From the point of view of monoidal category theory, string diagrams are first described (without actually being depicted, see the above comments) in
{#KellyLaplaza80} Max Kelly, M. L. Laplaza, Coherence for compact closed categories. Journal of Pure and Applied Algebra, 19:193–213, 1980 (doi:10.1016/0022-4049(80)90101-2, pdf)
(proving the coherence for compact closed categories)
following
{#Kelly72} Max Kelly, Many-variable functorial calculus I In: In: Max Kelly, M. Laplaza , L. Gaunce Lewis, Jr., Saunders Mac Lane (eds.) Coherence in Categories Lecture Notes in Mathematics, vol 281. Springer, Berlin, Heidelberg 1972 (doi:10.1007/BFb0059556)
(which does include the hand-drawn diagrams that are missing in Kelly-Laplaza 80!)
and in
André Joyal, Ross Street, The geometry of tensor calculus I, Advances in Math. 88 (1991) 55-112; MR92d:18011 (pdf, doi:10.1016/0001-8708(91)90003-P)
André Joyal and Ross Street, The geometry of tensor calculus II (pdf)
String diagram calculus was apparently popularized by its use in
Louis Kauffman, Knots and physics, Series on Knots and Everything, Volume 1, World Scientific, 1991 (doi:10.1142/1116)
(in the context of knot theory)
Probably David Yetter was the first (at least in public) to write string diagrams with “coupons” (a term used by Nicolai Reshitikhin and Turaev a few months later) to represent maps which are not inherent in the (braided or symmetric compact closed) monoidal structure.
If anyone has a reference that would go with “Probably David Yetter was the first…” that would be good to add.
Something I’ve been meaning to ask with Urs drawing these diagrams of late is where Cvitanovic’s bird tracks fit inside the family of diagrammatic notation.
Might you have a more specific link? I have been looking around there for a bit, but still haven’t seen any discussion of “bird tracks”.
added publication data and links for this one:
Okay, I found those birdtracks:
one needs to
1) go to birdtracks.eu
2) then choose “webbook” from the menu on the left,
3) then click on the words “hyperlinked pdf” (which is not self-evident, as it’s not underlined)
4) then (one does not get a pdf but) one has to wait for the web display to load…
5) then finally scroll forward to page 8.
There is an “Example” which is actually where the “birdtracks” seem to be defined, and, at least on this and the following pages, they are just the standard Penrose/string diagram notation for tensor calculus in $(FinVect, \otimes)$.
This should be added to the entry on string diagrams. But I won’t do it.
added publication data for this reference:
started an Examples-section (here)
For the moment it contains nothing but pointers to entries on Lie theory that show some string diagrammatics.
But if any entry deserves a good supply of graphical examples, it is this one here, and so maybe a stub entry named “Examples” reminds/motivates someone to add such.
added these pointers:
Jacob Biamonte, Ville Bergholm, Tensor Networks in a Nutshell, Contemporary Physics (arxiv:1708.00006)
Jacob Biamonte, Lectures on Quantum Tensor Networks (arXiv:1912.10049)
added pointer to
added to the list of examples (here) a pointer to ’t Hooft double line notation
Added
Added the reference,
I ended up editing the Idea section and re-instantiating pointer to Penrose notation right there at the beginning. I think this is what many readers who don’t already know about categories, let alone monoidal categories, will need to hear first to get, as it were, the idea of the subject. Also, this is the honest attribution of the idea (it’s easy to take any grand idea and generalize its context a tad more, and it’s close to trivial if the grand idea was all about abstracting away from its context in the first place!).
Now the idea-section starts out as follows, which should hopefully be uncontroversial:
String diagrams constitute a graphical calculus for expressing operations in monoidal categories. In the archetypical cases of the Cartesian monoidal category of finite sets this is Hotz’s notation (Hotz 65) for automata, while for finite-dimensional vector spaces with their usual tensor product this is Penrose’s notation (Penrose 71a, Penrose-Rindler 84) for tensor networks; but the same idea immediately applies more generally to any other monoidal category and yet more generally to bicategories, etc.
Also, I added captures of three figures from three original articles (Hotz, Penrose, Penrose-Rindler) flowing alongside the text. (All in the Idea-section here)
Finally, I took the liberty of reverting the renaming of the section title “Variants” (which John had changed to “Variants and Examples”): There is a section “Examples” right below, and there is a good point not to mix up variants of the general theory with examples.
So, instead, I now moved the example that John had silently added, namely “spin networks” to a new subsection “Examples – In representation theory” (here)
Added to the Examples-section a pointer to quantum circuit diagram.
added this pointer:
added pointer to today’s
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