# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

1. Add a reference for string diagrams in closed monoidal categories

Anonymous

• CommentRowNumber2.
• CommentAuthora_delpeuch
• CommentTimeJul 28th 2019
Wow, there is not a single string diagram shown on the page "string diagram".
Surely it cannot be accidental - am I missing something here? I guess I will try to add some myself and see what happens…
• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeJul 29th 2019

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.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeJul 29th 2019
• (edited Jul 29th 2019)

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.

• CommentRowNumber5.
• CommentAuthora_delpeuch
• CommentTimeJul 31st 2019
• (edited Jul 31st 2019)
Sorry, I did not mean to offend the previous contributors… but for me it is a bit symptomatic of the overall style of the house.

If my additions of diagrams are welcome then I will keep doing that. I just wanted to double check that I had not missed some reason why diagrams had been left out.
• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeJul 31st 2019

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.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeJul 31st 2019

“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.)

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJul 31st 2019

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.)

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeJul 31st 2019
• (edited Jul 31st 2019)

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?

• CommentRowNumber10.
• CommentAuthorTodd_Trimble
• CommentTimeJul 31st 2019

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.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeAug 1st 2019

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.

2. Added links to PDF files of 1965 habilitation thesis of G. Hotz

Armin Reichert

• CommentRowNumber13.
• CommentAuthorTodd_Trimble
• CommentTimeNov 15th 2019

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?

• CommentRowNumber14.
• CommentAuthorMike Shulman
• CommentTimeNov 15th 2019

Perhaps that question contains its own answer? (-:O

3. Added a link to Petri net in the related concept section.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeDec 6th 2019

in the process, I have merged the References-subsection that used to be separate as “Introductions” and “Surveys”.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeDec 6th 2019

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…

• CommentRowNumber18.
• CommentAuthorTodd_Trimble
• CommentTimeDec 6th 2019

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.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeDec 7th 2019
• (edited Dec 7th 2019)

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…

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeDec 7th 2019

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”.

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

• Günter Hotz, Eine Algebraisierung des Syntheseproblems von Schaltkreisen, EIK, Bd. 1, (185-205), Bd, 2, (209-231) 1965 (part I, part II)

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:

From the point of view of monoidal category theory, string diagrams are first described (without actually being depicted, see the above comments) in

following

and in

String diagram calculus was apparently popularized by its use in

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.

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeDec 7th 2019

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.

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeDec 7th 2019

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”.

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeDec 7th 2019

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeDec 7th 2019
• (edited Dec 7th 2019)

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.

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeDec 7th 2019

added publication data for this reference:

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeDec 7th 2019
• (edited Dec 7th 2019)

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.

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeDec 24th 2019

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeJan 6th 2020
• (edited Jan 6th 2020)

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeJan 8th 2020

added to the list of examples (here) a pointer to ’t Hooft double line notation

Matteo Durante

• CommentRowNumber31.
• CommentAuthorDavid_Corfield
• CommentTimeOct 14th 2020

• George Kaye, The Graphical Language of Symmetric Traced Monoidal Categories, (arXiv:2010.06319)
• CommentRowNumber32.
• CommentAuthorDavid_Corfield
• CommentTimeOct 27th 2020

• Cole Comfort, Antonin Delpeuch, Jules Hedges, Sheet diagrams for bimonoidal categories, (arXiv:2010.13361)
5. fix reference to CDH paper (thanks for adding it!)

Antonin Delpeuch

6. Added Jamie et al.’s paper as well as corrected some error involving on proof nets

Cole

• CommentRowNumber35.
• CommentAuthorJohn Baez
• CommentTimeApr 27th 2021

Clarified that tensor networks are a special case of string diagrams.

• CommentRowNumber36.
• CommentAuthorJohn Baez
• CommentTimeApr 27th 2021

Clarified that tensor networks are a special case of string diagrams.

• CommentRowNumber37.
• CommentAuthorUrs
• CommentTimeApr 28th 2021

Then let’s at least move it up to the top of the list of examples, certainly before mentioning of bicategories et al. I have moved the paragraph to here and adjusted a little.

• CommentRowNumber38.
• CommentAuthorUrs
• CommentTimeApr 28th 2021

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)

• CommentRowNumber39.
• CommentAuthorUrs
• CommentTimeApr 28th 2021

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)

• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeMay 13th 2021

Added to the Examples-section a pointer to quantum circuit diagram.

• CommentRowNumber41.
• CommentAuthorUrs
• CommentTimeMay 18th 2021
• (edited May 18th 2021)