minor addition to “variants” section

]]>added this pointer:

- Jeffrey Ellis Mandula,
*Diagrammatic techniques in group theory*, Southampton Univ. Phys. Dept. (1981) (cds:129911, pdf)

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

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)

]]>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 diagramsconstitute 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)

]]>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.

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

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

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

Cole

]]>fix reference to CDH paper (thanks for adding it!)

Antonin Delpeuch

]]>Added the reference,

- Cole Comfort, Antonin Delpeuch, Jules Hedges,
*Sheet diagrams for bimonoidal categories*, (arXiv:2010.13361)

Added

- George Kaye,
*The Graphical Language of Symmetric Traced Monoidal Categories*, (arXiv:2010.06319)

Broken link fixed.

Matteo Durante

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

added pointer to

- Predrag Cvitanović,
*Group Theory: Birdtracks, Lie’s, and Exceptional Groups*, Princeton University Press July 2008 (PUP, birdtracks.eu, pdf)

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)

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 publication data for this reference:

- Peter Selinger,
*A survey of graphical languages for monoidal categories*, in: Bob Coecke (ed.)*New Structures for Physics*, Lecture Notes in Physics, vol 813. Springer, Berlin, Heidelberg (2010) (arXiv:0908.334, doi:10.1007/978-3-642-12821-9_4)

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 and links for this one:

- Ross Street,
*Categorical structures*, in: M. Hazewinkel (ed.),*Handbook of algebra – Volume 1*, Elsevier 1996 (pdf, 978-0-444-82212-3)

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”.

]]>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.

]]>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

- 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*:

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

- Wikipedia,
*Penrose graphical notation*

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.

]]>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…

]]>