Categories: History of string diagrams (thread, 2017may02-...)

• 1. Pawel Sobocinski
• 2. Aleks Kissinger
• 3. Bob Coecke
• 4. John Baez
• 5. Ross Street
• 6. André Joyal
• 7. Eduardo Dubuc
• 8. Ross Duncan

If you use gmail and are a subscriber of the Categories mailing list then you may be able to access the thread below in your archives by searching for: categories diagrams after:2017/4/19 before:2017/5/18.

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.

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

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.

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

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

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

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.

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