History of string diagrams

[Pawel Sobocinski <sobocinski@gmail.com>]

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.


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