Re: Re: History of string diagrams

[Eduardo Julio Dubuc <edubuc@dm.uba.ar>]

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.

On 5/6/17 13:45, Joyal, Andr? wrote:
> 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, Andr?
>
>
>
> ________________________________________ From: Ross Street
> [ross.street@mq.edu.au] Sent: Friday, May 05, 2017 6:48 PM To: Aleks
> Kissinger Subject: categories: Re: History of string diagrams
>
> On 4 May 2017, at 1:19 AM, Aleks Kissinger
> <aleks0@gmail.com<mailto:aleks0@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
>
>
>



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