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Stream: learning: questions

Topic: who invented string diagrams

Matteo Capucci (he/him) (Apr 11 2024 at 15:32):

I thought I knew the answer to this question but today I realized Joyal and Street only defined string diagrams for monoidal categories. What about string diagrams for 2-categories? Who defined those first?

Amar Hadzihasanovic (Apr 11 2024 at 17:33):

Perhaps @Dan Marsden would know?

Dan Marsden (Apr 11 2024 at 18:01):

Unfortunately I don't know. Ralf and I did some digging around but couldn't find a definitive answer if I remember correctly. Of course, I would love to know if somebody does understand the history.

Mike Shulman (Apr 11 2024 at 18:03):

I think the earliest occurrence I'm aware of is in some paper by Ross Street, I don't remember which offhand.

Nathanael Arkor (Apr 11 2024 at 18:33):

Street certainly mentions string diagrams for 2-categories in Categorical structures (1996). Perhaps there are earlier references.

Dan Marsden (Apr 11 2024 at 18:55):

Street also mentions them in Higher Categories, Strings, Cubes and Simplex Equations (1995), which is very marginally earlier.

David Michael Roberts (Apr 12 2024 at 00:50):

Perhaps someone can email Ross to see if he had the idea independently, or it came from someone else and/or in collaboration/discussion with others. It may well be that the Sydney/Australian CT seminar prompted it in discussions at talks.

Todd Trimble (May 24 2024 at 16:43):

I thought I remembered Ross mentioning this in his Myhill Lectures at SUNY Buffalo, which if memory serves was in the spring of 1993, but I wasn't able to find his notes online. I'll try emailing.

Todd Trimble (May 25 2024 at 10:54):

Ross responded and couldn't think of anything in print earlier than what's been mentioned so far, but thinks I'm "probably right" about the Myhill Lectures. (I think the content of Categorical Structures was in large part already in those lectures.)

Matteo Capucci (he/him) (May 25 2024 at 11:20):

Thanks for asking him!

Eric M Downes (May 25 2024 at 13:01):

It's worth noting the significant contribution of the tensor diagram notation of Penrose (1971). Many string diagram people acknowledge this, its no secret, but still good to know about.

John Baez (May 25 2024 at 13:06):

And before Penrose came Feynman diagrams! Penrose must have been influenced by these, but it took people a while to understand how Feynman diagrams are string diagrams because they're actually pictures of morphisms in categories of infinite-dimensional group representations, in a not completely obvious way, and this makes diagrams with loops diverge, giving the famous infinities of quantum field theory.

Aaron Lauda and I outlined some of the early history of string diagrams in our prehistory of n-categorical physics.

Mike Shulman (May 25 2024 at 14:56):

The original question in this channel was specifically about string diagrams for 2-categories as opposed to monoidal categories, and I think Penrose and Feynman were just the monoidal case, right?

Todd Trimble (May 25 2024 at 15:03):

FWIW, I was in Australia starting in late 1993, and I can assure everyone that interpreting string diagrams in 2-categories was a commonplace at the time. I just don't know where it was first written down.

John Baez (May 25 2024 at 21:42):

Penrose and Feynman were only drawing diagrams of morphisms in specific symmetric monoidal categories, not 2-categories - and they weren't thinking about category theory, just the examples that concerned them.

Todd Trimble (May 26 2024 at 19:42):

On a hunch, I reached out to Iain Aitchison, who worked with Ross at Macquarie in the latter half of the 80's. I asked him specifically whether this reference

• I.R.Aitchison, String diagrams for non-abeIian cocycle conditions, Louvain-la-Neuve Category Theory Conference, 1987

could serve as an early example where string diagrams were interpreted in 2-categories. He very kindly wrote back, and said about this reference

Sometime in 1985 I think was the first time this was presented, which I suppose counts as the earliest reference. Kelly’s seminar

May or may not have had records of who did/would speak ..?

The diagrams were presented specifically to describe Street’s oriental cocycle conditions, so, yes, explicitly as presenting 2-categoricial objects.

I would strongly incline to accept his say-so on this matter. So there I think you probably have it.

Todd Trimble (May 27 2024 at 02:17):

But naturally, it would be good to have something visual! An event I remember from long ago is that I was with Ross in his office, poring over some sheets with diagrams hand-drawn by Iain, that exhibited his Pascal-triangle-like recurrences for diagrams for oriented simplexes [those are Street's orientals] and oriented cubes. Iain is now living in Japan and probably doesn't have access to that (he thinks it might be in storage in Melbourne).

Calling on @Amar Hadzihasanovic: do you have knowledge of what I'm referring to, these Pascal triangle things?

Amar Hadzihasanovic (May 27 2024 at 06:51):

There is one obvious connection between simplices and the Pascal triangle.
Since the simplices are generated by iterated cones, aka joins with a point $1 \star -$, this gives the following recurrence for the number of $k$-dimensional faces of the $n$-simplex: they are either of the form $1 \star x$ for $x$ a $(k-1)$-dimensional face of the $(n-1)$-simplex, or they are inclusions of $k$-dimensional faces of the $(n-1)$-simplex into the base of the cone.

The solution to this recurrence is the binomial coefficient $\binom{n+1}{k+1}$ and the recurrence is the Pascal rule $\binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}$.

Amar Hadzihasanovic (May 27 2024 at 06:55):

There is a similar recurrence for cubes, coming from their generation as iterated cylinders $I \times -$.

However neither of these have much to do with orientation, so I can't guarantee that these are the triangles that you have seen.

Todd Trimble (May 27 2024 at 14:58):

Thanks for responding, Amar!

Todd Trimble (May 27 2024 at 14:58):

Ross kindly sent me over email cell phone photos of the string diagrams that Aitchison used for his CT 1987 talk, describing in string diagram notation the oriented simplexes and oriented cubes and the recursive rules for generating the string diagrams. They're big tables, and so the individual entries in the table look a little small and the resolution is not great. But, despite the fact that they are hard to make out, I can tell they are string diagrams at least. I might upload them here, if I can get them to be more visible/legible.

Todd Trimble (May 27 2024 at 14:58):

As for the "2-categorical" meaning: the orientals that Ross and Iain were investigating are certain strict $n$-categories freely generated from $n$-computads. Meanwhile, a strict $n$-category can be given in terms of associative composition operations $\circ_i$, composing across an $i$-cell, such that any pair of distinct $\circ_i, \circ_j$ obeys the interchange law that holds for a 2-category. I expect those are the 2-categories under discussion in Iain's account.

Todd Trimble (May 27 2024 at 14:58):

Putting this all together, I think that the answer to @Matteo Capucci (he/him)'s question is that string diagrams were interpreted in 2-categories probably as early as 1985, and certainly as early as 1987, based on the citation from Iain's email that I gave here. And Ross has the visual goods to back that up.

Matteo Capucci (he/him) (May 28 2024 at 01:36):

Amazing!

Matteo Capucci (he/him) (May 28 2024 at 01:37):

So if I understand correctly, people knew about string diagrams even before that though. In particular it seems Street knew about these?

Todd Trimble (May 28 2024 at 01:43):

Street is often becomingly modest, but he's extremely intelligent, and I can't help but think this would have occurred to him very early in the game. (Not his style to say in email "of course I knew this all along" -- he kept strictly to verifiable facts as he recalled them, and I sort of had to nudge this along...)