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Jules Hedges (Oct 23 2020 at 10:18):A short (6 pages) paper on the foundations of string diagrams: https://arxiv.org/abs/2010.11582 - "Remark on topological nature of upward planarity" by Xuexing Lu
Jules Hedges (Oct 23 2020 at 10:18):Abstract: "The notion of an upward plane graph in graph theory and that of a progressive plane graph (or plane string diagram) in category theory are essentially the same thing. In this paper, we combine the ideas in graph theory and category theory to explain why and in what sense upward planarity is a topological property. The main result is that two upward planar drawings of an acyclic directed graph are equivalent (connected by a deformation) if and only if they are connected by a planar isotopy which preserves the orientation and polarization of G. This result gives a positive answer to Selinger's conjectue, whose strategy is different from the solution recently given by Delpeuch and Vicary. "
Matteo Capucci (he/him) (Oct 23 2020 at 10:46):What's an 'upward' plane graph? A directed graph whose edges all go upwards?
Nathaniel Virgo (Oct 23 2020 at 10:50):You'd think so
image.png
Matteo Capucci (he/him) (Oct 23 2020 at 10:53):Edges are downward oriented for an upward graph? :laughing:
Jules Hedges (Oct 23 2020 at 13:24):I guess "upward" must be standard terminology from topological graph theory. If the author had decided to follow the Oxford convention (which itself follows Feynman's convention) then it would have been consistent...
Morgan Rogers (he/him) (Oct 23 2020 at 13:30):That still begs the question of how the topological graph theorists chose that convention :joy:
Jules Hedges (Oct 23 2020 at 13:32):Well, it had to be some convention and there's only 4 to choose from
Nathaniel Virgo (Oct 23 2020 at 13:33):I think the moral of the story is, never name anything after a direction - someone somewhere will draw it the other way up :)
Jules Hedges (Oct 23 2020 at 13:33):Personally I'd prefer to think of a directed structure as going "downhill", so if you stick a particle at a source it will move "down" to a sink as though under gravity
Jules Hedges (Oct 23 2020 at 13:34):I remember somebody calling the upwards Feynman convention "from hell to heaven". I'd guess it's probably @Bob Coecke
Morgan Rogers (he/him) (Oct 23 2020 at 13:43):Jules Hedges said:
Well, it had to be some convention and there's only 4 to choose from
It would have been truly preposterous to call these "leftward plane graphs" :rolling_on_the_floor_laughing:
Jules Hedges (Oct 23 2020 at 13:45):Why? If you're a native speaker of Arabic for example then right-to-left should be the most intuitively natural choice for "forwards"
Jules Hedges (Oct 23 2020 at 13:46):This thread is an example of the following observation: It is not possible to talk about string diagrams without it degenerating into a flame war about orientation
Morgan Rogers (he/him) (Oct 23 2020 at 13:48):Okay, the confusion may be on my part here; are you saying that one would call the photo that Nathaniel shared an "upward planar graph" irrespective of how that diagram is oriented?
Jules Hedges (Oct 23 2020 at 13:49):Yes, I didn't hear that terminology before today but apparently it's standard
Jules Hedges (Oct 23 2020 at 13:49):The definition of a string diagram is exactly the same, irrespective of how you draw it on the page. Compare how nobody has flame wars over which direction cobordisms should go
Morgan Rogers (he/him) (Oct 23 2020 at 13:50):The thing that's preposterous to me is employing a certain name for a thing and then ignoring the orientation implicit in that name.
Jules Hedges (Oct 23 2020 at 13:52):My guess is the author was mostly citing people who draw string diagrams downwards, and was matching that convention rather than the one from graph theory
Jules Hedges (Oct 23 2020 at 13:52):In other words: I blame the topological graph theorists, rather than the author of that paper
John Baez (Oct 23 2020 at 14:36):Jules Hedges said:
This thread is an example of the following observation: It is not possible to talk about string diagrams without it degenerating into a flame war about orientation
I don't think the people are here to blame for exploding when someone draws an "upward plane graph" with edges pointing downward.
John Baez (Oct 23 2020 at 14:37):Though it is a compromise I hadn't considered: say the edges are pointing upward but draw them pointing downward.
Evan Patterson (Oct 23 2020 at 18:27):Unfortunately the terms "upward planar" and "upward drawing" are standard in topological graph theory and the literature on graph drawing. They should have just called them "monotone drawings" but oh well :)
Robert Seely (Oct 23 2020 at 19:20):This prompts me to ask a silly question: I've seen string (circuit) diagrams that are supposed to be read: (a) top-down, (b) bottom-up, (c) left-right, but I don't think I've ever seen (d) right-left - is there someone (esp in the monoidal category community - like linear logicians, etc ) or "someones" that do so? (I'm probably having a "senior moment", and the "answer" will be obvious to me once pointed out! - sorry!)
Jules Hedges (Oct 23 2020 at 20:00):So I was joking about exactly this on twitter and I was surprised to get some serious replies from people who've done right to left diagrams... I don't think anyone does it consistently though: https://twitter.com/_julesh_/status/1319634530131693569
It’s disappointing that only 3 of the 4 possible orientation conventions for a string diagram are in widespread use. We need an Arabic-speaking applied category theory group to start drawing all their string diagrams right-to-left
- julesh (@_julesh_) Dan Marsden (Oct 24 2020 at 08:06):If you're drawing string diagrams for bicategories, right to left is a good choice as it lines up with symbolic 1-cell composition notation. So in Cat for example, the picture for functor composition G F would have the wires in the right order. Of course, you could just switch to writing F;G, but that's less common than the other convention.
John Baez (Oct 24 2020 at 16:13):Robert: Peter Selinger's A survey of graphical languages for monoidal categories uses a left-to-right notation, and takes advantage of it to pack in diagrams without a lot of white space.
Robert Seely (Oct 24 2020 at 20:06):John Baez said:
Robert: Peter Selinger's A survey of graphical languages for monoidal categories uses a left-to-right notation, and takes advantage of it to pack in diagrams without a lot of white space.
?? I wondered about your comment; maybe you misread mine. I know left-to-right is pretty common, it was right-to-left I was asking about (and I don't see that in Peter's article, apart from cases where a "negation" or "star" operation reverses an arrow). Does anyone use that? The other three (obvious) possibilities are common - but this one seems untouched (maybe for a good reason?). (BTW - there are some 3D circuit - or perhaps "sheet" - diagrams "out there", so "front-back" and "back-front" could be added to the list of "things done".)
John Baez (Oct 24 2020 at 20:29):Yes, I misread your comment, since even left-to-right notation is pretty rare among the heavy-duty practitioners of string diagrams.
As others have pointed out, the main advantage of right-to-left notation is that it matches the usual of writing composition where means "do first , then ".