diagrams in Prof and optics · theory: applied category theory

absolutely livid to see my exact thoughts suddenly appearing on the arxiv, i was gonna write this paper dammit https://arxiv.org/abs/2004.04526

sarahzrf (May 01 2020 at 08:36):

sarahzrf (May 01 2020 at 08:37):

sarahzrf (May 01 2020 at 08:39):

Arthur Parzygnat (May 01 2020 at 09:20):

Arthur Parzygnat (May 01 2020 at 09:26):

And by the way, I've had this thought a few times in my life, too, @sarahzrf. Namely, I'm working on something, and suddenly I see a paper and I'm like "omg... did they just do everything I was doing and more!?" and I have a mini panic attack. But then I remind myself, "Hey, if I'm working on this, and so are other people, then that means I'm doing something right." And then after reading their work, I usually learn a lot more and often even find out it's not exactly the same as what I was doing and I have complimentary things to contribute to the overall discussion. Then I'm even more motivated to continue. Anyway, the point is that I've found these moments to actually do more help than harm.

Nathaniel Virgo (May 01 2020 at 09:47):

Nathaniel Virgo (May 01 2020 at 09:49):

Mario Román (May 01 2020 at 09:53):

Thank you, @sarahzrf ! :) I have been working on optics for a while now and I am happy to see this direction felt natural to you as well.

Sorry to hear if it caused a mini panic attack. I am preparing a better version of this right now that I hope I can share soon. If, in the future, you are still interested in the topic maybe we should put notes together and see what we can get.

Mario Román (May 01 2020 at 09:56):

Yes, thank you! Your question is referenced as a motivating example in the introduction :) I was happy to have an example outside optics showing that it was not only me asking this question to myself.

sarahzrf (May 01 2020 at 19:12):

sarahzrf (May 01 2020 at 19:13):

(=_=) (May 02 2020 at 01:57):

Serious question: how would a diagram(?) for the semantics of "yeah nah" and "nah yeah" look like?

Gershom (May 02 2020 at 02:26):

I am reminded of the famous story of a linguist who noted that the double negative could mean affirmation, but the double positive never meant negation, to which somebody in the audience (reputedly the mathematician Martin Kruskal) sarcastically responded "yeah, yeah."

John Baez (May 02 2020 at 02:27):

That is one of my favorite jokes. I didn't know there was a reputed actual guy who reputedly actually said that.

Verity Scheel (May 02 2020 at 02:49):

I usually see that joke with “yeah, right”, it’s also one of my favorites :D

Matteo Capucci (he/him) (May 02 2020 at 13:31):

Gotta shout out to @Mario Román for this paper, it's awesome. And gorgeous, with all those beautifully typeset diagrams.

Mike Shulman (May 02 2020 at 17:11):

FWIW, Wikipedia believes that the wag in the audience was Sidney Morgenbesser and the lecturer was the philosopher J. L. Austin, backing it up with three citations to eulogies or news articles (1, 2, 3).

Rich Hilliard (May 02 2020 at 18:47):

That's the way I heard it (Morgenbesser to Austin) back in early 80s around MIT Linguistics.

sarahzrf (May 13 2020 at 07:14):

i finally read the section w/ pointed profunctors where the diagrams get "inflated"

sarahzrf (May 13 2020 at 07:14):

sarahzrf (May 13 2020 at 07:49):

David Michael Roberts (May 13 2020 at 07:54):

Know the feeling. Drinfeld is now working with stuff I thought of 9 years ago. I couldn't get people to take me seriously at the time....

Mario Román (May 14 2020 at 10:50):

This one? with the inflated diagrams I guess it is easier to see just the initial and final results.
opendiagrams-rest.pdf

sarahzrf (May 14 2020 at 14:08):

jeez that looks like a pain to typeset, did you already have that laid out or did you make it just now o.O

Mario Román (May 14 2020 at 14:13):

sarahzrf (May 14 2020 at 14:16):

sarahzrf (Jun 15 2020 at 02:53):

@Mario Román i've just been reading about diagrams for double categories & thinking about what they might be good for expressing in, say, Prof—have you considered anything like that?

sarahzrf (Jun 15 2020 at 02:54):

i was thinking that a heteromorphism is exactly a square from the horizontal id : 1 ⇸ 1 to the horizontal P : C ⇸ D, with domain and codomain given by the vertical sides

sarahzrf (Jun 15 2020 at 02:55):

you can also kinda express this with bicategory diagrams, but it's a bit more awkward

sarahzrf (Jun 15 2020 at 02:59):

anyway, in the notation in that paper, a heteromorphism of P : C ⇸ D in P(A, B) would look like

    | A          C
    |
  f o ------P------
    |
    | B          D

    | A          C₁
    |
  f o ------P------
    |
    | B          C₂
    |
  g o ------Q------
    |
    | X          C₃
    |
  h o ------R------
    |
    | Y          C₄

sarahzrf (Jun 15 2020 at 03:00):

so you can quite naturally compose heteromorphisms of different profunctors end to end, and then they have these bits trailing off on the right marking the profunctor they came from, and you can read off immediately that the composite heteromorphism belongs to P;Q;R

sarahzrf (Jun 15 2020 at 03:00):

whereas in the bicategory diagrams, you'd need to mess around a bit more to make this sensible

Jules Hedges (Jun 15 2020 at 10:05):

It's possible I'm getting mixed up, but I'm pretty sure Guillaume is already using David Jaz's double category diagrams in the double category of Tambara modules to study optics

Jules Hedges (Jun 15 2020 at 10:06):

Ok, it's not in his paper, but I definitely remember spending an afternoon with him covering a whiteboard in double category diagrams

Mario Román (Jun 15 2020 at 15:36):

TL;DR: yes, thank you for bringing this, I agree. That should be a nice thing to do; probably not that much for the application I wanted but in general. I knew of David's work but had been paying more attention to the bicategorical case until Edward Morehouse and now you pointed this again to me. I guess that for my application the graphical calculus of symmetric monoidal double categories should work more or less similarly, and I guess a clear advantage of the double category is that it separates functors and profunctors nicely (avoiding the awkward: "left adjoint, but also Cauchy completion on the domain").

Long form: Thank you, @sarahzrf, for pointing that. I agree, I knew (the first part of) that paper; although I decided to go with monoidal bicategories because they were more familiar to me and they seemed to justify nicely the relation with monoidal diagrams. In any case, trying to use the double version of these things would be really nice. I understand that the idea would be to go to symmetric monoidal double categories. Also, Edward Morehouse (who I think is working on these) pointed me to the fact that I should be trying to see how the story worked in monoidal double categories; I think you both are right, that should be a clear "further work" on what I was doing. Let me try to link here what I know related to that, in case it helps anyone:

Maybe it was David Jaz Myers that proposed us to use these diagrams during the ACT school. I think David Dalrymple drew a lot of optics diagrams in double categories on Twitter. There is also this paper by Sprunger and Katsumata that I looked into and does something with "double categories with one object" (I think David Dalrymple also cites it). Also, Chad Nester may be able to say more on that, he was working with similar things at the same time (with different applications in mind).

But then I went back to monoidal bicategories for this paper. As you say, you can still repeat that in monoidal bicategories, it is just that you need to decide to bend some wire (which, in the particular application I wanted is fine, in general, it feels a bit awkward). The fact that it was awkward pushed me to consider the pointed profunctors. I learnt a trick like the one you mention from Marsden's "Category Theory using String Diagrams" (in Cat instead of Prof). I guess also an advantage of double categories over bicategories is to track which things are functors and profunctors. I was more focused on monoidal bicategories and getting that into a decent form, but it is true that a lot of people around me is using monoidal double categories.

sarahzrf (Jun 15 2020 at 16:03):

i do still think your open diagrams seem nicer than these for working with stuff atm—e.g., it's not clear to me how to deal with the monoidal structure on Prof here without having to go 3 dimensional

sarahzrf (Jun 15 2020 at 16:04):

just, i was struck by the fact that the first, really basic use cases for open diagrams seem to be automatically captured by double category diagrams very nicely

sarahzrf (Jun 15 2020 at 16:05):

and i was wondering if maybe that's a hint toward a more principled basis for open diagrams, or a generalization of them, etc etc

sarahzrf (Jun 15 2020 at 17:15):

Mario Román (Jun 15 2020 at 18:24):

Yes, exactly. These are cross sections. References to that should be on the paper (Schommer-Pries' thesis, and an article by Bartlett, if I remember correctly). An open diagram encodes a derivation from the empty diagram, I think that is maybe the parallelism you want (?) (both if you use bicategories like on that paper or double categories)

sarahzrf (Jun 15 2020 at 18:25):

Mario Román (Jun 15 2020 at 18:30):

A mixture of online editors, like Tikzit and mathcha, with then editing the Tikz code by hand. It takes patience, but it worked for me

Cole Comfort (Jun 15 2020 at 20:15):

One could alternatively use the tikz code of Bartlett et al where they use "bubbles" rather than "sheets". You can just download the source code and copy it.
https://arxiv.org/abs/1509.06811

I haven't fully read your paper @Mario Román , but I am curious what does your paper provide in addition to the paper which I mentioned. Is it that you are elaborating on the example of optics using their notation?

sarahzrf (Jun 15 2020 at 20:25):

sarahzrf (Jun 15 2020 at 20:32):

tangentially: i followed the citation to the bartlett et al paper earlier today—i hadnt bothered before—and from a quick scroll to check out what their diagrams are like, a couple of things they pointed out / drew seem rly interesting to me:
image.png
image.png

sarahzrf (Jun 15 2020 at 20:36):

sarahzrf (Jun 15 2020 at 20:38):

or, like... on point #2, that's not quite the right description, but it's at least along the same lines of what im trying to describe—like, u can see how in that 2nd diagram in the 2nd image i posted, that loop could be pulled up just fine if we were in a plane, but it can't go thru the hole as-is

sarahzrf (Jun 15 2020 at 20:39):

...except it doesnt rly make sense to suggest doing that, since bits of diagram that fit in one piece of pipe can't go in another piece of pipe—they're heteromorphisms of that particular profunctor—but it still feels like something is going on here...

sarahzrf (Jun 15 2020 at 20:40):

i meant to work out this bit in mario's paper and i forgot to come back to it and it never 100% clicked, especially the fact that it's apparently part of an adjunction
image.png
seems relevant?

sarahzrf (Jun 15 2020 at 20:41):

Mario Román (Jun 16 2020 at 09:40):

Let me be first totally clear that the answer is "not much" if the interest is on the monoidal bicategories themselves (for that I am using these references) instead of the application. The goal of the paper is not to create another tool but to apply the existing tool of the graphical calculus of monoidal bicategories to a problem (the proposal of pointed profunctors is what I am using to present this differently). The rest of the answer is longer:

The problem is not as much optics but all other "combs", "input-output machines" and diagrams-with-holes-like things (causality, games, learners, categories with feedback, there was even a discussion on zulip recently on how to try to approach these things (link)). I think the contribution of this paper is to make the distance from "this would be almost a diagram in a monoidal category but it is not and it is difficult to reason graphically about it" to "I could use monoidal bicategories and there this would be totally formal" smaller for an audience that, like me, was in principle not even thinking of being working with monoidal bicategories; and does not necessarily know any of the prerequisites for quantum field theory. So, I want to make clear that the goal (or the claim) was not to develop the graphical calculus of monoidal bicategories, internal diagrams for quantum, or to provide anything new on that (instead, I just send the reader to the previous work there).

Having said this, I only identified [BDSPV] and added the references to all of this after I had written most of the text. This is partly why the presentation of the same idea in this text is this different. In particular, they say:

We already had a convenient notation for open diagrams (e.g. Riley, on optics), but I also wanted to have a category in which they were a particular case of the calculus of monoidal bicategories. This is why I take the time to explain the monoidal bicategory of pointed profunctors and my notation is different, instead of just taking it from BDSPV. What I may be providing in addition to BDSPV is the proposal of using

\mathbf{Prof}^\ast

to reduce the justification of internal diagrams to the justification of calculi for monoidal bicategories. This seemed important for my application; it is a minor comment that seems to be missing on BDSPV and that, to the best of my limited knowledge, seems to justify the calculus even better than "a convenient mnemonic notation". This is why I use diagrams in the monoidal bicategory of pointed profunctors instead of internal diagrams. But again, this minor change is not the goal of the paper.

On notation: this may be psychological but, if I imagine the wires inside 3d bubbles, I think I can braid them (maybe it is just me, but isn't a braided category precisely that (?)). However, I remember to understand the goal on BDSPV is also general monoidal categories, I guess BDSPV would go on that concern by still using the same notation if the categories they deal with are not braided. I preferred to stick to a functor-box-like notation partly because of this and uniformity.

The shorter version for ACT does not make all of this discussion; it just makes everything fit as a particular case of the calculus of monoidal bicategories (Schommer-Pries, Bartlett, Vicary, Douglas; the references for this are still the same). This was because I just do not feel confident that I completely follow how they address those concerns; and I wanted to use the calculus of some monoidal bicategory. The 2nd arxiv'd version of my paper (no page limits, yay!) takes longer but does this discussion. It not only references the rest of the work by Bartlett, Douglas, Schommer-Pries and Vicary but specifically [BDSPV] as the same idea of inflating diagrams. I actually found it via Nick's Hu thesis, also referenced there. On retrospective, and after having received feedback, I think this discussion deserves to be also on the short version more prominently to make the readers also follow the link to quantum field theory.

Guillaume Boisseau (Jun 20 2020 at 14:46):

@Jules Hedges indeed I had (framed) double categories in mind when doing diagrams in Tamb. The oriented-wire notation is partly from there too

Stream: theory: applied category theory

Topic: diagrams in Prof and optics

sarahzrf (May 01 2020 at 08:36):

sarahzrf (May 01 2020 at 08:36):

sarahzrf (May 01 2020 at 08:37):

sarahzrf (May 01 2020 at 08:39):

Arthur Parzygnat (May 01 2020 at 09:20):

Arthur Parzygnat (May 01 2020 at 09:26):

Nathaniel Virgo (May 01 2020 at 09:47):

Nathaniel Virgo (May 01 2020 at 09:49):

Mario Román (May 01 2020 at 09:53):

Mario Román (May 01 2020 at 09:56):

sarahzrf (May 01 2020 at 19:12):

sarahzrf (May 01 2020 at 19:13):

(=_=) (May 02 2020 at 01:57):

Gershom (May 02 2020 at 02:26):

John Baez (May 02 2020 at 02:27):

Verity Scheel (May 02 2020 at 02:49):

Matteo Capucci (he/him) (May 02 2020 at 13:31):

Mike Shulman (May 02 2020 at 17:11):

Rich Hilliard (May 02 2020 at 18:47):

sarahzrf (May 13 2020 at 07:14):

sarahzrf (May 13 2020 at 07:14):

sarahzrf (May 13 2020 at 07:14):

sarahzrf (May 13 2020 at 07:49):

sarahzrf (May 13 2020 at 07:49):

David Michael Roberts (May 13 2020 at 07:54):

Mario Román (May 14 2020 at 10:50):

sarahzrf (May 14 2020 at 14:08):

sarahzrf (May 14 2020 at 14:08):

sarahzrf (May 14 2020 at 14:08):

Mario Román (May 14 2020 at 14:13):

sarahzrf (May 14 2020 at 14:16):

sarahzrf (May 14 2020 at 14:16):

sarahzrf (Jun 15 2020 at 02:53):

sarahzrf (Jun 15 2020 at 02:54):

sarahzrf (Jun 15 2020 at 02:55):

sarahzrf (Jun 15 2020 at 02:59):

sarahzrf (Jun 15 2020 at 03:00):

sarahzrf (Jun 15 2020 at 03:00):

Jules Hedges (Jun 15 2020 at 10:05):

Jules Hedges (Jun 15 2020 at 10:06):

Mario Román (Jun 15 2020 at 15:36):

sarahzrf (Jun 15 2020 at 16:03):

sarahzrf (Jun 15 2020 at 16:04):

sarahzrf (Jun 15 2020 at 16:05):

sarahzrf (Jun 15 2020 at 17:15):

Mario Román (Jun 15 2020 at 18:24):

sarahzrf (Jun 15 2020 at 18:25):

Mario Román (Jun 15 2020 at 18:30):

Cole Comfort (Jun 15 2020 at 20:15):

sarahzrf (Jun 15 2020 at 20:25):

sarahzrf (Jun 15 2020 at 20:32):

sarahzrf (Jun 15 2020 at 20:36):

sarahzrf (Jun 15 2020 at 20:38):

sarahzrf (Jun 15 2020 at 20:39):

sarahzrf (Jun 15 2020 at 20:40):

sarahzrf (Jun 15 2020 at 20:41):

Mario Román (Jun 16 2020 at 09:40):

Guillaume Boisseau (Jun 20 2020 at 14:46):