Symmetric Monoidal Categories: a Rosetta Stone

The Topos Institute is in business! I’m really excited about visiting there this summer and working on applied category theory.

They recently had a meeting with some people concerned about AI risks, called Finding the Right Abstractions, organized by Scott Garrabrant, David Spivak, and Andrew Critch. I gave a gentle introduction to the uses of symmetric monoidal categories:

• Symmetric monoidal categories: a Rosetta Stone.

To describe systems composed of interacting parts, scientists and engineers draw diagrams of networks: flow charts, Petri nets, electrical circuit diagrams, signal-flow graphs, chemical reaction networks, Feynman diagrams and the like. All these different diagrams fit into a common framework: the mathematics of symmetric monoidal categories. While originally the morphisms in such categories were mainly used to describe processes, we can also use them to describe open systems.

You can see the slides here, and watch a video here:

For a lot more detail on these ideas, see:

• John Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta Stone, in New Structures for Physics, ed. Bob Coecke, Lecture Notes in Physics vol. 813, Springer, Berlin, 2011, pp. 95—174.

15 Responses to Symmetric Monoidal Categories: a Rosetta Stone

  1. The link to the slides is broken for me.

  2. Keith Harbaugh says:

    Since your talk is aimed at a fairly general audience, let me point out an introduction to symmetic monoidal categories as a framework for understanding linear algebra:
    https://graphicallinearalgebra.net/
    by Pawel Sobocinski.

    The downside of this for many will be the extent to which he talks about things extraneous to mathematics. But the very significant upside is his introduction of such significant mathematical topics as PROPs, interacting Hopf algebras, Frobenius algebras, string diagrams, and the hot topic of coalgebras in a very familiar, easily understandable, setting.

    • John Baez says:

      Yes, Pawel’s introduction is great! I can’t resist adding that my student Jason Erbele developed a lot of the same mathematics independently in his study of control theory, with help from me:

      • John Baez and Jason Erbele, Categories in control, Theory and Applications of Categories 30 (2015), 836–881.

      and in more detail in his thesis:

      • Jason Erbele, Categories in Control: Applied PROPS, Ph.D. thesis, U. C. Riverside, 2016.

      I blogged about this stuff, too. However, Pawel’s blog develops the material in a way that’s much easier for nonexperts to follow, so I recommend it highly to anyone getting started on monoidal categories or string diagrams!

  3. repagulum says:

    Hi, John! I’ve seen your Rosetta stone paper something like 10 years maybe ago, it’s such a great and inspiring paper, thanks. Serious question though – you mention ecosystems in the end as an example. Could you elaborate more on this diagram/example? Is topos theory helpful there?

    So the question is – in what sense does your (or your students) research in Network Theory helps in ecology or ecosystems? I understand, that it helps in creating more theoretically sound approach, but is there any other impact, maybe applications? Does it maybe speed up algorithms, or improves causality/explainability?

    • John Baez says:

      For the last ten years I’ve been setting up network theory as a general approach to systems, starting with easy examples like electrical circuits, control theory, and chemistry. My goal was always to use this theory to study biology and ecology. Unfortunately it took a long time to get the basics worked out. Also, having math grad students, I keep tending to get pulled into developing the mathematical infrastructure rather than applying it. We actually had some seminars on Odum’s work, but we never got around to writing papers on it.

      Luckily I am retiring at the end of June, and not taking new students for a while, so I’ll have more time to think on my own. I really do want to get into applications to ecology. So, please ask me this question again in another 5 or 10 years.

      You might be interested in the “manifesto” I wrote when I began this project, since it explains my original goal:

      Network theory (part 1), 2011.

      For what’s been done so far, you can go here:

      Network theory.

      However, there’s very little about biology or ecology so far—just a general theory of networks.

      In parallel I’ve been thinking about information and thermodynamics in biology and ecology, and I’m helping run a session about this soon:

      Nonequilibrium thermodynamics in biology.

      I plan to combine these ideas with the network theory ideas.

      • Keith Harbaugh says:

        “I am retiring at the end of June”

        Certainly an interesting and important milestone!
        I am just curious, and possibly others are as well, what that process, in its various aspects, is like.
        E.g., What were/are the last courses you taught?
        What events does UCR arrange for the sendoff?
        What are your various thoughts at this time? (I am sure you have many)
        I just checked your Diary, and didn’t see anything on this.
        Maybe that’s a good place for such information, or maybe here at Azimuth?

      • John Baez says:

        Keith wrote:

        E.g., What were/are the last courses you taught?

        I’m currently teaching two courses; next week is the last week of class!

        One is Math 7B, Calculus for the Life Sciences, a course where kids learn integration, with a lot of separable differential equations describing population models. The other is Math 241, Classical Mechanics, a graduate course where I explain Lagrangian and Hamiltonian mechanics and a bit of symplectic geometry.

        What events does UCR arrange for the sendoff?

        They were going to have a Zoom party but I asked them not to. My students were going to arrange a 60th birthday conference but then COVID-19 came along, so we postponed it to next year.

        What are your various thoughts at this time?

        Whew, that’s a long story. I’ve been planning this for years, and I find it both exciting and a bit scary.

        I am eager to quit working at UCR because while I enjoy teaching and research, I find the environment increasingly bureaucratic, and as a senior faculty member I’m expected to do more and more administrative work, which I hate.

        I am also somewhat burnt out on working with lots of grad students. I love them, and they make research go faster in some ways, but they make it hard to suddenly switch gears and jump from one project to another: lately they’ve all been doing category theory, which has really boosted my understanding of that subject, but slowed my work on physics, and biology, and various areas of pure math.

        So, I’m looking forward to a more peaceful existence where I spend more time thinking and writing. I want to do a lot of expository writing on math and physics. Research questions tend to come up in the process of trying to explain things well, so I’ll be doing research too. I also want to come up with some new ideas connecting category theory to thermodynamics and ecology.

        I also want to spend more time composing music. I’ve recently had a kind of breakthrough getting to know Renaissance polyphonic music, and I can imagine combining some of those harmonic and melodic ideas with modern electronic music. I find modern music to be a bit boxed in right now.

        Soon I will spend a month at the Topos Institute, talking to people about practical applications of category theory. I will try to persuade them of the importance of ecological issues, and see if we can figure out some ways to help. It’ll be lots of fun, I bet!

        • Sounds like you have a full schedule. I’m reminded of Abraham Pais’s last visit to Einstein. Pais knocked on the door, Einstein told him to come in, then looked up from his writing pad. They chatted for about 15 minutes then Pais left. As he did so, he looked over his shoulder, even though it took just a couple of seconds to get back to the door. Einstein was back at work.

          Since J. Baez is already taken in the music world, will your nom de plume be Azimuth?

        • John Baez says:

          I think accepting any comparison to Einstein would give me crackpot points.

          I’m thinking of releasing my albums under the name JOHN BAEZ, with the vertical lines in the H slanted together so that some people buy my stuff by accident.

        • Seriously, have you thought about a collaboration? How well do you know her?

          As a fan of folk-rock, fellow fans will know why I am annoyed by the bands Fairground Attraction and Steely Dan. :-)

          I think the comparison is OK as long as you don’t make it yourself. :-)

          Of course, Einstein was also a reasonably good musician.

        • John Baez says:

          I’m not really in touch with Joan Baez and would never dream of trying a collaboration; we have completely different ideas on music. What I really need is to find some friendly musicians who know a lot about modern digital composition and recording and can somehow stand teaching me about it—maybe in return for lessons on math and physics?

        • That’s one possibility. Or maybe get in contact with Brian May, one of the few real rock stars with a doctorate in (astro)physics.

  4. This is the greatest blog ever. I wonder about the applications to ecology. It would mean getting something done. But I’ve heard one person describe the situation like this: Businesses control government; government controls the people; and there’s nothing we can do about that. Except maybe, to get the leaders of business to be more rational about ecology. After all, in any business organization it’s not the engineers and scientists who control the business. It’s finance, sales, marketing. So if network theory is to help ecology given this unchangeable situation, network theory would have to become useful to them. In that case, everything would have to be translated into the ordinary human language they use. But take the word ‘morphism’ as you use it in your video. A powerful mathematician such as yourself and the many who subscribe to your blog will have acquired a sense of this word based on years of experience in the hardest intellectual work possible. But this is not going to be the case for, say, a CEO who’s started out in sales. However– everybody in business should be acquainted with the word ‘process’ (because of the popular focus on ‘business processes’ resulting from the ‘quality’ revolution. e.g. Dr. Deming, for a short while in the beginning– until he started saying that management was the problem.) So instead of ‘morphism’, what about the arrow in a category theory diagram, and in your video, being called a ‘possible event’ (A process comprises events)?

    • John Baez says:

      Lee wrote:

      So if network theory is to help ecology given this unchangeable situation, network theory would have to become useful to them. In that case, everything would have to be translated into the ordinary human language they use.

      Sure! Right now I’m just trying to figure out whether, and how, category theory can be useful in biology or ecology. At this stage, anyone who wants to do this needs to think as freely as possible, using all the resources—terminology, ideas, etc.—that they have. But if it ever amounts to something, the discoveries can surely be translated into language that others can act on.

      This sort of thing is happening faster in other applications. For example, tomorrow we’ll be having a conference on category theory in robotics. Even though they use math, it can be hard for people working in robotics to learn category theory. But some have, and they’re already using it! Together we will try to get some more roboticists interested.

      Another example: for the first half of May the category theorists at the Topos Institute, sort of near Silicon Valley, had a meeting with some people concerned about AI risks, called Finding the Right Abstractions. It’s too early to be sure that category theory will be helpful in developing safer AI, but it’s a good thing to think about.

      So, people are trying various things. Some will work, some won’t. The ones that work can be explained in simple language to a large audience. Before something works, it doesn’t make much sense to popularize it.

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