## The Best Work on Sustainability?

7 April, 2020

Some people want to give a \$1,000,000 prize for a “discovery of high scientific value that has significant repercussions in the field of environmental sustainability in order to improve the quality of life, in harmony with the production system and the transition to new development models.”

So, they’re looking for people and organizations who have done the very best recent work in these area:

• energy transition towards renewable sources

• sustainable mobility

• clean energy and renewable resources

• energy efficiency

• clean technologies for the exploitation of fossil fuels

• sustainable management and efficient use of natural resources

• eco-friendly management of materials during their entire life cycle

• reduction of CO2 emissions

• innovative systems for the exploitation of solar energy

• discovery and development of new materials for the production

• storage and distribution of clean energy

## Category Theory Calendar

6 April, 2020

There are now enough online events in category theory that a calendar is needed. And here it is!

It should show the times in your time zone, at least if you don’t prevent it from getting that information.

## Structured Cospans and Petri Nets

6 April, 2020

$\;$

This talk on structured cospans and Petri nets is the second of a two-part series, but it should be understandable on its own. The first part is on structured cospans and double categories.

I’m giving this second talk at the MIT Categories Seminar. It’ll be on Thursday April 9th, 12 noon Eastern Time.

The MIT seminar is going to try something new this time: my talk will be live streamed both on Zoom and YouTube:

Zoom Meeting: https://zoom.us/j/988858130
Meeting ID: 988 858 130

You can chat during and after the talk at the Category Theory Community Server. (To join this, click here.) The talk will be recorded and remain available on YouTube.

You can already see the slides here.

Structured cospans and Petri nets

Abstract. “Structured cospans” are a general way to study networks with inputs and outputs. Here we illustrate this using a type of network popular in theoretical computer science: Petri nets. An “open” Petri net is one with certain places designated as inputs and outputs. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Using the formalism of structured cospans, open Petri nets can be treated as morphisms of a symmetric monoidal category—or better, a symmetric monoidal double category. We explain two forms of semantics for open Petri nets using symmetric monoidal double functors out of this double category. The first, an operational semantics, gives for each open Petri net a category whose morphisms are the processes that this net can carry out. The second, a “reachability” semantics, simply says what these processes can accomplish. This is joint work with Kenny Courser and Jade Master.

The talk is based on these papers:

• John Baez and Kenny Courser, Structured cospans.

• John Baez and Jade Master, Open Petri nets.

• Jade Master, Generalized Petri nets.

I’ve blogged about open Petri nets before, and these articles might be a good way to start learning about them:

Part 1: the double category of open Petri nets.

Part 2: the reachability semantics for open Petri nets.

Part 3: the free symmetric monoidal category on a Petri net.

## A Categorical View of Conditional Expectation

2 April, 2020

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I always like to see categories combined with probability theory and analysis. So I’m glad Prakash Panangaden did that in his talk at the ACT@UCR seminar. Afterwards we had discussions at the Category Theory Community Server, and you can see them here if you’re a member:

https://categorytheory.zulipchat.com/

You can see his slides here, or download a video here, or watch the video here:

A categorical view of conditional expectation

Abstract. This talk is a fragment from a larger work on approximating Markov processes. I will focus on a functorial definition of conditional expectation without talking about how it was used. We define categories of cones—which are abstract versions of the familiar cones in vector spaces—of measures and related categories cones of Lp functions. We will state a number of dualities and isomorphisms between these categories. Then we will define conditional expectation by exploiting these dualities: it will turn out that we can define conditional expectation with respect to certain morphisms. These generalize the standard notion of conditioning with respect to a sub-sigma algebra. Why did I use the plural? Because it turns out that there are two kinds of conditional expectation, one of which looks like a left adjoint (in the matrix sense not the categorical sense) and the other looks like a right adjoint. I will review concepts like image measure, Radon-Nikodym derivatives and the traditional definition of conditional expectation. This is joint work with Philippe Chaput, Vincent Danos and Gordon Plotkin.

For more, see:

• Philippe Chaput, Vincent Danos, Prakash Panangaden and Gordon Plotkin, Approximating Markov processes by averaging, in International Colloquium on Automata, Languages, and Programming, Springer, Berlin, 2009.

## How Scientists Can Fight COVID-19

31 March, 2020

A friend listed some calls for help:

• The UK urgently needs help from modellers. You must be willing to work on specified tasks and meet deadlines. Previous experience in epidemic modelling is not required.

https://epcced.github.io/ramp/

• MIT is having a “Beat the Pandemic” hackathon online April 3-5. You can help them develop solutions that address the most pressing technical, social, and financial issues caused by the COVID-19 outbreak:

https://covid19challenge.mit.edu/

• The COVID-19 National Scientist Volunteer Database Coordination Team is looking for scientists to help local COVID-19 efforts in the US:

• The Real Time Epidemic datathon, which started March 30, is collective open source project for developing real-time and large-scale epidemic forecasting models:

https://www.epidemicdatathon.com/

• Crowdfight COVID-19 is a mailing list that sends lists of tasks for which help is needed:

## Structured Cospans and Double Categories

31 March, 2020

$\; \phantom{a}$

This talk on structured cospans and double categories is the first of a two-part series; the second part is about structured cospans and Petri nets.

I gave the first talk at the ACT@UCR seminar. Afterwards we had discussions at the Category Theory Community Server, and you can see them here if you’re a member:

https://categorytheory.zulipchat.com/

You can see my slides here, or download a video here, or watch the video here:

Structured cospans and double categories

Abstract. One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce “structured cospans” as a way to study networks with inputs and outputs. Given a functor L: A → X, a structured cospan is a diagram in X of the form

If A and X have finite colimits and L is a left adjoint, we obtain a symmetric monoidal category whose objects are those of A and whose morphisms are certain equivalence classes of structured cospans. However, this arises from a more fundamental structure: a symmetric monoidal double category where the horizontal 1-cells are structured cospans, not equivalence classes thereof. We explain the mathematics and illustrate it with an example from epidemiology.

This talk was based on work with Kenny Courser and Christina Vasilakopoulou, some of which appears here:

• John Baez and Kenny Courser, Structured cospans.

• Kenny Courser, Open Systems: a Double Categorical Perspective.

Yesterday Rongmin Lu told me something amazing: structured cospans were invented in 2007 by José Luiz Fiadeiro and Vincent Schmit. It’s pretty common for simple ideas to be discovered several times. The amazing thing is that these other authors also called these things ‘structured cospans’!

• José Luiz Fiadeiro and Vincent Schmitt, Structured co-spans: an algebra of interaction protocols, in International Conference on Algebra and Coalgebra in Computer Science, Springer, Berlin, 2007.

These earlier authors did not do everything we’ve done, so I’m not upset. Their work proves I chose the right name.

## Online Worldwide Seminar on Logic and Semantics

30 March, 2020

Someone should make a grand calendar, readable by everyone, of all the new math seminars that are springing into existence. Here’s another! It’s a bit outside the core concerns of Azimuth, but it’ll have a lot of category theory, and it features some good practices that I hope more seminars adopt, or tweak.

Online Worldwide Seminar on Logic and Semantics, organized by Alexandra Silva, Pawel Sobocinski and Jamie Vicary.

There will be talks fortnightly at 1 pm UTC, which is currently 2 pm British Time, thanks to daylight savings time. Here are the first few:

• Wednesday, April 1, — Kevin Buzzard, Imperial College London: “Is HoTT the way to do mathematics?”

• Wednesday, April 15 — Joost-Pieter Katoen, Aachen University: “Termination of probabilistic programs”.

• Wednesday, April 29 — Daniela Petrisan, University of Paris: “Combining probabilistic and non-deterministic choice via weak distributive laws”.

• Wednesday, May 13 — Bartek Klin, Warsaw University: “Monadic monadic second order logic”.

• Wednesday, May 27 — Dexter Kozen, Cornell University: “Brzozowski derivatives as distributive laws”.

Joining the seminar.To join any OWLS seminar, visit the following link, up to 15 minutes before the posted start time:

You will be given the option to join through your web browser, or to launch the Zoom client if it is installed on your device. For the best experience, we recommend using the client.

Audio and video. We encourage all participants to enable their audio and video at all times (click “Use Device Audio” in the Zoom interface.) Don’t worry about making noise and disrupting the proceedings accidentally; the Chairperson will ensure your audio is muted by default during the seminar. Having your audio and video enabled will allow other participants to see your face in the “Gallery” view, letting them know that you’re taking part. It also gives you the option of asking a question, and of making best use of the “coffee break” sessions. For most users with good network access (such as a fast home broadband connection), there is no need to worry that having your audio and video enabled will degrade the experience; the technology platform ensures that the speaker’s audio/video stream is prioritised at all times. However, those on slow connections may find it better to disable their audio and video.

Coffee breaks. Every OWLS seminar has two “coffee breaks”, one starting 15 minutes before the posted start time of the seminar, and the second starting after the seminar is finished. To participate in these, feel free to join the meeting early, or to keep the meeting window open after the end of the talk. During these coffee break periods, participants will be automatically gathered into small groups, assigned at random; please introduce yourself to the other members of your group, and chat just like you would at a real conference. Remember to bring your own coffee!

During the seminar. If you’d like to ask a question, either during the seminar or in the question period at the end, click the “Participants” menu and select “Raise hand”. The Chairperson may choose to interrupt the speaker and give your audio/video feed the focus, giving you the opportunity to ask your question verbally, or may instead decide to let the seminar continue. You may click “Lower hand” at any time to show you no longer wish to ask a question. To preserve the experience of a real face-to-face conference, there is no possibility of giving a written question, and the chat room is disabled at all times. You also have the opportunity to give nonverbal feedback to the speaker by clicking the “speed up” or “slow down” buttons, also in the “Participants” menu.

Recordings. All OWLS seminars are recorded and uploaded to YouTube after the event. Only the audio/video of the chairperson, speaker, and questioners will be captured. If you prefer not to be recorded, do not ask a question. Of course, the organizers do not make any recordings of the coffee break sessions.