Applied Category Theory 2020 — Adjoint School

23 December, 2019

Like last year and the year before, there will be a school associated to this year’s conference on applied category theory! If you’re trying to get into applied category theory, this is the best possible way.

Applied Category Theory 2020 — Adjoint School.

The school will consist of online meetings from February to June 2020, followed by a research week June 29–July 3, 2020 at MIT in Cambridge Massachusetts. The conference follows on July 6–10, 2020, and if you attend the school you should also go to the conference.

The deadline to apply is January 15 2020; apply here.

There will be 4 mentors teaching courses at the school:

• Michael Johnson, Categories of maintainable relations.

• Nina Otter, Diagrammatic and algebraic approaches to distances between persistence modules.

• Valeria de Paiva, Dialectica categories of Petri nets.

• Michael Shulman, A practical type theory for symmetric monoidal categories.

Click on the links for more detailed information!

Who should apply?

Anyone, from anywhere in the world, who is interested in applying category-theoretic methods to problems outside of pure mathematics. This is emphatically not restricted to math students, but one should be comfortable working with mathematics. Knowledge of basic category-theoretic language—the definition of monoidal category for example—is encouraged.

We will consider advanced undergraduates, PhD students, post-docs, as well as people working outside of academia. Members of minorities, and of any groups which are underrepresented in the mathematics and computer science communities, are especially encouraged to apply.

Structure of the school

Every participant will be assigned to one of the groups above, according to their preference (and to the availability of places within the groups). Each group will consist of a mentor, a TA, and 4-5 students.

Online meetings

Between February and June 2020 there will be an online reading seminar. Each group will have a reading list of two papers, which they will study, and then present to the rest of the school during weekly online meetings. Every member of the school is encouraged to take part in the discussion of every paper, first during the meeting via live chat, and then, in written form, on an online forum. After the presentation and the forum discussion the students of each group will write a blog post about their assigned paper on the n-Category Café.

During this period, the TAs will be there to help the students, answer any question they might have, and moderate the discussions. This way, all the participants will build the necessary background to take part in the research activities during the week at MIT.

Research week

After the online meetings, there will be a two-week event at MIT, from June 29th to July 10th 2020. The first week is dedicated exclusively to the participants of the school. They will work in groups on the research projects outlined above, led by their mentors, with the help of their TAs.

During the second week the ACT 2020 Conference will take place, which is open to a wider audience. The member of each group of the school will have the possibility to present their activity to the audience of the conference, and share their ideas. The conference is not technically part of the school, but is about very similar topics, and participation is very much encouraged. The online meetings should prepare students to be able to follow some of the conference presentations to a reasonable degree, and introduce them to the main problems and techniques of the field.


For any questions or doubts please write us at the address act adjoint school at gmail dot com.


Carmen Constantin

Eliana Lorch

Paolo Perrone

Applied Category Theory Postdocs at NIST

13 December, 2019

An advertisement:

We are looking to expand our group of applied category theorists at the National Institute of Standards and Technology (NIST). Our group develops use cases, tools and methodology to apply category theory and related methods in a broad range of disciplines centered around the design, implementation, operation and evolution of engineered systems.

We encourage those eligible and interested to apply for the National Research Council Research Associateship Program. The upcoming deadline is February 1st, for those looking to start by December 2020.

The relevant postdoctoral opportunities can be found here:

Mathematical Foundations for System Interoperability
Research in Cyber-Physical Systems

These 2-year postdoctoral positions are only open to US citizens, come with a base stipend around $72k (12 month), great benefits, and travel support.

For non-US citizens, NIST has mechanisms to host foreign guest researchers (undergrad through professor). Typically, such researchers propose their own projects to be completed in collaboration with researchers and use of facilities at NIST.

For more information, contact Spencer Breiner (, Blake Pollard (, and/or Eswaran Subrahmanian (

Applied Category Theory Meeting at UCR (Part 3)

15 November, 2019


We had a special session on applied category theory here at UCR:

Applied category theory, Fall Western Sectional Meeting of the AMS, 9–10 November 2019, U.C. Riverside.

I was bowled over by the large number of cool ideas. I’ll have to blog about some of them. A bunch of people stayed for a few days afterwards, and we had lots of great conversations.

The biggest news was that Brendan Fong and David Spivak definitely want to set up an applied category theory in the San Francisco Bay Area, which they’re calling the Topos Institute. They are now in the process of raising funds for this institute! I plan to be involved, so I’ll be saying more about this later.

But back to the talks. We didn’t make videos, but here are the slides. Click on talk titles to see abstracts of the talks. For a multi-author talk, the person whose name is in boldface is the one who gave the talk. You also might enjoy comparing the 2017 talks.

Saturday November 9, 2019

8:00 a.m.
Fibrations as generalized lens categoriestalk slides.
David I. Spivak, Massachusetts Institute of Technology

9:00 a.m.
Supplying bells and whistles in symmetric monoidal categoriestalk slides.
Brendan Fong, Massachusetts Institute of Technology
David I. Spivak, Massachusetts Institute of Technology

9:30 a.m.
Right adjoints to operadic restriction functorstalk slides.
Philip Hackney, University of Louisiana at Lafayette
Gabriel C. Drummond-Cole, IBS Center for Geometry and Physics

10:00 a.m.
Duality of relationstalk slides.
Alexander Kurz, Chapman University

10:30 a.m.
A synthetic approach to stochastic maps, conditional independence, and theorems on sufficient statisticstalk slides.
Tobias Fritz, Perimeter Institute for Theoretical Physics

3:00 p.m.
Constructing symmetric monoidal bicategories functoriallytalk slides.
Michael Shulman, University of San Diego
Linde Wester Hansen, University of Oxford

3:30 p.m.
Structured cospanstalk slides.
Kenny Courser, University of California, Riverside
John C. Baez, University of California, Riverside

4:00 p.m.
Generalized Petri netstalk slides.
Jade Master, University of California, Riverside

4:30 p.m.
Formal composition of hybrid systemstalk slides and website.

Paul Gustafson, Wright State University
Jared Culbertson, Air Force Research Laboratory
Dan Koditschek, University of Pennsylvania
Peter Stiller, Texas A&M University

5:00 p.m.
Strings for cartesian bicategoriestalk slides.
M. Andrew Moshier, Chapman University

5:30 p.m.
Defining and programming generic compositions in symmetric monoidal categoriestalk slides.
Dmitry Vagner, Los Angeles, CA

Sunday November 10, 2019

8:00 a.m.
Mathematics for second quantum revolutiontalk slides.
Zhenghan Wang, UCSB and Microsoft Station Q

9:00 a.m.
A compositional and statistical approach to natural languagetalk slides.
Tai-Danae Bradley, CUNY Graduate Center

9:30 a.m.
Exploring invariant structure in neural activity with applied topology and category theorytalk slides.
Brad Theilman, UC San Diego
Krista Perks, UC San Diego
Timothy Q Gentner, UC San Diego

10:00 a.m.
Of monks, lawyers and villages: new insights in social network science — talk cancelled due to illness.
Nina Otter, Mathematics Department, UCLA
Mason A. Porter, Mathematics Department, UCLA

10:30 a.m.
Functorial cluster embeddingtalk slides.

Steve Huntsman, BAE Systems FAST Labs

2:00 p.m.
Quantitative equational logictalk slides.
Prakash Panangaden, School of Computer Science, McGill University
Radu Mardare, Strathclyde University
Gordon D. Plotkin, University of Edinburgh

3:00 p.m.
Brakes: an example of applied category theorytalk slides in PDF and Powerpoint.
Eswaran Subrahmanian, Carnegie Mellon University / National Institute of Standards and Technology

3:30 p.m.
Intuitive robotic programming using string diagramstalk slides.
Blake S. Pollard, National Institute of Standards and Technology

4:00 p.m.
Metrics on functor categoriestalk slides.
Vin de Silva, Department of Mathematics, Pomona College

4:30 p.m.
Hausdorff and Wasserstein metrics on graphs and other structured datatalk slides.
Evan Patterson, Stanford University

Why Is Category Theory a Trending Topic?

8 November, 2019

I wrote something for the Spanish newspaper El País, which has a column on mathematics called “Café y Teoremas”. Ágata Timón helped me a lot with writing this, and she also translated it into Spanish:

• John Baez, Qué es la teoría de categorías y cómo se ha convertido en tendencia, El País, 8 November 2019.

Here’s the English-language version I wrote. It’s for a general audience so don’t expect hard-core math!

Why has “category theory” become a trending topic?

Recently, various scientific media have been paying attention to a branch of mathematics called “category theory” that has become pretty popular inside the mathematical community in recent years. Some mathematicians are even starting to complain on Twitter that more people are tweeting about category theory than their own specialties. But what is this branch of mathematics, and why is it becoming so fashionable?

Category theory was invented in 1945 as a general technique to transform problems in one field of pure mathematics into problems in another field, where they could be solved. For example, we know that at any moment there must be a location on the surface of the Earth there where the wind velocity is zero. This is a marvelous result—but to prove this result, we must translate it into a fact about algebra, and a bit of category theory is very helpful here. More difficult results often require more category theory. The proof of Fermat’s Last Theorem, for example, builds on a vast amount of 20th-century mathematics, in which category theory plays a crucial role.

Category theory is sometimes called “the mathematics of mathematics”, since it stands above many other fields of mathematics, connecting and uniting them. Unfortunately even mathematicians have a limited tolerance for this high level of abstraction. So, for a long time many mathematicians called category theory “abstract nonsense”—using it reluctantly when it was necessary for their work, but not really loving it.

On the other hand, other mathematicians embraced the beauty and power of category theory. Thus, its influence has gradually been spreading. Since the 1990s, it has been infiltrating computer science: for example, new programming languages like Haskell and Scala use ideas from this subject. But now we are starting to see people apply category theory to chemistry, electrical engineering, and even the design of brakes in cars! “Applied category theory”, once an oxymoron, is becoming a real subject.

To understand this we need a little taste of the ideas. A category consists of a set of “objects” together with “morphisms”—some kind of processes, or paths—going between these objects. For example, we could take the objects to be cities, and the morphisms to be routes from one city to another. The key requirement is that if we have a morphism from an object x to an object y and a morphism from y to an object z, we can “compose” them and get a morphism from x to z. For example, if you have a way to drive from Madrid to Seville and a way to drive from Seville to Faro, that gives a way to drive from Madrid to Faro. Thus there is a category of cities and routes between them.

In mathematics, this focus on morphisms represented a radical shift of viewpoint. Starting around 1900, logicians tried to build the whole of mathematics on solid foundations. This turned out to be a difficult and elusive task, but their best attempt at the time involved “set theory”. A set is simply a collection of elements. In set theory as commonly practiced by mathematicians, these elements are also just sets. In this worldview, everything is just a set. It is a static worldview, as if we had objects but no morphisms. On the other hand, category theory builds on set theory by emphasizing morphisms—ways of transforming things—as equal partners to things themselves. It is not incompatible with set theory, but it offers new ways of thinking.

The idea of a category is simple. Exploiting it is harder. A loose group of researchers are starting to apply category theory to subjects beyond pure mathematics. The key step is to focus a bit less on things and a bit more on morphisms, which are ways to go between things, or ways to transform one thing into another. This is attitude is well suited to computer programming: a program is a way to transform input data into output data, and composing programs is the easiest way to build complicated programs from simpler ones. But personally, I am most excited by applications to engineering and the natural sciences, because these are newer and more surprising.

I was very pleased when two of my students got internships at the engineering firm Siemens, applying category theory to industrial processes. The first, Blake Pollard, now has a postdoctoral position at the National Institute of Standards and Technology in the USA. Among other things, he has used a programming method based on category theory to help design a “smart grid”—an electrical power network that is flexible enough to handle the ever-changing power generated by thousands of homes equipped with solar panels.

Rumors say that soon there may even be an institute of applied category theory, connecting mathematicians to programmers and businesses who need this way of thinking. It is too early to tell if this is the beginning of a trend, but my friends and colleagues on Twitter are very excited.

Applied Category Theory Meeting at UCR (Part 2)

30 September, 2019


Joe Moeller and I have finalized the schedule of our meeting on applied category theory:

Applied Category Theory, special session of the Fall Western Sectional Meeting of the AMS, U. C. Riverside, Riverside, California, 9–10 November 2019.

It’s going to be really cool, with talks on everything from brakes to bicategories, from quantum physics to social networks, and more—with the power of category theory as the unifying theme!

You can get information on registration, hotels and such here. If you’re coming, you might also want to attend Eugenia Cheng‘s talk on the afternoon of Friday November 8th.   I’ll announce the precise title and time of her talk, and also the location of all the following talks, as soon as I know!

In what follows, the person actually giving the talk has an asterisk by their name. You can click on talk titles to see abstracts of the talks.

Saturday November 9, 2019, 8:00 a.m.-10:50 a.m.

Saturday November 9, 2019, 3:00 p.m.-5:50 p.m.

Sunday November 10, 2019, 8:00 a.m.-10:50 a.m.

Sunday November 10, 2019, 2:00 p.m.-4:50 p.m.

2020 Category Theory Conferences

9 August, 2019


Yes, my last post was about ACT2019, but we’re already planning next year’s applied category theory conference and school! I’m happy to say that Brendan Fong and David Spivak have volunteered to run it at MIT on these dates:

• Applied Category Theory School: June 29–July 3, 2020.
• Applied Category Theory Conference: July 6–10, 2020.

The precise dates for the other big category theory conference, CT2020, have not yet been decided. However, it will take place in Genoa sometime in the interval June 18–28, 2020.

And don’t forget to submit your abstracts for the November 2019 applied category theory special session at U. C. Riverside by September 3rd! We’ve got a great lineup of speakers, but anyone who wants to give a talk—including the invited speakers—needs to submit an abstract to the AMS website by September 3rd. The AMS has no mercy about this.

Applied Category Theory 2019 Talks

20 July, 2019

Applied Category Theory 2019 happened last week! It was very exciting: about 120 people attended, and they’re pushing forward to apply category theory in many different directions. The topics ranged from ultra-abstract to ultra-concrete, sometimes in the same talk.

The talks are listed above — click for a more readable version. Below you can read what Jules Hedges and I wrote about all those talks:

• Jules Hedges, Applied Category Theory 2019.

I tend to give terse summaries of the talks, with links to the original papers or slides. Jules tends to give his impressions of their overall significance. They’re nicely complementary.

You can also see videos of some talks, created by Jelle Herold with help from Fabrizio Genovese:

• Giovanni de Felice, Functorial question answering.

• Antonin Delpeuch, Autonomization of monoidal categories.

• Colin Zwanziger, Natural model semantics for comonadic and adjoint modal type theory.

• Nicholas Behr, Tracelets and tracelet analysis Of compositional rewriting systems.

• Dan Marsden, No-go theorems for distributive laws.

• Christian Williams, Enriched Lawvere theories for operational semantics.

• Walter Tholen, Approximate composition.

• Erwan Beurier, Interfacing biology, category theory & mathematical statistics.

• Stelios Tsampas, Categorical contextual reasoning.

• Fabrizio Genovese, idris-ct: A library to do category theory in Idris.

• Michael Johnson, Machine learning and bidirectional transformations.

• Bruno Gavranović, Learning functors using gradient descent

• Zinovy Diskin, Supervised learning as change propagation with delta lenses.

• Bryce Clarke, Internal lenses as functors and cofunctors.

• Ryan Wisnewsky, Conexus AI.

• Ross Duncan, Cambridge Quantum Computing.

• Beurier Erwan, Memoryless systems generate the class of all discrete systems.

• Blake Pollard, Compositional models for power systems.

• Martti Karvonen, A comonadic view of simulation and quantum resources.

• Quanlong Wang, ZX-Rules for 2-qubit Clifford+T quantum circuits, and beyond.

• James Fairbank, A Compositional framework for scientific model augmentation.

• Titoan Carette, Completeness of graphical languages for mixed state quantum mechanics.

• Antonin Delpeuch, A complete language for faceted dataflow languages.

• John van der Wetering, An effect-theoretic reconstruction of quantum mechanics.

• Vladimir Zamdzhiev, Inductive datatypes for quantum programming.

• Octavio Malherbe, A categorical construction for the computational definition of vector spaces.

• Vladimir Zamdzhiev, Mixed linear and non-linear recursive types.

Applied Category Theory 2019 Program

3 July, 2019

Bob Coecke, David Spivak, Christina Vasilakopoulou and I are running a conference on applied category theory:

Applied Category Theory 2019, 15–19 July, 2019, Lecture Theatre B of the Department of Computer Science, 10 Keble Road, Oxford.

You can now see the program here, or below. Hope to see you soon!

Applied Category Theory Meeting at UCR (Part 1)

16 June, 2019


The American Mathematical Society is having their Fall Western meeting here at U. C. Riverside during the weekend of November 9th and 10th, 2019. Joe Moeller and I are organizing a session on Applied Category Theory! We already have some great speakers lined up:

• Tai-Danae Bradley
• Vin de Silva
• Brendan Fong
• Nina Otter
• Evan Patterson
• Blake Pollard
• Prakash Panangaden
• David Spivak
• Brad Theilman
• Dmitry Vagner
• Zhenghan Wang

Alas, we have no funds for travel and lodging. If you’re interested in giving a talk, please submit an abstract here:

General information about abstracts, American Mathematical Society.

More precisely, please read the information there and then click on the link on that page to submit an abstract. It should then magically fly through the aether to me! Abstracts are due September 3rd, but the sooner you submit one, the greater the chance that we’ll have space.

For the program of the whole conference, go here:

Fall Western Sectional Meeting, U. C. Riverside, Riverside, California, 9–10 November 2019.

I will also be running a special meeting on diversity and excellence in mathematics on Friday November 8th. There will be a banquet that evening, and at some point I’ll figure out how tickets for that will work.

We had a special session like this in 2017, and it’s fun to think about how things have evolved since then.

David Spivak had already written Category Theory for the Sciences, but more recently he’s written another book on applied category theory, Seven Sketches, with Brendan Fong. He already had a company, but now he’s helping run Conexus, which plans to award grants of up to $1.5 million to startups that use category theory (in exchange for equity). Proposals are due June 30th, by the way!

I guess Brendan Fong was already working with David Spivak at MIT in the fall of 2017, but since then they’ve written Seven Sketches and developed a graphical calculus for logic in regular categories. He’s also worked on a functorial approach to machine learning—and now he’s using category theory to unify learners and lenses.

Blake Pollard had just finished his Ph.D. work at U.C. Riverside back in 2018. He will now talk about his work with Spencer Breiner and Eswaran Subrahmanian at the National Institute of Standards and Technology, using category theory to help develop the “smart grid”—the decentralized power grid we need now. Above he’s talking to Brendan Fong at the Centre for Quantum Technologies, in Singapore. I think that’s where they first met.

Eswaran Subrahmanian will also be giving a talk this time! He works at NIST and Carnegie Mellon; he’s an engineer who specializes in using mathematics to help design smart networks and other complex systems.

Nina Otter was a grad student at Oxford in 2017, but now she’s at UCLA and the University of Leipzig. She worked with Ulrike Tillmann and Heather Harrington on stratifying multiparameter persistent homology, and is now working on a categorical formulation of positional and role analysis in social networks. Like Brendan, she’s on the executive board of the applied category theory journal Compositionality.

I first met Tai-Danae Bradley at ACT2018. Now she will talk about her work at Tunnel Technologies, a startup run by her advisor John Terilla. They model sequences—of letters from an alphabet, for instance—using quantum states and tensor networks.

Vin de Silva works on topological data analysis using persistent cohomology so he’ll probably talk about that. He’s studied the “interleaving distance” between persistence modules, using category theory to treat it and the Gromov-Hausdorff metric in the same setting. He came to the last meeting and it will be good to have him back.

Evan Patterson is a statistics grad student at Stanford. He’s worked on knowledge representation in bicategories of relations, and on teaching machines to understand data science code by the semantic enrichment of dataflow graphs. He too came to the last meeting.

Dmitry Vagner was also at the last meeting, where he spoke about his work with Spivak on open dynamical systems and the operad of wiring diagrams. He is now working on mathematically defining and implementing (in Idris) wiring diagrams for symmetric monoidal categories.

Prakash Panangaden has long been a leader in applied category theory, focused on semantics and logic for probabilistic systems and languages, machine learning, and quantum information theory.

Brad Theilman is a grad student in computational neuroscience at U.C. San Diego. I first met him at ACT2018. He’s using algebraic topology to design new techniques for quantifying the spatiotemporal structure of neural activity in the auditory regions of the brain of the European starling. (I bet you didn’t see those last two words coming!)

Last but not least, Zhenghan Wang works on condensed matter physics and modular tensor categories at U.C. Santa Barbara. At Microsoft’s Station Q, he is using this research to help design topological quantum computers.

In short: a lot has been happening in applied category theory, so it will be good to get together and talk about it!

Applied Category Theory 2019

7 February, 2019

I hope to see you at this conference, which will occur right before the associated school meets in Oxford:

Applied Category Theory 2019, July 15-19, 2019, Oxford, UK.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying systems of all sorts using category-theoretic tools. These systems are found in the natural sciences and social sciences, as well as in computer science, linguistics, and engineering. The background and experience of our members is as varied as the systems being studied. The goal of the ACT2019 Conference is to bring the majority of researchers in the field together and provide a platform for exposing the progress in the area. Both original research papers as well as extended abstracts of work submitted/accepted/published elsewhere will be considered.

There will be best paper award(s) and selected contributions will be awarded extended keynote slots.

The conference will include a business showcase and tutorials, and there also will be an adjoint school, the following week (see webpage).

Important dates

Submission of contributed papers: 3 May
Acceptance/Rejection notification: 7 June


Prospective speakers are invited to submit one (or more) of the following:

• Original contributions of high quality work consisting of a 5-12 page extended abstract that provides sufficient evidence of results of genuine interest and enough detail to allow the program committee to assess the merits of the work. Submissions of works in progress are encouraged but must be more substantial than a research proposal.

• Extended abstracts describing high quality work submitted/published elsewhere will also be considered, provided the work is recent and relevant to the conference. These consist of a maximum 3 page description and should include a link to a separate published paper or preprint.

The conference proceedings will be published in a dedicated Proceedings issue of the new Compositionality journal:

Only original contributions are eligible to be published in the proceedings.

Submissions should be prepared using LaTeX, and must be submitted in PDF format. Use of the Compositionality style is encouraged. Submission is done via EasyChair:

Program chairs

John Baez (U.C. Riverside)
Bob Coecke (University of Oxford)

Program committee

Bob Coecke (chair)
John Baez (chair)
Christina Vasilakopoulou
David Moore
Josh Tan
Stefano Gogioso
Brendan Fong
Steve Lack
Simona Paoli
Joachim Kock
Kathryn Hess Bellwald
Tobias Fritz
David I. Spivak
Ross Duncan
Dan Ghica
Valeria de Paiva
Jeremy Gibbons
Samuel Mimram
Aleks Kissinger
Jamie Vicary
Martha Lewis
Nick Gurski
Dusko Pavlovic
Chris Heunen
Corina Cirstea
Helle Hvid Hansen
Dan Marsden
Simon Willerton
Pawel Sobocinski
Dominic Horsman
Nina Otter
Miriam Backens

Steering committee

John Baez (U.C. Riverside)
Bob Coecke (University of Oxford)
David Spivak (M.I.T.)
Christina Vasilakopoulou (U.C. Riverside)