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.

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.

My student Christian Williams has started a community server for category theory, computer science, logic, as well as general science and industry. In just a few days, it has grown into a large and lively place, with people of many backgrounds and interests. Please feel free to join!

Join here:

https://categorytheory.zulipchat.com/join/zf9mmiai0xracqt7gxdeboxb/

(this link will expire in a while) and from then on you can just go here:

http://categorytheory.zulipchat.com

Due to the coronavirus outbreak, many universities are moving activities online. This is a great opportunity to open up ACT2020 to a broader audience, with speakers from around the world.

The conference will take place July 6-10 online, coordinated by organizers in Boston USA. Each day there will be around six hours of live talks, which will be a bit more spaced out than usual to accommodate the different time zones of our speakers. All the talks will be both live streamed and recorded on YouTube. We will also have chat rooms and video chats in which participants can discuss various themes in applied category theory.

We will give more details as they become available and post updates on our official webpage:

Since there is no need to book travel, we were able to postpone the acceptance notification, and hence the submission deadline. If you would like to speak, please prepare an abstract or a conference paper according to the instructions here:

http://act2020.mit.edu/#papers

**Important dates (all in 2020)**

• Submission of contributed papers: May 10

• Acceptance/Rejection notification: June 7

• Tutorial day: July 5

• Main conference: July 6-10

Registration will now be free; please register for the conference ahead of time here:

http://act2020.mit.edu/#registration

We will send registering participants links to the live stream, the recordings, and the chat rooms, and we’ll use the list to inform participants of any changes.

**Submissions**

To give a talk at ACT2020, you have to submit a paper. You can submit either original research papers or extended abstracts of work submitted/accepted/published elsewhere. Accepted original research papers will be invited for publication in a proceedings volume.

Here’s how to submit papers. Two types of submissions are accepted, which will be reviewed to the same standards:

**Proceedings Track.** Original contributions of high quality work consisting of a 5–12 page extended abstract that provides evidence for results of genuine interest, and with 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.

**Non-Proceedings Track.** Descriptions of high-quality work submitted or published elsewhere will also be considered, provided the work is recent and relevant to the conference. The work may be of any length, but the program committee members may only look at the first 3 pages of the submission, so you should ensure these pages contain sufficient evidence of the quality and rigor of your work.

Submissions should be prepared using LaTeX, and must be submitted in PDF format. Submission is currently open, and can be perfomed at the following web page:

https://easychair.org/conferences/?conf=act2020

One or more best paper awards may be given out at the discretion of the PC chairs. Selected contributions will be offered extended keynote slots in the program.

**Organizers**

Here are the local organizers:

• Destiny Chen (administration)

• Brendan Fong

• David Jaz Myers (logistics)

• Paolo Perrone (publicity)

• David Spivak

Here is the committee running the school:

• Carmen Constantin

• Eliana Lorch

• Paolo Perrone

Here is the steering committee:

• John Baez

• Bob Coecke

• David Spivak

• Christina Vasilakopoulou

Here is the program committee:

• Mathieu Anel, CMU

• John Baez, University of California, Riverside

• Richard Blute, University of Ottawa

• Tai-Danae Bradley, City University of New York

• Andrea Censi, ETC Zurich

• Bob Coecke, University of Oxford

• Valeria de Paiva, Samsung Research America and University of Birmingham

• Ross Duncan, University of Strathclyde

• Eric Finster, University of Birmingham

• Brendan Fong, Massachusetts Institute of Technology

• Tobias Fritz, Perimeter Institute for Theoretical Physics

• Richard Garner, Macquarie University

• Fabrizio Romano Genovese, Statebox

• Amar Hadzihasanovic, IRIF, Université de Paris

• Helle Hvid Hansen, Delft University of Technology

• Jules Hedges, Max Planck Institute for Mathematics in the Sciences

• Kathryn Hess Bellwald, Ecole Polytechnique Fédérale de Lausanne

• Chris Heunen, The University of Edinburgh

• Joachim Kock, UAB

• Tom Leinster, The University of Edinburgh

• Martha Lewis, University of Amsterdam

• Daniel R. Licata, Wesleyan University

• David Jaz Myers, Johns Hopkins University

• Paolo Perrone, MIT

• Vaughan Pratt, Stanford University

• Peter Selinger, Dalhousie University

• Michael Shulman, University of San Diego

• *David I. Spivak, MIT (co-chair)*

• Walter Tholen, York University

• Todd Trimble, Western Connecticut State University

• *Jamie Vicary, University of Birmingham (co-chair)*

• Maaike Zwart, University of Oxford

Here’s the big annual conference on applied category theory:

• ACT2020, 2020 July 6–10, online worldwide. Organized by Brendan Fong and David Spivak.

This happens right after the applied category theory school, which will take place June 29 – July 3. There will also be a tutorial day on Sunday July 5, with talks by Paolo Perrone, Emily Riehl, David Spivak and others.

To give a talk at ACT2020, you have to submit a paper. You can submit either original research papers or extended abstracts of work submitted/accepted/published elsewhere. Accepted original research papers will be invited for publication in a proceedings volume. Some contributions will be invited to become keynote addresses, and best paper awards may also be given. The conference will also include a business showcase.

Here’s how to submit papers. Two types of submissions are accepted, which will be reviewed to the same standards:

**Proceedings Track.** Original contributions of high quality work consisting of a 5–12 page extended abstract that provides evidence for results of genuine interest, and with 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.

**Non-Proceedings Track.** Descriptions of high-quality work submitted or published elsewhere will also be considered, provided the work is recent and relevant to the conference. The work may be of any length, but the program committee members may only look at the first 3 pages of the submission, so you should ensure these pages contain sufficient evidence of the quality and rigor of your work.

Submissions should be prepared using LaTeX, and must be submitted in PDF format. Submission is currently open, and can be perfomed at the following web page:

https://easychair.org/conferences/?conf=act2020

Here are some important dates, all in 2020:

• Submission of contributed papers: April 26

• Acceptance/rejection notification: May 17

• Early bird registration deadline: May 20

• Final registration deadline: June 26

• Tutorial day: July 5

• Main conference: July 6–10

Here is the program committee:

• Mathieu Anel, CMU

• John Baez, University of California, Riverside

• Richard Blute, University of Ottawa

• Tai-Danae Bradley, City University of New York

• Andrea Censi, ETC Zurich

• Bob Coecke, University of Oxford

• Valeria de Paiva, Samsung Research America and University of Birmingham

• Ross Duncan, University of Strathclyde

• Eric Finster, University of Birmingham

• Brendan Fong, Massachusetts Institute of Technology

• Tobias Fritz, Perimeter Institute for Theoretical Physics

• Richard Garner, Macquarie University

• Fabrizio Romano Genovese, Statebox

• Amar Hadzihasanovic, IRIF, Université de Paris

• Helle Hvid Hansen, Delft University of Technology

• Jules Hedges, Max Planck Institute for Mathematics in the Sciences

• Kathryn Hess Bellwald, Ecole Polytechnique Fédérale de Lausanne

• Chris Heunen, The University of Edinburgh

• Joachim Kock, UAB

• Tom Leinster, The University of Edinburgh

• Martha Lewis, University of Amsterdam

• Daniel R. Licata, Wesleyan University

• David Jaz Myers, Johns Hopkins University

• Paolo Perrone, MIT

• Vaughan Pratt, Stanford University

• Peter Selinger, Dalhousie University

• Michael Shulman, University of San Diego

• *David I. Spivak, MIT (co-chair)*

• Walter Tholen, York University

• Todd Trimble, Western Connecticut State University

• *Jamie Vicary, University of Birmingham (co-chair)*

• Maaike Zwart, University of Oxford

Here is the steering committee:

• John Baez

• Bob Coecke

• David Spivak

• Christina Vasilakopoulou

Here is the committee running the school:

• Carmen Constantin

• Eliana Lorch

• Paolo Perrone

And here are the local organizers:

• Destiny Chen (administration)

• Brendan Fong

• David Jaz Myers (logistics)

• Paolo Perrone (publicity)

• David Spivak

More news will follow!

**Sadly, this workshop has been cancelled due to the coronavirus pandemic. It may be postponed to a later date.**

My former student Blake Pollard is working at the National Institute of Standards and Technology. He’s working with Spencer Breiner and Eswaran Subrahmanian, who are big advocates of using category theory to organize design and manufacturing processes. In the spring of 2018 they had a workshop on applied category theory with a lot of honchos from industry and government in attendance—you can see videos by clicking the link.

This spring they’re having another workshop on this topic!

• Applied Category Theory Workshop, April 8-9, 2020, National Institute of Standards and Technology, Gaithersburg, Maryland. Organized by Spencer Breiner, Blake Pollard and Eswaran Subrahmanian.

The focus of this workshop in on fostering the development of tooling and use-cases supporting the applied category theory community. We are particularly interested in bringing together practitioners who are engaged with susceptible domains as well as those involved in the implementation, support, and utilization of software and other tools. There will be a number of talks/demos showcasing existing approaches as well as ample time for discussion.

Here are the speakers listed so far:

• John Baez, University of California, Riverside

• Arquimedes Canedo, Siemens

• Daniel Cicala, New Haven University

• James Fairbanks, Georgia Tech Research Institute

• Jules Hedges, Max Planck Institute for the Mathematical Sciences

• Jelle Herold, Statebox

• Evan Patterson, Stanford University

• Qunfen Qi, University of Huddersfield

• Christian Williams, University of California, Riverside

• Ryan Wisnesky, Conexus.ai

I’ll also be giving a separate talk on “ecotechnology” at NIST on Friday April 10th; more about that later!

An interesting situation has arisen. Some people working on applied category theory have been seeking a ‘killer app’: that is, an application of category theory to practical tasks that would be so compelling it would *force* the world to admit categories are useful. Meanwhile, the UMAP algorithm, based to some extent on category theory, has become very important in genomics:

• Leland McInnes, John Healy and James Melville, UMAP: uniform manifold approximation and projection for dimension reduction.

But while practitioners have embraced the algorithm, they’re still puzzled by its category-theoretic underpinnings, which are discussed in Section 2 of the paper. (You can read the remaining sections, which describe the algorithm quite concretely, without understanding Section 2.)

I first heard of this situation on Twitter when James Nichols wrote:

Wow! My first sighting of applied category theory: the UMAP algorithm. I’m a category novice, but the resulting adjacency-graph algorithm is v simple, so surely the theory boils down to reasonably simple arguments in topology/Riemannian geometry?

Do any of you prolific ACT tweeters know much about UMAP? I understand the gist of the linked paper, but not say why we need category theory to define this “fuzzy topology” concept, as opposed to some other analytic defn.

Junhyong Kim added:

What was gained by CT for UMAP? (honest question, not trying to be snarky)

Leland McInnes, one of the inventors of UMAP, responded:

It is my math background, how I think about the problem, and how the algorithm was derived. It wasn’t something that was added, but rather something that was always there—for me at least. In that sense what was gained was the algorithm.

I don’t really understand UMAP; for a good introduction to it see the original paper above and also this:

• Nikolay Oskolkov, How Exactly UMAP Works—and Why Exactly It Is Better Than tSNE, 3 October 2019.

tSNE is an older algorithm for taking clouds of data points in high dimensions and mapping them down to fewer dimensions so we can understand what’s going on. From the viewpoint of those working on genomics, the main good thing about UMAP is that it solves a bunch of problems that plagued tSNE. Oskolkov explains what these problems are and how UMAP deals with them. But he also alludes to the funny disconnect between these practicalities and the underlying theory:

My first impression when I heard about UMAP was that this was a completely novel and interesting dimension reduction technique which is based on solid mathematical principles and hence very different from tSNE which is a pure Machine Learning semi-empirical algorithm. My colleagues from Biology told me that the original UMAP paper was “too mathematical”, and looking at the Section 2 of the paper I was very happy to see strict and accurate mathematics finally coming to Life and Data Science. However, reading the UMAP docs and watching Leland McInnes talk at SciPy 2018, I got puzzled and felt like UMAP was

another neighbor graphtechnique which is so similar to tSNE thatI was struggling to understand how exactly UMAP is different from tSNE.

He then goes on and attempts to explain exactly why UMAP does so much better than tSNE. None of his explanation mentions category theory.

Since I don’t really understand UMAP or why it does better than tSNE, I can’t add anything to this discussion. In particular, I can’t say how much the category theory really helps. All I can do is explain a bit of the category theory. I’ll do that now, *very briefly*, just as a way to get a conversation going. I will try to avoid category-theoretic jargon as much as possible—not because I don’t like it or consider it unimportant, but because that jargon is precisely what’s stopping certain people from understanding Section 2.

I think it all starts with this paper by Spivak, which McInnes, Healy and Melville cite but for some reason don’t provide a link to:

• David Spivak, Metric realization of fuzzy simplicial sets.

Spivak showed how to turn a ‘fuzzy simplicial set’ into an ‘uber-metric space’ and vice versa. What are these things?

An ‘uber-metric space’ is very simple. It’s a slight generalization of a metric space that relaxes the usual definition in just two ways: it lets distances be infinite, and it lets distinct points have distance zero from each other. This sort of generalization can be very useful. I could talk about it a lot, but I won’t.

A fuzzy simplicial set is a generalization of a simplicial set.

A simplicial set starts out as a set of vertices (or 0-simplices), a set of edges (or 1-simplices), a set of triangles (or 2-simplices), a set of tetrahedra (or 3-simplices), and so on: in short, a set of n-simplices for each n. But there’s more to it. Most importantly, each n-simplex has a bunch of faces, which are lower-dimensional simplices.

I won’t give the whole definition. To a first approximation you can visualize a simplicial set as being like this:

But of course it doesn’t have to stop at dimension 3—and more subtly, you can have things like two different triangles that have exactly the same edges.

In a ‘fuzzy’ simplicial set, instead of a *set* of n-simplices for each n, we have a *fuzzy set* of them. But what’s a fuzzy set?

Fuzzy set theory is good for studying collections where membership is somewhat vaguely defined. Like a set, a fuzzy set has elements, but each element has a ‘degree of membership’ that is a number 0 < x ≤ 1. (If its degree of membership were zero, it wouldn't be an element!)

A map f: X → Y between fuzzy sets is an ordinary function, but obeying this condition: it can only send an element x ∈ X to an element f(x) ∈ Y whose degree of membership is greater than or equal to that of x. In other words, we don't want functions that send things to things with a lower degree of membership.

Why? Well, if I'm quite sure something is a dog, and every dog has a nose, then I'm must be at least equally sure that this dog has a nose! (If you disagree with this, then you can make up some other concept of fuzzy set. There are a number of such concepts, and I'm just describing one.)

So, a fuzzy simplicial set will have a set of n-simplices for each n, with each n-simplex having a degree of membership… but the degree of membership of its faces can't be less than its own degree of membership.

This is not the precise definition of fuzzy simplicial set, because I'm leaving out some distracting nuances. But you can get the precise definition by taking a nuts-and-bolts definition of simplicial set, like Definition 3.2 here:

• Greg Friedman, An elementary illustrated introduction to simplicial sets.

and replacing all the sets by fuzzy sets, and all the maps by maps between fuzzy sets.

If you like visualizing things, you can visualize a fuzzy simplicial set as an ordinary simplicial set, as in the picture above, but where an n-simplex is shaded darker if its degree of membership is higher. An n-simplex can’t be shaded darker than any of its faces.

How can you turn a fuzzy simplicial set into an uber-metric space? And how can you turn an uber-metric space into a fuzzy simplicial set?

Spivak focuses on the first question, because the answer is simpler, and it determines the answer to the second using some category theory. (Psst: *adjoint functors!*)

The answer to the first question goes like this. Say you have a fuzzy simplicial set. For each n-simplex whose degree of membership equals you turn it into a copy of this uber-metric space:

This is really just an ordinary metric space: an n-simplex that’s a subspace of Euclidean (n+1)-dimensional space with its usual Euclidean distance function. Then you glue together all these uber-metric spaces, one for each simplex in your fuzzy simplical set, to get a big fat uber-metric space.

This process is called ‘realization’. The key here is that if an n-simplex has a *high* degree of membership, it gets ‘realized’ as a metric space shaped like a *small* n-simplex. I believe the basic intuition is that an n-simplex with a high degree of membership describes an (n+1)-tuple of things—its vertices—that are *close* to each other.

In theory, I should try to describe the reverse process that turns an uber-metric space into a fuzzy simplicial set. If I did, I believe we would see that whenever an (n+1)-tuple of things—that is, points of our uber-metric space—are *close*, they give an n-simplex with a high degree of membership.

If so, then both uber-metric spaces and fuzzy simplicial sets are just ways of talking about which collections of data points are close, and we can translate back and forth between these descriptions.

But I’d need to think about this a bit more to do a good job of going further, and reading the UMAP paper a bit more I’m beginning to suspect that’s not the main thing that practitioners need to understand. I’m beginning to think the most useful thing is to get a feeling for fuzzy simplicial sets! I hope I’ve helped a bit in that direction. They are very simple things. They are also closely connected to an idea from topological data analysis:

• Nina Otter, Magnitude meets persistence. Homology theories for filtered simplicial sets.

I should admit that McInnes, Healy and Melville tweak Spivak’s formalism a bit. They call Spivak’s uber-metric spaces ‘extended-pseudo-metric spaces’, but they focus on a special kind, which they call ‘finite’. Unfortunately, I can’t find where they define this term. They also only consider a special sort of fuzzy simplicial set, which they call ‘bounded’, but I can’t find the definition of this term either! Without knowing these definitions, I can’t comment on how these tweaks change things.

Like last year and the year before, there will be a *school* associated to this year’s international 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!

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.

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.

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.

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*.

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 SystemsThese 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 (spencer.breiner@nist.gov), Blake Pollard (blake.pollard@nist.gov), and/or Eswaran Subrahmanian (sub@cmu.edu).

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.

• 8:00 a.m.

Fibrations as generalized lens categories — talk slides.

**David I. Spivak**, Massachusetts Institute of Technology

• 9:00 a.m.

Supplying bells and whistles in symmetric monoidal categories — talk slides.

**Brendan Fong**, Massachusetts Institute of Technology

David I. Spivak, Massachusetts Institute of Technology

• 9:30 a.m.

Right adjoints to operadic restriction functors — talk slides.

**Philip Hackney**, University of Louisiana at Lafayette

Gabriel C. Drummond-Cole, IBS Center for Geometry and Physics

• 10:00 a.m.

Duality of relations — talk slides.

**Alexander Kurz**, Chapman University

• 10:30 a.m.

A synthetic approach to stochastic maps, conditional independence, and theorems on sufficient statistics — talk slides.

**Tobias Fritz**, Perimeter Institute for Theoretical Physics

• 3:00 p.m.

Constructing symmetric monoidal bicategories functorially — talk slides.

**Michael Shulman**, University of San Diego

Linde Wester Hansen, University of Oxford

• 3:30 p.m.

Structured cospans — talk slides.

**Kenny Courser**, University of California, Riverside

John C. Baez, University of California, Riverside

• 4:00 p.m.

Generalized Petri nets — talk slides.

**Jade Master**, University of California, Riverside

• 4:30 p.m.

Formal composition of hybrid systems — talk 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 bicategories — talk slides.

**M. Andrew Moshier**, Chapman University

• 5:30 p.m.

Defining and programming generic compositions in symmetric monoidal categories — talk slides.

**Dmitry Vagner**, Los Angeles, CA

• 8:00 a.m.

Mathematics for second quantum revolution — talk slides.

**Zhenghan Wang**, UCSB and Microsoft Station Q

• 9:00 a.m.

A compositional and statistical approach to natural language — talk slides.

**Tai-Danae Bradley**, CUNY Graduate Center

• 9:30 a.m.

Exploring invariant structure in neural activity with applied topology and category theory — talk 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 embedding — talk slides.

**Steve Huntsman**, BAE Systems FAST Labs

• 2:00 p.m.

Quantitative equational logic — talk 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 theory — talk 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 diagrams — talk slides.

**Blake S. Pollard**, National Institute of Standards and Technology

• 4:00 p.m.

Metrics on functor categories — talk slides.

**Vin de Silva**, Department of Mathematics, Pomona College

• 4:30 p.m.

Hausdorff and Wasserstein metrics on graphs and other structured data — talk slides.

**Evan Patterson**, Stanford University

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!

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.