Applied Category Theory at UCR (Part 3)

13 November, 2017

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

Applied category theory, Fall Western Sectional Meeting of the AMS, 4-5 November 2017, U.C. Riverside.

A bunch of people stayed for a few days afterwards, and we had a lot of great discussions. I wish I could explain everything that happened, but I’m too busy right now. Luckily, even if you couldn’t come here, you can now see slides of almost all the talks… and videos of many!

Click on talk titles to see abstracts. For multi-author talks, the person whose name is in boldface is the one who gave the talk. For videos, go here: I haven’t yet created links to all the videos.

Saturday November 4, 2017

9:00 a.m.A higher-order temporal logic for dynamical systemstalk slides.
David I. Spivak, MIT.

10:00 a.m.
Algebras of open dynamical systems on the operad of wiring diagramstalk slides.
Dmitry Vagner, Duke University
David I. Spivak, MIT
Eugene Lerman, University of Illinois at Urbana-Champaign

10:30 a.m.
Abstract dynamical systemstalk slides.
Christina Vasilakopoulou, UCR
David Spivak, MIT
Patrick Schultz, MIT

3:00 p.m.
Decorated cospanstalk slides.
Brendan Fong, MIT

4:00 p.m.
Compositional modelling of open reaction networkstalk slides.
Blake S. Pollard, UCR
John C. Baez, UCR

4:30 p.m.
A bicategory of coarse-grained Markov processestalk slides.
Kenny Courser, UCR

5:00 p.m.
A bicategorical syntax for pure state qubit quantum mechanicstalk slides.
Daniel M. Cicala, UCR

5:30 p.m.
Open systems in classical mechanicstalk slides.
Adam Yassine, UCR

Sunday November 5, 2017

9:00 a.m.
Controllability and observability: diagrams and dualitytalk slides.
Jason Erbele, Victor Valley College

9:30 a.m.
Frobenius monoids, weak bimonoids, and corelationstalk slides.
Brandon Coya, UCR

10:00 a.m.
Compositional design and tasking of networks.
John D. Foley, Metron, Inc.
John C. Baez, UCR
Joseph Moeller, UCR
Blake S. Pollard, UCR

10:30 a.m.
Operads for modeling networkstalk slides.
Joseph Moeller, UCR
John Foley, Metron Inc.
John C. Baez, UCR
Blake S. Pollard, UCR

2:00 p.m.
Reeb graph smoothing via cosheavestalk slides.
Vin de Silva, Department of Mathematics, Pomona College

3:00 p.m.
Knowledge representation in bicategories of relationstalk slides.
Evan Patterson, Stanford University, Statistics Department

3:30 p.m.
The multiresolution analysis of flow graphstalk slides.
Steve Huntsman, BAE Systems

4:00 p.m.
Data modeling and integration using the open source tool Algebraic Query Language (AQL)talk slides.
Peter Y. Gates, Categorical Informatics
Ryan Wisnesky, Categorical Informatics

Complex Adaptive Systems (Part 6)

31 October, 2017

I’ve been slacking off on writing this series of posts… but for a good reason: I’ve been busy writing a paper on the same topic! In the process I caught a couple of mistakes in what I’ve said so far. But more importantly, there’s a version out now, that you can read:

• John Baez, John Foley, Blake Pollard and Joseph Moeller, Network models.

There will be two talks about this at the AMS special session on Applied Category Theory this weekend at U. C. Riverside: one by John Foley of Metron Inc., and one by my grad student Joseph Moeller. I’ll try to get their talk slides someday. But for now, here’s the basic idea.

Our goal is to build operads suited for designing networks. These could be networks where the vertices represent fixed or moving agents and the edges represent communication channels. More generally, they could be networks where the vertices represent entities of various types, and the edges represent relationships between these entities—for example, that one agent is committed to take some action involving the other. This paper arose from an example where the vertices represent planes, boats and drones involved in a search and rescue mission in the Caribbean. However, even for this one example, we wanted a flexible formalism that can handle networks of many kinds, described at a level of detail that the user is free to adjust.

To achieve this flexibility, we introduced a general concept of ‘network model’. Simply put, a network model is a kind of network. Any network model gives an operad whose operations are ways to build larger networks of this kind by gluing smaller ones. This operad has a ‘canonical’ algebra where the operations act to assemble networks of the given kind. But it also has other algebras, where it acts to assemble networks of this kind equipped with extra structure and properties. This flexibility is important in applications.

What exactly is a ‘kind of network’? That’s the question we had to answer. We started with some examples, At the crudest level, we can model networks as simple graphs. If the vertices are agents of some sort and the edges represent communication channels, this means we allow at most one channel between any pair of agents.

However, simple graphs are too restrictive for many applications. If we allow multiple communication channels between a pair of agents, we should replace simple graphs with ‘multigraphs’. Alternatively, we may wish to allow directed channels, where the sender and receiver have different capabilities: for example, signals may only be able to flow in one direction. This requires replacing simple graphs with ‘directed graphs’. To combine these features we could use ‘directed multigraphs’.

But none of these are sufficiently general. It’s also important to consider graphs with colored vertices, to specify different types of agents, and colored edges, to specify different types of channels. This leads us to ‘colored directed multigraphs’.

All these are examples of what we mean by a ‘kind of network’, but none is sufficiently general. More complicated kinds, such as hypergraphs or Petri nets, are likely to become important as we proceed.

Thus, instead of separately studying all these kinds of networks, we introduced a unified notion that subsumes all these variants: a ‘network model’. Namely, given a set C of ‘vertex colors’, a network model is a lax symmetric monoidal functor

F: \mathbf{S}(C) \to \mathbf{Cat}

where \mathbf{S}(C) is the free strict symmetric monoidal category on C and \mathbf{Cat} is the category of small categories.

Unpacking this somewhat terrifying definition takes a little work. It simplifies in the special case where F takes values in \mathbf{Mon}, the category of monoids. It simplifies further when C is a singleton, since then \mathbf{S}(C) is the groupoid \mathbf{S}, where objects are natural numbers and morphisms from m to n are bijections

\sigma: \{1,\dots,m\} \to \{1,\dots,n\}

If we impose both these simplifying assumptions, we have what we call a one-colored network model: a lax symmetric monoidal functor

F : \mathbf{S} \to \mathbf{Mon}

As we shall see, the network model of simple graphs is a one-colored network model, and so are many other motivating examples. If you like André Joyal’s theory of ‘species’, then one-colored network models should be pretty fun, since they’re species with some extra bells and whistles.

But if you don’t, there’s still no reason to panic. In relatively down-to-earth terms, a one-colored network model amounts to roughly this. If we call elements of F(n) ‘networks with n vertices’, then:

• Since F(n) is a monoid, we can overlay two networks with the same number of vertices and get a new one. We call this operation

\cup \colon F(n) \times F(n) \to F(n)

• Since F is a functor, the symmetric group S_n acts on the monoid F(n). Thus, for each \sigma \in S_n, we have a monoid automorphism that we call simply

\sigma \colon F(n) \to F(n)

• Since F is lax monoidal, we also have an operation

\sqcup \colon F(m) \times F(n) \to F(m+n)

We call this operation the disjoint union of networks. In examples like simple graphs, it looks just like what it sounds like.

Unpacking the abstract definition further, we see that these operations obey some equations, which we list in Theorem 11 of our paper. They’re all obvious if you draw pictures of examples… and don’t worry, our paper has a few pictures. (We plan to add more.) For example, the ‘interchange law’

(g \cup g') \sqcup (h \cup h') = (g \sqcup h) \cup (g' \sqcup h')

holds whenever g,g' \in F(m) and h, h' \in F(n). This is a nice relationship between overlaying networks and taking their disjoint union.

In Section 2 of our apper we study one-colored network models, and give lots of examples. In Section 3 we describe a systematic procedure for getting one-colored network models from monoids. In Section 4 we study general network models and give examples of these. In Section 5 we describe a category \mathbf{NetMod} of network models, and show that the procedure for getting network models from monoids is functorial. We also make \mathbf{NetMod} into a symmetric monoidal category, and give examples of how to build new networks models by tensoring old ones.

Our main result is that any network model gives a typed operad, also known as a ‘colored operad’. This operad has operations that describe how to stick networks of the given kind together to form larger networks of this kind. This operad has a ‘canonical algebra’, where it acts on networks of the given kind—but the real point is that it has lots of other algebra, where it acts on networks of the given kind equipped with extra structure and properties.

The technical heart of our paper is Section 6, mainly written by Joseph Moeller. This provides the machinery to construct operads from network models in a functorial way. Category theorists should find this section interesting, because because it describes enhancements of the well-known ‘Grothendieck construction’ of the category of elements \int F of a functor

F: \mathbf{C} \to \mathbf{Cat}

where \mathbf{C} is any small category. For example, if \mathbf{C} is symmetric monoidal and F : \mathbf{C} \to (\mathbf{Cat}, \times) is lax symmetric monoidal, then we show \int F is symmetric monoidal. Moreover, we show that the construction sending the lax symmetric monoidal functor F to the symmetric monoidal category \int F is functorial.

In Section 7 we apply this machinery to build operads from network models. In Section 8 we describe some algebras of these operads, including an algebra whose elements are networks of range-limited communication channels. In future work we plan to give many more detailed examples, and to explain how these algebras, and the homomorphisms between them, can be used to design and optimize networks.

I want to explain all this in more detail—this is a pretty hasty summary, since I’m busy this week. But for now you can read the paper!

Applied Category Theory 2018 — Adjoint School

22 October, 2017

The deadline for applying to this ‘school’ on applied category theory is Wednesday November 1st.

Applied Category Theory: Adjoint School: online sessions starting in January 2018, followed by a meeting 23–27 April 2018 at the Lorentz Center in Leiden, the Netherlands. Organized by Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford).

The name ‘adjoint school’ is a bad pun, but the school should be great. Here’s how it works:


The Workshop on Applied Category Theory 2018 takes place in May 2018. A principal goal of this workshop is to bring early career researchers into the applied category theory community. Towards this goal, we are organising the Adjoint School.

The Adjoint School will run from January to April 2018. By the end of the school, each participant will:

  • be familiar with the language, goals, and methods of four prominent, current research directions in applied category theory;
  • have worked intensively on one of these research directions, mentored by an expert in the field; and
  • know other early career researchers interested in applied category theory.

They will then attend the main workshop, well equipped to take part in discussions across the diversity of applied category theory.


The Adjoint School comprises (1) an Online Reading Seminar from January to April 2018, and (2) a four day Research Week held at the Lorentz Center, Leiden, The Netherlands, from Monday April 23rd to Thursday April 26th.

In the Online Reading Seminar we will read papers on current research directions in applied category theory. The seminar will consist of eight iterations of a two week block. Each block will have one paper as assigned reading, two participants as co-leaders, and three phases:

  • A presentation (over WebEx) on the assigned reading delivered by the two block co-leaders.
  • Reading responses and discussion on a private forum, facilitated by Brendan Fong and Nina Otter.
  • Publication of a blog post on the n-Category Café written by the co-leaders.

Each participant will be expected to co-lead one block.

The Adjoint School is taught by mentors John Baez, Aleks Kissinger, Martha Lewis, and Pawel Sobocinski. Each mentor will mentor a working group comprising four participants. During the second half of the Online Reading Seminar, these working groups will begin to meet with their mentor (again over video conference) to learn about open research problems related to their reading.

In late April, the participants and the mentors will convene for a four day Research Week at the Lorentz Center. After opening lectures by the mentors, the Research Week will be devoted to collaboration within the working groups. Morning and evening reporting sessions will keep the whole school informed of the research developments of each group.

The following week, participants will attend Applied Category Theory 2018, a discussion-based 60-attendee workshop at the Lorentz Center. Here they will have the chance to meet senior members across the applied category theory community and learn about ongoing research, as well as industry applications.

Following the school, successful working groups will be invited to contribute to a new, to be launched, CUP book series.

Reading list

Meetings will be on Mondays; we will determine a time depending on the locations of the chosen participants.

Research projects

John Baez: Semantics for open Petri nets and reaction networks
Petri nets and reaction networks are widely used to describe systems of interacting entities in computer science, chemistry and other fields, but the study of open Petri nets and reaction networks is new, and raise many new questions connected to Lawvere’s “functorial semantics”.
Reading: Fong; Baez and Pollard.

Aleks Kissinger: Unification of the logic of causality
Employ the framework of (pre-)causal categories to unite notions of causality and techniques for causal reasoning which occur in classical statistics, quantum foundations, and beyond.
Reading: Kissinger and Uijlen; Henson, Lal, and Pusey.

Martha Lewis: Compositional approaches to linguistics and cognition
Use compact closed categories to integrate compositional models of meaning with distributed, geometric, and other meaning representations in linguistics and cognitive science.
Reading: Coecke, Sadrzadeh, and Clark; Bolt, Coecke, Genovese, Lewis, Marsden, and Piedeleu.

Pawel Sobocinski: Modelling of open and interconnected systems
Use Carboni and Walters’ bicategories of relations as a multidisciplinary algebra of open and interconnected systems.
Reading: Carboni and Walters; Willems.


We hope that each working group will comprise both participants who specialise in category theory and in the relevant application field. As a prerequisite, those participants specialising in category theory should feel comfortable with the material found in Categories for the Working Mathematician or its equivalent; those specialising in applications should have a similar graduate-level introduction.

To apply, please fill out the form here. You will be asked to upload a single PDF file containing the following information:

  • Your contact information and educational history.
  • A brief paragraph explaining your interest in this course.
  • A paragraph or two describing one of your favorite topics in category theory, or your application field.
  • A ranked list of the papers you would most like to present, together with an explanation of your preferences. Note that the paper you present determines which working group you will join.

You may add your CV if you wish.

Anyone is welcome to apply, although preference may be given to current graduate students and postdocs. Women and members of other underrepresented groups within applied category theory are particularly encouraged to apply.

Some support will be available to help with the costs (flights, accommodation, food, childcare) of attending the Research Week and the Workshop on Applied Category Theory; please indicate in your application if you would like to be considered for such support.

If you have any questions, please feel free to contact Brendan Fong (bfo at mit dot edu) or Nina Otter (otter at maths dot ox dot ac dot uk).

Application deadline: November 1st, 2017.

Vladimir Voevodsky, 1966 — 2017

6 October, 2017

Vladimir Voevodsky died last week. He won the Fields Medal in 2002 for proving the Milnor conjecture in a branch of algebra known as algebraic K-theory. He continued to work on this subject until he helped prove the more general Bloch–Kato conjecture in 2010.

Proving these results—which are too technical to easily describe to nonmathematicians!—required him to develop a dream of Grothendieck: the theory of motives. Very roughly, this is a way of taking the space of solutions of some polynomial equations and chopping it apart into building blocks. But this process of ‘chopping’ and also these building blocks, called ‘motives’, are very abstract—nothing easy to visualize.

It’s a bit like how a proton is made of quarks. You never actually see a quark in isolation, so you have to think very hard to realize they are there at all. But once you know this, a lot of things become clear.

This is wonderful, profound mathematics. But in the process of proving the Bloch-Kato conjecture, Voevodsky became tired of this stuff. He wanted to do something more useful… and more ambitious. He later said:

It was very difficult. In fact, it was 10 years of technical work on a topic that did not interest me during the last 5 of these 10 years. Everything was done only through willpower.

Since the autumn of 1997, I already understood that my main contribution to the theory of motives and motivic cohomology was made. Since that time I have been very conscious and actively looking for. I was looking for a topic that I would deal with after I fulfilled my obligations related to the Bloch-Kato hypothesis.

I quickly realized that if I wanted to do something really serious, then I should make the most of my accumulated knowledge and skills in mathematics. On the other hand, seeing the trends in the development of mathematics as a science, I realized that the time is coming when the proof of yet another conjecture won’t have much of an effect. I realized that mathematics is on the verge of a crisis, or rather, two crises.

The first is connected with the separation of “pure” and applied mathematics. It is clear that sooner or later there will be a question about why society should pay money to people who are engaged in things that do not have any practical applications.

The second, less obvious, is connected with the complication of pure mathematics, which leads to the fact that, sooner or later, the articles will become too complicated for detailed verification and the process of accumulating undetected errors will begin. And since mathematics is a very deep science, in the sense that the results of one article usually depend on the results of many and many previous articles, this accumulation of errors for mathematics is very dangerous.

So, I decided, you need to try to do something that will help prevent these crises. For the first crisis, this meant that it was necessary to find an applied problem that required for its solution the methods of pure mathematics developed in recent years or even decades.

He looked for such a problem. He studied biology and found an interesting candidate. He worked on it very hard, but then decided he’d gone down a wrong path:

Since childhood I have been interested in natural sciences (physics, chemistry, biology), as well as in the theory of computer languages, and since 1997, I have read a lot on these topics, and even took several student and post-graduate courses. In fact, I “updated” and deepened the knowledge that had to a very large extent. All this time I was looking for that I recognized open problems that would be of interest to me and to which I could apply modern mathematics.

As a result, I chose, I now understand incorrectly, the problem of recovering the history of populations from their modern genetic composition. I took on this task for a total of about two years, and in the end, already by 2009, I realized that what I was inventing was useless. In my life, so far, it was, perhaps, the greatest scientific failure. A lot of work was invested in the project, which completely failed. Of course, there was some benefit, of course—I learned a lot of probability theory, which I knew badly, and also learned a lot about demography and demographic history.

But he bounced back! He came up with a new approach to the foundations of mathematics, and helped organize a team at the Institute of Advanced Studies at Princeton to develop it further. This approach is now called homotopy type theory or univalent foundations. It’s fundamentally different from set theory. It treats the fundamental concept of equality in a brand new way! And it’s designed to be done with the help of computers.

It seems he started down this new road when the mathematician Carlos Simpson pointed out a serious mistake in a paper he’d written.

I think it was at this moment that I largely stopped doing what is called “curiosity-driven research” and started to think seriously about the future. I didn’t have the tools to explore the areas where curiosity was leading me and the areas that I considered to be of value and of interest and of beauty.

So I started to look into what I could do to create such tools. And it soon became clear that the only long-term solution was somehow to make it possible for me to use computers to verify my abstract, logical, and mathematical constructions. The software for doing this has been in development since the sixties. At the time, when I started to look for a practical proof assistant around 2000, I could not find any. There were several groups developing such systems, but none of them was in any way appropriate for the kind of mathematics for which I needed a system.

When I first started to explore the possibility, computer proof verification was almost a forbidden subject among mathematicians. A conversation about the need for computer proof assistants would invariably drift to Gödel’s incompleteness theorem (which has nothing to do with the actual problem) or to one or two cases of verification of already existing proofs, which were used only to demonstrate how impractical the whole idea was. Among the very few mathematicians who persisted in trying to advance the field of computer verification in mathematics during this time were Tom Hales and Carlos Simpson. Today, only a few years later, computer verification of proofs and of mathematical reasoning in general looks completely practical to many people who work on univalent foundations and homotopy type theory.

The primary challenge that needed to be addressed was that the foundations of mathematics were unprepared for the requirements of the task. Formulating mathematical reasoning in a language precise enough for a computer to follow meant using a foundational system of mathematics not as a standard of consistency to establish a few fundamental theorems, but as a tool that can be employed in ­everyday mathematical work. There were two main problems with the existing foundational systems, which made them inadequate. Firstly, existing foundations of mathematics were based on the languages of predicate logic and languages of this class are too limited. Secondly, existing foundations could not be used to directly express statements about such objects as, for example, the ones in my work on 2-theories.

Still, it is extremely difficult to accept that mathematics is in need of a completely new foundation. Even many of the people who are directly connected with the advances in homotopy type theory are struggling with this idea. There is a good reason: the existing foundations of mathematics—ZFC and category theory—have been very successful. Overcoming the appeal of category theory as a candidate for new foundations of mathematics was for me personally the most challenging.

Homotopy type theory is now a vital and exciting area of mathematics. It’s far from done, and to make it live up to Voevodsky’s dreams will require brand new ideas—not just incremental improvements, but actual sparks of genius. For some of the open problems, see Mike Shulman’s comment on the n-Category Café, and some replies to that.

I only met him a few times, but as far as I can tell Voevodsky was a completely unpretentious person. You can see that in the picture here.

He was also a very complex person. For example, you might not guess that he took great wildlife photos:

You also might not guess at this side of him:

In 2006-2007 a lot of external and internal events happened to me, after which my point of view on the questions of the “supernatural” changed significantly. What happened to me during these years, perhaps, can be compared most closely to what happened to Karl Jung in 1913-14. Jung called it “confrontation with the unconscious”. I do not know what to call it, but I can describe it in a few words. Remaining more or less normal, apart from the fact that I was trying to discuss what was happening to me with people whom I should not have discussed it with, I had in a few months acquired a very considerable experience of visions, voices, periods when parts of my body did not obey me, and a lot of incredible accidents. The most intense period was in mid-April 2007 when I spent 9 days (7 of them in the Mormon capital of Salt Lake City), never falling asleep for all these days.

Almost from the very beginning, I found that many of these phenomena (voices, visions, various sensory hallucinations), I could control. So I was not scared and did not feel sick, but perceived everything as something very interesting, actively trying to interact with those “beings” in the auditorial, visual and then tactile spaces that appeared around me (by themselves or by invoking them). I must say, probably, to avoid possible speculations on this subject, that I did not use any drugs during this period, tried to eat and sleep a lot, and drank diluted white wine.

Another comment: when I say “beings”, naturally I mean what in modern terminology are called complex hallucinations. The word “beings” emphasizes that these hallucinations themselves “behaved”, possessed a memory independent of my memory, and reacted to attempts at communication. In addition, they were often perceived in concert in various sensory modalities. For example, I played several times with a (hallucinated) ball with a (hallucinated) girl—and I saw this ball, and felt it with my palm when I threw it.

Despite the fact that all this was very interesting, it was very difficult. It happened for several periods, the longest of which lasted from September 2007 to February 2008 without breaks. There were days when I could not read, and days when coordination of movements was broken to such an extent that it was difficult to walk.

I managed to get out of this state due to the fact that I forced myself to start math again. By the middle of spring 2008 I could already function more or less normally and even went to Salt Lake City to look at the places where I wandered, not knowing where I was, in the spring of 2007.

In short, he was a genius akin to Cantor or Grothendieck, at times teetering on the brink of sanity, yet gripped by an immense desire for beauty and clarity, engaging in struggles that gripped his whole soul. From the fires of this volcano, truly original ideas emerge.

This last quote, and the first few quotes, are from some interviews in Russian, done by Roman Mikhailov, which Mike Stay pointed out to me. I used Google Translate and polished the results a bit:

Интервью Владимира Воеводского (часть 1), 1 July 2012. English version via Google Translate: Interview with Vladimir Voevodsky (Part 1).

Интервью Владимира Воеводского (часть 2), 5 July 2012. English version via Google Translate: Interview with Vladimir Voevodsky (Part 2).

The quote about the origins of ‘univalent foundations’ comes from his nice essay here:

• Vladimir Voevodsky, The origins and motivations of univalent foundations, 2014.

There’s also a good obituary of Voevodsky explaining its relation to Grothendieck’s idea in simple terms:

• Institute for Advanced Studies, Vladimir Voevodsky 1966–2017, 4 October 2017.

The photograph of Voevodsky is from Andrej Bauer’s website:

• Andrej Bauer, Photos of mathematicians.

To learn homotopy type theory, try this great book:

Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program, Institute for Advanced Study.

Applied Category Theory at UCR (Part 2)

21 September, 2017

I’m running a special session on applied category theory, and now the program is available:

Applied category theory, Fall Western Sectional Meeting of the AMS, 4-5 November 2017, U.C. Riverside.

This is going to be fun.

My former student Brendan Fong is now working with David Spivak at MIT, and they’re both coming. My collaborator John Foley at Metron is also coming: we’re working on the CASCADE project for designing networked systems.

Dmitry Vagner is coming from Duke: he wrote a paper with David and Eugene Lerman on operads and open dynamical systems. Christina Vaisilakopoulou, who has worked with David and Patrick Schultz on dynamical systems, has just joined our group at UCR, so she will also be here. And the three of them have worked with Ryan Wisnesky on algebraic databases. Ryan will not be here, but his colleague Peter Gates will: together with David they have a startup called Categorical Informatics, which uses category theory to build sophisticated databases.

That’s not everyone—for example, most of my students will be speaking at this special session, and other people too—but that gives you a rough sense of some people involved. The conference is on a weekend, but John Foley and David Spivak and Brendan Fong and Dmitry Vagner are staying on for longer, so we’ll have some long conversations… and Brendan will explain decorated corelations in my Tuesday afternoon network theory seminar.

Here’s the program. Click on talk titles to see abstracts. For a multi-author talk, the person with the asterisk after their name is doing the talking. All the talks will be in Room 268 of the Highlander Union Building or ‘HUB’.

Saturday November 4, 2017, 9:00 a.m.-10:50 a.m.

9:00 a.m.
A higher-order temporal logic for dynamical systems.
David I. Spivak, MIT

10:00 a.m.
Algebras of open dynamical systems on the operad of wiring diagrams.
Dmitry Vagner*, Duke University
David I. Spivak, MIT
Eugene Lerman, University of Illinois at Urbana-Champaign

10:30 a.m.
Abstract dynamical systems.
Christina Vasilakopoulou*, University of California, Riverside
David Spivak, MIT
Patrick Schultz, MIT

Saturday November 4, 2017, 3:00 p.m.-5:50 p.m.

3:00 p.m.
Black boxes and decorated corelations.
Brendan Fong, MIT

4:00 p.m.
Compositional modelling of open reaction networks.
Blake S. Pollard*, University of California, Riverside
John C. Baez, University of California, Riverside

4:30 p.m.
A bicategory of coarse-grained Markov processes.
Kenny Courser, University of California, Riverside

5:00 p.m.
A bicategorical syntax for pure state qubit quantum mechanics.
Daniel M. Cicala, University of California, Riverside

5:30 p.m.
Open systems in classical mechanics.
Adam Yassine, University of California Riverside

Sunday November 5, 2017, 9:00 a.m.-10:50 a.m.

9:00 a.m.
Controllability and observability: diagrams and duality.
Jason Erbele, Victor Valley College

9:30 a.m.
Frobenius monoids, weak bimonoids, and corelations.
Brandon Coya, University of California, Riverside

10:00 a.m.
Compositional design and tasking of networks.
John D. Foley*, Metron, Inc.
John C. Baez, University of California, Riverside
Joseph Moeller, University of California, Riverside
Blake S. Pollard, University of California, Riverside

10:30 a.m.
Operads for modeling networks.
Joseph Moeller*, University of California, Riverside
John Foley, Metron Inc.
John C. Baez, University of California, Riverside
Blake S. Pollard, University of California, Riverside

Sunday November 5, 2017, 2:00 p.m.-4:50 p.m.

2:00 p.m.
Reeb graph smoothing via cosheaves.
Vin de Silva, Department of Mathematics, Pomona College

3:00 p.m.
Knowledge representation in bicategories of relations.
Evan Patterson*, Stanford University, Statistics Department

3:30 p.m.
The multiresolution analysis of flow graphs.
Steve Huntsman*, BAE Systems

4:00 p.m.
Data modeling and integration using the open source tool Algebraic Query Language (AQL).
Peter Y. Gates*, Categorical Informatics
Ryan Wisnesky, Categorical Informatics

Applied Category Theory 2018

12 September, 2017

There will be a conference on applied category theory!

Applied Category Theory (ACT 2018). School 23–27 April 2018 and workshop 30 April–4 May 2018 at the Lorentz Center in Leiden, the Netherlands. Organized by Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford).

The plenary speakers will be:

• Samson Abramsky (Oxford)
• John Baez (UC Riverside)
• Kathryn Hess (EPFL)
• Mehrnoosh Sadrzadeh (Queen Mary)
• David Spivak (MIT)

There will be a lot more to say as this progresses, but for now let me just quote from the conference website:

Applied Category Theory (ACT 2018) is a five-day workshop on applied category theory running from April 30 to May 4 at the Lorentz Center in Leiden, the Netherlands.

Towards an Integrative Science: in this workshop, we want to instigate a multi-disciplinary research program in which concepts, structures, and methods from one scientific discipline can be reused in another. The aim of the workshop is to (1) explore the use of category theory within and across different disciplines, (2) create a more cohesive and collaborative ACT community, especially among early-stage researchers, and (3) accelerate research by outlining common goals and open problems for the field.

While the workshop will host discussions on a wide range of applications of category theory, there will be four special tracks on exciting new developments in the field:

1. Dynamical systems and networks
2. Systems biology
3. Cognition and AI
4. Causality

Accompanying the workshop will be an Adjoint Research School for early-career researchers. This will comprise a 16 week online seminar, followed by a 4 day research meeting at the Lorentz Center in the week prior to ACT 2018. Applications to the school will open prior to October 1, and are due November 1. Admissions will be notified by November 15.

The organizers

Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford)

We welcome any feedback! Please send comments to this link.

About Applied Category Theory

Category theory is a branch of mathematics originally developed to transport ideas from one branch of mathematics to another, e.g. from topology to algebra. Applied category theory refers to efforts to transport the ideas of category theory from mathematics to other disciplines in science, engineering, and industry.

This site originated from discussions at the Computational Category Theory Workshop at NIST on Sept. 28-29, 2015. It serves to collect and disseminate research, resources, and tools for the development of applied category theory, and hosts a blog for those involved in its study.

The proposal: Towards an Integrative Science

Category theory was developed in the 1940s to translate ideas from one field of mathematics, e.g. topology, to another field of mathematics, e.g. algebra. More recently, category theory has become an unexpectedly useful and economical tool for modeling a range of different disciplines, including programming language theory [10], quantum mechanics [2], systems biology [12], complex networks [5], database theory [7], and dynamical systems [14].

A category consists of a collection of objects together with a collection of maps between those objects, satisfying certain rules. Topologists and geometers use category theory to describe the passage from one mathematical structure to another, while category theorists are also interested in categories for their own sake. In computer science and physics, many types of categories (e.g. topoi or monoidal categories) are used to give a formal semantics of domain-specific phenomena (e.g. automata [3], or regular languages [11], or quantum protocols [2]). In the applied category theory community, a long-articulated vision understands categories as mathematical workspaces for the experimental sciences, similar to how they are used in topology and geometry [13]. This has proved true in certain fields, including computer science and mathematical physics, and we believe that these results can be extended in an exciting direction: we believe that category theory has the potential to bridge specific different fields, and moreover that developments in such fields (e.g. automata) can be transferred successfully into other fields (e.g. systems biology) through category theory. Already, for example, the categorical modeling of quantum processes has helped solve an important open problem in natural language processing [9].

In this workshop, we want to instigate a multi-disciplinary research program in which concepts, structures, and methods from one discipline can be reused in another. Tangibly and in the short-term, we will bring together people from different disciplines in order to write an expository survey paper that grounds the varied research in applied category theory and lays out the parameters of the research program.

In formulating this research program, we are motivated by recent successes where category theory was used to model a wide range of phenomena across many disciplines, e.g. open dynamical systems (including open Markov processes and open chemical reaction networks), entropy and relative entropy [6], and descriptions of computer hardware [8]. Several talks will address some of these new developments. But we are also motivated by an open problem in applied category theory, one which was observed at the most recent workshop in applied category theory (Dagstuhl, Germany, in 2015): “a weakness of semantics/CT is that the definitions play a key role. Having the right definitions makes the theorems trivial, which is the opposite of hard subjects where they have combinatorial proofs of theorems (and simple definitions). […] In general, the audience agrees that people see category theorists only as reconstructing the things they knew already, and that is a disadvantage, because we do not give them a good reason to care enough” [1, pg. 61].

In this workshop, we wish to articulate a natural response to the above: instead of treating the reconstruction as a weakness, we should treat the use of categorical concepts as a natural part of transferring and integrating knowledge across disciplines. The restructuring employed in applied category theory cuts through jargon, helping to elucidate common themes across disciplines. Indeed, the drive for a common language and comparison of similar structures in algebra and topology is what led to the development category theory in the first place, and recent hints show that this approach is not only useful between mathematical disciplines, but between scientific ones as well. For example, the ‘Rosetta Stone’ of Baez and Stay demonstrates how symmetric monoidal closed categories capture the common structure between logic, computation, and physics [4].

[1] Samson Abramsky, John C. Baez, Fabio Gadducci, and Viktor Winschel. Categorical methods at the crossroads. Report from Dagstuhl Perspectives Workshop 14182, 2014.

[2] Samson Abramsky and Bob Coecke. A categorical semantics of quantum protocols. In Handbook of Quantum Logic and Quantum Structures. Elsevier, Amsterdam, 2009.

[3] Michael A. Arbib and Ernest G. Manes. A categorist’s view of automata and systems. In Ernest G. Manes, editor, Category Theory Applied to Computation and Control. Springer, Berlin, 2005.

[4] John C. Baez and Mike STay. Physics, topology, logic and computation: a Rosetta stone. In Bob Coecke, editor, New Structures for Physics. Springer, Berlin, 2011.

[5] John C. Baez and Brendan Fong. A compositional framework for passive linear networks. arXiv e-prints, 2015.

[6] John C. Baez, Tobias Fritz, and Tom Leinster. A characterization of entropy in terms of information loss. Entropy, 13(11):1945–1957, 2011.

[7] Michael Fleming, Ryan Gunther, and Robert Rosebrugh. A database of categories. Journal of Symbolic Computing, 35(2):127–135, 2003.

[8] Dan R. Ghica and Achim Jung. Categorical semantics of digital circuits. In Ruzica Piskac and Muralidhar Talupur, editors, Proceedings of the 16th Conference on Formal Methods in Computer-Aided Design. Springer, Berlin, 2016.

[9] Dimitri Kartsaklis, Mehrnoosh Sadrzadeh, Stephen Pulman, and Bob Coecke. Reasoning about meaning in natural language with compact closed categories and Frobenius algebras. In Logic and Algebraic Structures in Quantum Computing and Information. Cambridge University Press, Cambridge, 2013.

[10] Eugenio Moggi. Notions of computation and monads. Information and Computation, 93(1):55–92, 1991.

[11] Nicholas Pippenger. Regular languages and Stone duality. Theory of Computing Systems 30(2):121–134, 1997.

[12] Robert Rosen. The representation of biological systems from the standpoint of the theory of categories. Bulletin of Mathematical Biophysics, 20(4):317–341, 1958.

[13] David I. Spivak. Category Theory for Scientists. MIT Press, Cambridge MA, 2014.

[14] David I. Spivak, Christina Vasilakopoulou, and Patrick Schultz. Dynamical systems and sheaves. arXiv e-prints, 2016.

Postdoc in Applied Category Theory

8 September, 2017

guest post by Spencer Breiner

One Year Postdoc Position at Carnegie Mellon/NIST

We are seeking an early-career researcher with a background in category theory, functional programming and/or electrical engineering for a one-year post-doctoral position supported by an Early-concept Grant (EAGER) from the NSF’s Systems Science program. The position will be managed through Carnegie Mellon University (PI: Eswaran Subrahmanian), but the position itself will be located at the US National Institute for Standards and Technology (NIST), located in Gaithersburg, Maryland outside of Washington, DC.

The project aims to develop a compositional semantics for electrical networks which is suitable for system prediction, analysis and control. This work will extend existing methods for linear circuits (featured on this blog!) to include (i) probabilistic estimates of future consumption and (ii) top-down incentives for load management. We will model a multi-layered system of such “distributed energy resources” including loads and generators (e.g., solar array vs. power plant), different types of resource aggregation (e.g., apartment to apartment building), and across several time scales. We hope to demonstrate that such a system can balance local load and generation in order to minimize expected instability at higher levels of the electrical grid.

This post is available full-time (40 hours/5 days per week) for 12 months, and can begin as early as October 1st.

For more information on this position, please contact Dr. Eswaran Subrahmanian ( or Dr. Spencer Breiner (